Statistics Question

1.  H1 is provided on p.85 of the article.  What would be H0 for this study?2.  Of all participants in the study (described in the Participants section), what percentage were Instagram users?  Provide the number of Instagram users and total number of participants in the answer.3.  In Table 1, what does the frequency value of “13” represent?  Variable names from the table alone will not be sufficient.4.  According to Table 1, how many people with public accounts had personal information on their bio?  How many did not?5.  Of the 11 Grandiose variables listed in Table 2, which have scores that are negatively skewed?  (OK, I can hear you groaning!  Hint:  There are 4 of them; refer back to your Measures of Central Tendency handout/notes to refresh as to which should be larger, the mean or the median).6.  Of the 11 Grandiose variables listed in Table 2, which has the lowest variability?7.  In the Correlations results of Section 3.7, which Instagram behavior had the highest correlation with grandiose narcissism?  With vulnerable narcissism?  (Hint: they are in para. 2 of the Correlations section; look for rho values).8.  In Figure 2, which two variables had the strongest correlation?  The weakest?  (This figure has correlation values on the top part of the diagonal; along the diagonal are the histograms for each variable and below the diagonal are the scatterplots for each relationship.)  Social Networking, 2016, 5, 82-92
Published Online April 2016 in SciRes. http://www.scirp.org/journal/sn
http://dx.doi.org/10.4236/sn.2016.52009
An Exploratory Study of the Relationships
between Narcissism, Self-Esteem and
Instagram Use
Olga Paramboukis, Jason Skues, Lisa Wise
Department of Psychological Sciences, Faculty of Health, Arts and Design, Swinburne University of Technology,
Hawthorn, Australia
Received 21 March 2016; accepted 25 April 2016; published 28 April 2016
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
The aim of this mixed-methods exploratory study was to examine the relationship between narcissism, self-esteem and Instagram usage and was motivated by unsubstantiated media claims of
increasing narcissism due to excessive use of social networks. A sample of 200 participants responded to an online survey which consisted of the Five Factor Narcissism Inventory (FFNI), the
Rosenberg Self-Esteem scale, and the Instagram Usage, Behaviours, and Affective Responses Questionnaire (IUBARQ) constructed specifically for the purposes of this study. There was only weak
evidence for any relationship between narcissism and Instagram usage, suggesting that media
concerns are somewhat exaggerated. However the negative correlation between vulnerable narcissism and self-esteem warrants further examination.
Keywords
Narcissism, Vulnerable Narcissism, Grandiose Narcissism, Instagram, Self-Esteem, Social Media
1. Narcissism, Self-Esteem and Social Media
Recent research has suggested that young people today are more narcissistic compared to previous generations
[1]. This statistical increase in scores on narcissism measures has coincided with the introduction, uptake, and
widespread use of social networking sites such as Facebook and Twitter. Several researchers have investigated
the simultaneous increase in narcissism and social media use and note that self-reported narcissism tends to be
associated with different motivations and patterns of usage for social media [2]-[10]. Yet few studies have investigated whether social networking sites other than Facebook or Twitter are related to high levels of selfHow to cite this paper: Paramboukis, O., Skues, J. and Wise, L. (2016) An Exploratory Study of the Relationships between
Narcissism, Self-Esteem and Instagram Use. Social Networking, 5, 82-92. http://dx.doi.org/10.4236/sn.2016.52009
O. Paramboukis et al.
reported narcissism. This study contributes to the current research by investigating how Instagram, a social networking site focusing on editing, posting, and commenting on images, is associated with the personality trait of
narcissism.
1.1. Measures of Narcissism
Although narcissism has been investigated for around 40 years (see [11] for a review), there is still ongoing debate about whether narcissism should be conceptualised as a psychiatrically diagnosed personality disorder or a
subclinical personality trait [12]-[14]. Narcissism has been viewed from a social and personality psychology
perspective as a trait comprising multiple dimensions shaped from the earlier clinical construct. A common distinction in both the clinical and social/personality psychology literature is between grandiose and vulnerable
narcissism [15]. The grandiose dimension refers to traits such as exhibitionism, callousness, extraversion, manipulativeness, superiority, aggression, indifference and seeking of acclaim, whereas the vulnerable dimension is
believed to reflect feelings of inadequacy, emptiness and shame, reactive anger, helplessness, hypervigilance to
insult, excessive shyness and interpersonal avoidance [16] [17]. In general, more emphasis has been placed on
the grandiose aspect of narcissism compared to the vulnerable aspect. The current study defines narcissism from
the social and personality perspective as a sub-clinical trait with two factors, grandiose and vulnerable narcissism respectively.
1.2. Narcissism and Self-Esteem
Some researchers have posited significant conceptual overlap between narcissism and self-esteem with individuals high in both traits having a higher opinion of themselves [18]-[20]. In contrast, others have argued that,
while self-esteem is considered to be an intrapersonal trait, narcissism is primarily interpersonal [21]. Narcissistic individuals may present a false mask of high self-esteem, scoring high on explicit measures of self-esteem,
but showing much lower scores on implicit measures of the same trait [21].
Another possible explanation for the equivocal findings relates to the notion of different aspects of the self,
e.g., agentic versus communal self-views [18] [19]. Campbell et al. [19] found that narcissists and people with
high self-esteem report positive, albeit distinct, self-views. That is, narcissists perceive themselves as better than
average primarily on traits reflecting agency (e.g., competence), whereas individuals with high self-esteem hold
superior beliefs regarding both agency and communal traits. In this regard, the self-regulatory strategies employed by narcissists involve seeking attention and admiration by comparing themselves to others, and by defending their competence to others. Given the multiple alternative explanations relating to the relationship between narcissism and self-esteem, this study will contribute to this research by exploring the relationship between these two constructs in the context of social network use.
1.3. Narcissism and Social Networking Sites
Several researchers have investigated the relationship between narcissism and social media with studies ranging
from testing simple correlations between narcissism scores and basic usage and descriptive data, to studies that
have examined how different dimensions of narcissism relate to motivations and behaviours associated with
different social network sites including MySpace, Facebook, and Twitter. Not surprisingly, the findings have
been mixed. The early studies focused primarily on Facebook and reported significant correlations between narcissism and time per day spent using Facebook, number of Facebook friends, numbers of photos and the selection of specific profile photos, and status updates [3] [8]. Bergman et al. [2] did not find narcissism to be related
to actual social media usage, but instead that it was positively associated with motivations such as wanting to
have a lot of online friends, believing others are interested in what they are doing, and wanting to show others
what they were doing. This highlights that researchers need to look beyond simplistic quantitative variables in
relation to social media use.
In one of the first studies to separate narcissism into different factors and relate these factors to Facebook use,
Carpenter [4] found Grandiose-Exhibitionism to be associated with self-promoting behaviours, number of
Facebook friends, seeking social support and retaliating against perceived mean comments, while Entitlement/
Exploitativeness was related to more anti-social behaviours such as retaliation and checking up on whether he or
she is being talked about by others.
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Panek et al. [10] also examined how different aspects of narcissism were related to Facebook and Twitter use
in a study of both undergraduate students and adults. For university students, exhibitionism was related to time
spent and number of Facebook status posts, and entitlement was related to time spent per day. For adults, superiority and authority were related to Facebook checking, and vanity was associated with Facebook posting and
Twitter checking. According to Panek et al., university students use Twitter as a “technologically augmented
megaphone” that allows them to demonstrate their superiority to others. For adults, it appears to be Facebook
that is used in this manner. Davenport et al. [5] also examined the role of narcissism and motives in relation to
Facebook and Twitter in university and adult samples. Similarly to Panek et al. [10], Twitter was found to be
associated with narcissistic university students for whom tweeting was the preferred mode of communication.
However, it was more important for narcissists in both samples to have more Facebook friends than Twitter followers, which is probably a reflection of the different affordances and relationships with people in the audience
for both of the social network sites. Ong et al. [9] also found in their sample of secondary school students that
after controlling for extraversion, narcissism was positively associated with self-generated content on Facebook
(e.g., profile picture selection etc), but not system-generated content (e.g., number of friends or photos etc).
However, Ong et al. acknowledged that they did not consider privacy settings, which can limit the size and intended audience.
It may be that the expression of narcissistic behaviour through social networking sites is more of a by-product
of a society that is becoming increasingly more “self-centred”, and social media merely provide another arena in
which narcissistic tendencies can be displayed. Alternatively, social media may facilitate, encourage and applaud narcissistic behaviours in a problematic spiral that magnifies the degree of narcissism even further. A further possibility is that this association is merely the result of changes in the way people respond to narcissism
and self-esteem scale items on the respective scales (i.e., with less false modesty than in previous generations),
rather than a actual change in the nature of the personality traits themselves.
1.4. Instagram
Instagram is a photo and video sharing social networking site that is becoming increasingly popular among
young people1. Instagram prompts users to edit photos using inbuilt, easily-applied filters and special effects,
before posting these images onto the Instagram site [22]. Instagram differs from Facebook and Twitter through
being entirely focused on images. According to Instagram Press [22], 300 million of its users have an Instagram
account that they regularly use (monthly). There is also an average of 70 million photos being posted daily
worldwide, attracting 2.5 billion “likes” [22]. Nonetheless, in spite of its widespread usage and specific focus on
posting images, there is a dearth of research on Instagram, and how it relates to narcissism.
Based on the integration of previous research on other social networking sites and the affordances that Instagram provides to users, it is argued that narcissistic tendencies such as attention-seeking and exhibitionism may
be facilitated by Instagram usage due to its specific image-based applications and functions. Firstly, Instagram
facilitates the selection and editing of photos that can be used to make a specific impression to others by glamorizing their portrayal of themselves or their lives. Such behaviour aligns with grandiose narcissism traits such as
attention-seeking, vanity, self-promotion and exhibitionism. Secondly, “liking” and “commenting” functions are
available on Instagram for followers and do not require the formation of a deeper relationship (which may be
achieved via instant messaging functions). This aspect of the site may greatly appeal to highly narcissistic individuals (both grandiose and vulnerable) as they tend to not retain close relationships despite their desire for social contact [23] [24]. Thirdly, “hash tagging” may also be used as a form of self-promotion by both highly vulnerable and grandiose narcissistic individuals as a user may choose to hash tag their photo with popular search
terms with the intention of their photo being seen by a larger audience.
1.5. The Current Study
The current study aims to explore the relationships between narcissism and its subtypes, self-esteem, and Instagram use. The main motivation for the study was to investigate the relationship between narcissism and social
1
Although typically considered a separate social networking site in its own right, it is important to highlight that Instagram is often linked to
Facebook and Twitter, which means that content and communication on these sites is not necessarily mutually exclusive and will depend on
whether one has linked the respective profiles. Indeed since this study was undertaken, Facebook has purchased Instagram, making the
boundary between them even more difficult to define.
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media use by targeting a social networking site (i.e., Instagram) that specifically facilitates the type of behaviour
that has been shown to be associated with narcissism in previous research (i.e., photo-sharing).
H1: Individuals who score high on narcissism will engage in more Instagram behaviours.
We anticipate that some aspects of narcissism may be related to self-esteem and that both narcissism and
self-esteem may influence patterns of Instagram use. These relationships are best investigated as research questions, in keeping with the exploratory nature of the study.
RQ1: How do different aspects of narcissism relate to self-esteem and Instagram use?
RQ2: Is there a difference in the pattern of Instagram use for individuals who are classified as grandiose narcissists compared to those classified as vulnerable narcissists?
2. Method
2.1. Participants
After preliminary data screening, during which participants with missing data were deleted listwise, a total of
200 participants completed the study. There were 148 female and 52 male participants, with ages ranging from
18 to 51 (M = 22.41, Med = 21, SD = 6.15). Of these participants, only 154 had Instagram accounts, and the
majority of data will be reported from the demographic of interest, constituting 141 of the 154 Instagram users
who were under the age of 26.
2.2. Measures
Five Factor Narcissism Inventory (FFNI [25]). The Five Factor Narcissism Inventory is a 148-item narcissism
personality trait measure that was designed from a theoretical framework that views narcissistic traits as maladaptive extensions of traits from the Five Factor Model of personality. The FFNI contains 15 different facets
which form two subtypes of narcissism, namely Grandiose Narcissism (Indifference, Exhibitionism, Thrill
Seeking, Authoritativeness, Grandiose Fantasies, Manipulativeness, Exploitativeness, Entitlement, Arrogance,
Lack of Empathy and Acclaim Seeking) and Vulnerable Narcissism (Reactive Anger, Shame, Need for Admiration and Cynicism/Distrust). Participants were asked to respond on a five point Likert scale ranging from (1)
Strongly Disagree to (5) Strongly Agree with higher scores corresponding to more of a particular trait.
Rosenberg Self-Esteem Scale (RSS [26]). Participants completed the Rosenberg self-esteem scale which is a
measure of global self-esteem that consists of 10 items which are measured on a four point Likert scale ranging
from (1) Strongly Disagree to (4) Strongly Agree [26].
Instagram Usage, Behaviour, and Emotional Reactions Questionnaire (IUBRQ). Participants completed
the Instagram Usage, Behaviour, and Emotional Reactions Questionnaire (IUBRQ), which was designed for the
purpose of the current study. Since no specific scale is currently available to operationalise Instagram usage, the
following questionnaire was created to investigate only the areas of interest in this study, which included frequency of usage, frequency of Instagram-specific behaviours and the attitudes and affective reactions towards
Instagram usage. The selection of the content for this scale was formed via informal focus groups with a small
cohort of research students from our laboratory, along observations of online forums regarding behaviours on
Instagram undertaken by the first author.
Instagram Usage. This section comprised 12 questions ranging from open-ended estimates of time or frequency to yes/no responses or 4-point likert scale responses.
Instagram Behaviours. This section consisted of 16 questions relating to ways of interacting with Instagram.
Participants responded on a five point scale (1 = Never; 2 = Rarely; 3 = Sometimes, 4 = Often; and 5 = Very
Often) based on their recollection of activities over the past month.
Instagram Attitudes. This section consisted of 5 questions relating to motivations for interacting with Instagram. Participants responded on a four point scale (1 = Not at all important; 2 = Slightly important; 3 = Kind of
important; and 4 = Very important).
Instagram Emotional Reactions. The section consisted of 3 open-ended questions where participants were
invited to respond to questions about how they characterised some aspects of Instagram usage, and also how
they emotionally reacted to positive and potentially negative feedback they may receive on Instagram posts.
2.2. Procedure
Participants were invited to complete a 30-minute online survey which consisted of the FFNI, the RSES, and the
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IBURQ. Participants completed the survey at a time and location that was convenient to them, and they were
provided with a debriefing statement immediately upon the completion of the survey. The study was approved
by Swinburne University’s ethical review committee. Data were analysed using Microsoft Excel, SPSS Version
23.0, and R Version 3.2.3.
3. Results and Discussion
3.1. Instagram and Other Social Networks
Of the 200 participants recruited for the study, 154 had Instagram accounts, of whom 122 were female and 32
were male. Of the 154 participants with Instagram accounts, 62 participants linked their Instagram account to
another social network. Of these, 53 linked their Instagram account to Facebook, 9 to Twitter, 8 to Tumblr.
Some linked to more than one social network, with 8 linking to both Facebook and Twitter, 6 linking to both
Facebook and Tumbler, 3 linking to both Twitter and Tumblr. No participants linked to all three. Thirty six of
the remaining 46 non-Instagram users reported having a Facebook account, and only 5 did not report any use of
social networking sites at all.
Within our sample, it appeared that Facebook was still the most popular social networking site, and that Instagram was often used in conjunction with other social media sites. Since our data were collected for this study,
Facebook has purchased Instagram, underscoring the transient nature of usage/behaviour patterns for specific
social network sites.
3.2. Instagram Privacy Settings
We asked participants three questions relating to privacy: 1) was their account publically available?; 2) did they
accept follower requests from people unknown to them?; and 3) did their bio page contain personal information?
(see Table 1). While it might be expected that users would maintain consistency between different privacy settings (for example, keeping their content private, accepting follower requests only from people they know and
restricting personal information in their bio), this was not always the case. For example, as can be seen from
Table 1, 24 participants who kept their accounts private accepted follower requests from strangers. The inconsistency in privacy settings is more likely to reflect a lack of knowledge of account settings and how they operate than any deliberate strategy for information dissemination.
3.3. Instagram Usage, Behaviours and Emotional Reactions Questionnaire
The vast majority of the sample of Instagram users were university students under the age of 26 (141 of 154 Instagram users) and this was the demographic targeted for our analysis of relationships between personality traits
and Instagram behaviours and attitudes. The IUBRQ survey attempted to quantify aspects of Instagram usage in
terms of frequency of interaction with Instagram, the types of behaviours engaged in through Instagram, and the
motives and attitudes surrounding Instagram interactions.
3.4. Instagram Usage
While half of the participants (51%) visited their Instagram site often or very often, the majority of the sample
posted photos occasionally or rarely (77%). This suggests that the majority of Instagram users are consumers
rather than creators of content.
Table 1. Privacy settings on Instagram: cross-tabulation of participants who accept followers
who are unknown to them, have personal information in their bio sections and have their
accounts publically available.
Accept Unknown
Bio with Personal Info
Account Public
Yes
No
Yes
No
Yes
69
13
33
49
No
24
48
19
53
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3.5. Instagram Behaviours
The Behaviours section of the IUBRQ survey addressed ways in which users could interact via Instagram and
reflects features of Instagram at the time when the research was undertaken.
As can be seen from the top panel of Figure 1, most participants rarely used the Instagram features listed,
consistent with finding that very few Instagram users regularly post photos. The most frequent behaviour was
use of hashtags, which was the only listed behaviour that the majority of Instagram users performed at least
sometimes.
In order to generate possible metrics to capture the degree of interaction with Instagram, a Principal Components Analysis was conducted on the 16 Instagram Behaviour items of the IUBRQ. A one-factor solution was
revealed to be the best fitting model in which 13 of the 16 items had primary loadings above 0.3 on a single factor and explained 25% of the variance. On the basis of this analysis, we compiled a single Instagram Behaviours
score from all the items of this section for use in the correlational analyses with personality traits.
3.6. Instagram Attitudes
The Attitudes section of the IUBRQ survey addressed attitudes towards Instagram usage. As can be seen from
the bottom panel of Figure 1, being portrayed and recognised in a positive light was important to the majority of
participants. However it should be noted that positive acclaim was not purely reflected through getting “likes”,
something that 35% of the participants did not find particularly important. Achieving symmetry in the layout of
images on the screen was also not particularly important to most participants.
3.7. Narcissism, Self-Esteem and Instagram Behaviours
In order to test whether narcissism relates to self-esteem and Instagram use, summary data were first calculated
for each of the narcissism and self-esteem scales and relevant subscales used in this study and reported in Table
2.
Narcissism. The median values for both grandiose and vulnerable narcissism scores were approximately at
the midpoint of the scoring, with some participants scoring toward the upper limit of possible scores. At the trait
level, the highest-scoring trait was “acclaim seeking” (M = 36.33, SD = 6.85), whilst the lowest scoring trait was
Figure 1. Likert scales for each item on the Instagram Behaviours and Instagram
Attitudes sections of the IUBARQ.
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Table 2. Means, SDs, medians and ranges for scales and subscales from the FFNI and the
Rosenberg Self-Esteem Scale.
Descriptive Statistics
Personality variable
Mean
SD
Median
Range
Indifference
26.90
8.55
26
10 – 49
Exhibitionism
32.35
6.81
33
15 – 49
Thrill seeking
21.13
6.87
21
8 – 36
Arrogance
22.10
6.19
21
10 – 48
Entitlement
20.23
5.41
20
10 – 48
Manipulativeness
24.80
7.16
23
10 – 47
Exploitativeness
20.94
7.29
20
10 – 47
Authoritativeness
31.99
7.62
33
10 – 50
Grandiose fantasies
31.44
7.02
32
11 – 49
Lack of empathy
17.72
5.22
17
10 – 42
Acclaim seeking
36.33
6.85
37
11 – 50
Grandiose Total
285.80
283
155 – 462
Reactive anger
27.53
6.56
28
11 – 46
Shame
31.78
8.12
32
12 – 50
Need for admiration
28.39
6.22
28
10 – 44
Distrust
26.46
5.72
27
12 – 39
Vulnerable Total
114.20
116
51 – 168
FFNI Total
400.00
401
269 – 622
Rosenberg Self-Esteem
29.33
29
11 – 39
5.34
“lack of empathy” (M = 17.72, SD = 5.22).
Self-esteem. Self-esteem scores were found to be high in the sample with more than 75% of the sample scoring more than the midpoint of the scale.
Correlations. Spearman’s rank order correlations were used to test the significance of associations between
narcissism, self-esteem, Instagram Behaviours as measured by a composite score from the IUBRQ Behaviours
items and Instagram Attitudes as measured by a composite score from the IUBRQ Attitudes items. As can be
seen in Figure 2, overall narcissism (FFNI Total) did not correlate with self-esteem (RSS). However when narcissism was separated into grandiose and vulnerable dimensions, a weak positive correlation was found between
grandiose narcissism and self-esteem (ρ = 0.35, p < 0.001), whilst a negative moderate strength correlation was found between vulnerable narcissism and self-esteem (ρ = −0.59, p < 0.001). That is, those Instagram users with higher levels of grandiose narcissism tended to report higher self-esteem levels, whilst vulnerable narcissists reported lower self-esteem levels. All three narcissism scores were positively associated with both Instagram Attitudes and Instagram Behaviours, but the relationships between self-esteem and both Instagram Attitudes and Instagram Behaviours were not significant. An inspection of the item level data for selected Instagram behaviours revealed significant correlations between grandiose narcissism and Instagram behaviours such as “Hashtagging popular or expensive brands” (ρ = 0.29, p < 0.001), “Posting photos of things you want, but do not have” (ρ = 0.19, p < 0.001), “Posting photos of celebrities or people you admire” (ρ = 0.18, p < 0.05), and “Posting photos of progress towards physical health, fitness and wellbeing” (ρ = 0.20, p < 0.05). Moreover, vulnerable narcissism correlated with “Hashtagging popular or expensive brands (ρ = 0.24, p < 0.001), “Posting photos of things you want, but do not have” (ρ = 0.21, p < 0.05), “Posting photos of celebrities or people you admire” (ρ = 0.34, p < 0.001), “Posting photos of yourself at impressive events or functions” (ρ = 0.23, p < 0.001), and “Request for followers” (e.g. #follow4follow) 88 O. Paramboukis et al. Figure 2. Scatterplots, frequency histograms and correlations for narcissism scores, self-esteem scores, Instagram Behaviours and Instagram Attitudes. Ranges of values for scatterplots and histograms can be identified from the range column of Table 2. (ρ = 0.17, p < 0.05). For the most part, the pattern of correlations were similar for grandiose and vulnerable narcissism, with the only differences being a significant association between grandiose narcissism and posting photos of physical health, fitness and wellbeing, and a significant relationship between vulnerable narcissism and requests for followers. 3.8. Qualitative Analyses The similarities and differences between grandiose and vulnerable narcissism were also explored using openended questions exploring emotional reactions to feedback received from other people in response to a participant’s Instagram behaviours. Prior to performing a content analysis on the open-ended responses, three groups of participants were identified according to their combined Grandiose and Vulnerable Narcissism scores, namely 1) a High Grandiose/Low Vulnerable group; 2) a High Vulnerable/Low Grandiose; group; and 3) a High Grandiose/High Vulnerable group. Responses to three questions were compared across the three groups. The first question was 1) “Do you feel happy or elated when you receive a lot of feedback on a post?” The most common response to this question across all three groups was “Yes, because it gives a sense of validation and/or approval”. According to one participant, “…I feel that people are ‘liking’ an aspect of my life and further more are showing approval of me as a person”. In response to the second question, “If your post receives no feedback, how does that make you feel?”, participants classified in the high grandiose groups typically responded with indifference, whilst the most common responses for participants in the high vulnerable/low grandiose group were experienced negative emotions (32.1%) and deleted the post (17.9%). For the last question, “If someone is critical of your post, how does that make you feel?”, the most common response for the high vulnerable, low grandiose group was “Block/Delete the post and/or the person whom was critical” (28.6%), with the second most common response relating to feeling defensive (25%). Indeed, one participant states “I react negatively and defensively. It makes me feel judged and unintelligent”. In contrast, both high grandiose groups tended to be neutral in their responses. In sum, the content analysis revealed that participants who were simultaneously high in vulnerable narcissism and low in grandiose narcissism reacted more strongly to both negative and positive interactions on Instagram in comparison to participants who scored either high on grandiose narcissism exclusively or scored high on both grandiose and vulnerable narcissism. 89 O. Paramboukis et al. 4. General Discussion This study aimed to explore the interrelationships between narcissism, self-esteem and Instagram usage. Although there were differences in how grandiose and vulnerable narcissism related to self-esteem, neither types of narcissism were strongly associated with Instagram usage as identified via the IUBRQ. 4.1. Narcissism and Self-Esteem The FFNI total score for narcissism was not correlated with self-esteem, however vulnerable narcissism was negatively correlated with self-esteem, while grandiose narcissism was positively correlated with self-esteem, albeit at a lower level. This finding provides further evidence for the distinction between grandiose and vulnerable narcissism and supports the view that self-esteem is an important factor that distinguishes between those subtypes of grandiose and vulnerable narcissism (see [18] [19]). Moreover the data suggest that vulnerable narcissism may have more of an intrapersonal component compared to grandiose narcissism, and that internal feelings of inadequacy and helplessness are more closely related to self-esteem compared to feelings of superiority over others. However, it still remains unclear whether those with high grandiose narcissism levels do in fact have higher self-esteem levels, or whether their underlying need for superiority and exhibitionism results in presenting an overly inflated self view on a self-report measure of self-esteem. 4.2. Narcissism and Instagram Behaviours Vulnerable narcissism also had the highest degree of correlation with both Instagram Behaviours and Instagram Attitudes although these relationships were not strong. In terms of specific Instagram behaviours, posting up photos of one’s physical appearance was more associated with grandiose narcissism, while requests for followers was more associated with vulnerable narcissism. The qualitative data identified those who were high in vulnerable but low in grandiose narcissism showed stronger emotional reactions to Instagram feedback. These findings are broadly consistent with past research that suggests that narcissism is related to self-generated content on social network sites, rather than system-generated content [9]. Those high on vulnerable narcissism appeared to be more interested in increasing their popularity and seeking the approval of others, whereas grandiose narcissists used overt attempts of drawing attention to themselves in order to be admired. That vulnerable narcissists use Instagram as a platform to seek out positive feedback aligns with the notion that people seek validation from others in order to help boost self-esteem [27]. This would also help to explain the strong emotional reactions to negative feedback for individuals high on vulnerable narcissism, which includes the dimensions of reactive anger, need for admiration and shame. In contrast, grandiose narcissists appear to seek out opportunities to engage in behaviours that afford self-promotion (see [4] [6]) in order to maintain their elevated positive selfview [3], which fits with the grandiose dimensions of exhibitionism and seeking of acclaim. 4.3. Implications The findings from this study have provided theoretical, methodological and practical implications. Firstly, this study provides further evidence for distinguishing between grandiose and vulnerable narcissism, and that these two subtypes are differentially related to self-esteem. Secondly, this study has expanded the research on narcissism subtypes and social network use, targeting a social network that specifically focuses on posting images. However despite this specific focus, the study has shown only weak evidence for any relationship between narcissism and Instagram use. This study also developed a survey that aimed to quantify specific Instagram behaviours in an attempt to uncover more nuanced patterns of behaviour relating both to reasons for posting content, and reactions to feedback on content. Even with detailed quantitative and qualitative data on Instagram usage, the weak evidence for any association between narcissism and Instagram suggests that Instagram offers a platform for expression of existing narcissistic tendencies rather than a medium that encourages extremes of narcissistic behaviour in people who do not normally show such tendencies. 4.4. Limitations and Future Research Although this study was one of the very studies to have attempted to operationalise Instagram behaviours by developing a specific measure that quantifies some of the common Instagram behaviours, it should be noted that 90 O. Paramboukis et al. the frequency of particular Instagram behaviours may not accurately reflect the importance placed on these behaviours by the individual participants. The qualitative data collected in this study provided some insights into the way individual participants interpret and react to Instagram content but did not really provide a coherent narrative on why people use Instagram, nor is it clear that such a narrative exists at a psychological level (i.e., at a deeper level than that it offers an easy way to share photos with friends and family). It is important to note that this study targeted Instagram for its specific emphasis on posting and sharing photos rather than on providing instant messaging for communicating with others. However our data suggest that Instagram is often used in conjunction with other social network sites and most participants had more followers on Facebook than on Instagram. The fact that Facebook has now purchased Instagram may result in a tighter integration of Instagram features into Facebook, or a change in Instagram features that renders some items of the IUBRQ irrelevant. Indeed the primary limitation of most studies on social networking sites is that they only provide a transient snapshot of behaviour patterns as both the technologies and user bases evolve. 4.5. Conclusion In conclusion, the primary finding from this study is that there is only weak evidence for any relationship between narcissism and Instagram usage, suggesting that media concerns about social media giving rise to unprecedented narcissistic behaviour are somewhat exaggerated. However, it appears that there is a complex relationship between narcissism and self-esteem such that vulnerable narcissism is negatively correlated with selfesteem and grandiose narcissism is positively associated with self-esteem albeit more weakly. Vulnerable narcissism appears to be more strongly associated with Instagram usage, with vulnerable narcissists seeking acclaim and being more sensitive to feedback on their posts. In contrast grandiose narcissists appear to use Instagram to exhibit their superiority over others but are not overly sensitive to feedback. Despite the limitations of the study, the findings have provided a better understanding of the associations between narcissism subtypes, self-esteem and Instagram use, and highlight the need for further exploration of the relationship between selfesteem and vulnerable narcissism, which has typically received less attention by researchers than grandiose narcissism. References [1] Twenge, J.M., Konrath, S., Foster, J.D., Campbell, W.K. and Bushman, B.J. (2008) Egos Inflating over TIme: A Cross Temporal Meta-Analysis of the Narcissistic Personality Inventory. Journal of Personality, 76, 875-902. http://dx.doi.org/10.1111/j.1467-6494.2008.00507.x [2] Bergman, S., Fearrington, M.E., Davenport, S.W. and Bergman, J.Z. (2011) Millenials, Narcissism and Social Networking: What Narcissists do on Social Networks and Why. Personality and Individual Differences, 50, 706-711. http://dx.doi.org/10.1016/j.paid.2010.12.022 [3] Buffardi, L.E. and Campbell, W.K. (2008) Narcissism and Social Networking Web Sites. Personality and Social Psychology Bulletin, 34, 1303-1314. http://dx.doi.org/10.1177/0146167208320061 [4] Carpenter, C.J. (2012) Narcissism on Facebook: Self-Promotional and Anti-Social Behaviour. Personality and Individual Differences, 52, 482-486. http://dx.doi.org/10.1016/j.paid.2011.11.011 [5] Davenport, S.W., Bergman, S.M., Bergman, J.Z. and Fearrington, M.E. (2014) Twitter versus Facebook: Exploring the Role of Narcissism in Motives and Usage of Different Social Media Platforms. Computers in Human Behaviour, 32, 212-220. http://dx.doi.org/10.1016/j.chb.2013.12.011 [6] Fox, J. and Rooney, M.C. (2015) The Dark Triad and Trait Self-Objectification as Predictors of Men’s Use and SelfPresentation Behaviours on Social Networking Sites. Personality and Individual Differences, 76, 161-165. http://dx.doi.org/10.1016/j.paid.2014.12.017 [7] Gentile, B., Twenge, J.M., Freeman, E.C. and Campbell, W.K. (2012) The Effect of Social Networking Websites on Positive Self-Views: An Experimental Investigation. Computers in Human Behaviour, 28, 1929-1933. http://dx.doi.org/10.1016/j.chb.2012.05.012 [8] Mehdizadeh, S. (2010) Self-Presentation 2.0: Narcissism and Self-Esteem on Facebook. Cyberpsychology, Behaviour and Social Networking, 13, 357-364. http://dx.doi.org/10.1089/cyber.2009.0257 [9] Ong, E.Y.L., Ang, R.P., Ho, J.C.M., Lim, J.C.Y., Goh, D.H. and Lee, C.S. (2011) Narcissism, Extraversion and Adolescents’ Self-Presentation on Facebook. Personality and Individual Differences, 50, 180-185. http://dx.doi.org/10.1016/j.paid.2010.09.022 91 O. Paramboukis et al. [10] Panek, E.T., Nardis, Y. and Konrath, S. (2013) Mirror or Megaphone?: How Relationships between Narcissism and Social Networking Site Use Differ on Facebook and Twitter. Computers in Human Behaviour, 29, 2004-2012. http://dx.doi.org/10.1016/j.chb.2013.04.012 [11] Cain, N.M., Pincus, A.L. and Ansell, E.B. (2008) Narcissism at the Crossroads: Phenotypic Description of Pathological Narcissism across Clinical Theory, Social/Personality Psychology and Psychiatric Diagnosis. Clinical Psychology Review, 28, 638-656. http://dx.doi.org/10.1016/j.cpr.2007.09.006 [12] Kohut, H. (1971) The Analysis of the Self: A Systematic Psychoanalytic Approach to the Treatment of Narcissistic Personality Disorders. International Press, New York. [13] Kernberg, O.F., (1984) Severe Personality Disorders: Psychotherapeutic Strategies. Yale University, New Haven. [14] Kernberg, O.F. (1985) Borderline Conditions and Pathological Narcissism. Rowman and Littlefield, Lanham, MD. [15] Pincus, A.L, and Lukowitsky, M.R. (2010) Pathological Narcissism and Narcissistic Personality Disorder. Annual Review of Clinical Psychology, 6, 421-446. http://dx.doi.org/10.1146/annurev.clinpsy.121208.131215 [16] Ackerman, R.A., Witt, E.A., Donnellan, M.B., Tzresniewski, K.H., Robbins, R.W. and Kashy, D.A. (2011) What Does the Narcissistic Personality Inventory Really Measure? Assessment, 18, 67-87. http://dx.doi.org/10.1177/1073191110382845 [17] Miller, J.D. and Campbell, K. (2008) Comparing Clinical and Social Personality Conceptualizations of Narcissism. Journal of Personality, 76, 449-476. http://dx.doi.org/10.1111/j.1467-6494.2008.00492.x [18] Bosson, J.K., Lakey, C. E., Campbell, W.K., Zeigler-Hill, V., Jordan, C.H. and Kernis, M.H. (2008) Untangling the Links between Narcissism and Self-Esteem: A Theoretical and Empirical Review. Personality and Social Psychology Compass, 2, 1415-1439. http://dx.doi.org/10.1111/j.1751-9004.2008.00089.x [19] Campbell, W.K., Rudich, E.A. and Sedikides, C. (2002) Narcissism, Self-Esteem, and the Positivity of Self-Views: Two Portraits of Self-Love. Personality and Social Psychology Bulletin, 28, 358-368. http://dx.doi.org/10.1177/0146167202286007 [20] Hovrath, S. and Morf, C.C. (2010) To Be Grandiose or Not To Be Worthless: Different Routes to Self-Enhancement for Narcissism and Self-Esteem. Journal of Research in Personality, 44, 585-592. http://dx.doi.org/10.1016/j.jrp.2010.07.002 [21] Zeigler-Hill, V. (2006) Discrepancies between Implicit and Explicit Self-Esteem: Implications for Narcissism and SelfEsteem Instability. Journal of Personality, 74, 119-144. http://dx.doi.org/10.1111/j.1467-6494.2005.00371.x [22] Instagram Press (2015, 13th of January) Retrieved from http://Instagram.com/press/ [23] Kealy, D. and Rasmussen, B. (2012) Veiled and Vulnerable: The Other Side of Grandiose Narcissism. Clinical Social Work Journal, 40, 356-365. http://dx.doi.org/10.1007/s10615-011-0370-1 [24] McGregor, I., Nail, P.R., Kocalar, D. and Haji, R. (2013) I’m OK, I’m OK: Praise Makes Narcissists with Low Implicit Self-Esteem Indifferent to the Suffering of Others. Personality and Individual Differences, 55, 655-659. http://dx.doi.org/10.1016/j.paid.2013.05.007 [25] Glover, N., Miller, J.D., Lynam, D.R., Crego, C. and Widiger, T.A. (2012) The Five-Factor Narcissism Inventory: A Five-Factor Measure of Narcissistic Personality Traits. Journal of Personality Assessment, 94, 500-512. http://dx.doi.org/10.1080/00223891.2012.670680 [26] Rosenberg, M. (1965) Society and the Adolescent Self-image. Princeton University Press, Princeton, NJ. [27] Krämer, N.C. and Winter, S. (2008) Impression Management 2.0: The Relationship of Self-Esteem, Extraversion, SelfEfficacy, and Self-Presentation within Social Networking Sites. Journal of Media Psychology, 20, 106-116. http://dx.doi.org/10.1027/1864-1105.20.3.106 92 Ed iti o n © Deborah Batt 10 Statistics for the Behavioral Sciences Frederick J Gravetter The College at Brockport, State University of New York Larry B. WaLLnau The College at Brockport, State University of New York Australia ● Brazil ● Mexico ● Singapore ● United Kingdom ● United States This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version. Statistics for the Behavioral Sciences, Tenth Edition Frederick J Gravetter and Larry B. Wallnau Product director: Jon-david Hague Product Manager: timothy Matray Content developer: Stefanie Chase Product Assistant: Kimiya Hojjat Marketing Manager: Melissa Larmon Content Project Manager: Michelle Clark © 2017, 2013 Cengage Learning WCn: 02-200-203 ALL RiGHtS RESERVEd. no part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. 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Find your local representative at www.cengage.com Cengage Learning products are represented in Canada by nelson Education, Ltd. to learn more about Cengage Learning Solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Printed in Canada Print Number: 01 Print Year: 2015 B RiEF Co n tEn t S CHAPtER 1 Introduction to Statistics 1 CHAPtER 2 Frequency Distributions CHAPtER 3 Central Tendency CHAPtER 4 Variability CHAPtER 5 z-Scores: Location of Scores and Standardized Distributions 131 CHAPtER 6 Probability CHAPtER 7 Probability and Samples: The Distribution of Sample Means 193 CHAPtER 8 Introduction to Hypothesis Testing 223 CHAPtER 9 Introduction to the t Statistic C H A P t E R 10 CHAPtER 11 33 67 99 159 267 The t Test for Two Independent Samples 299 The t Test for Two Related Samples 335 C H A P t E R 12 Introduction to Analysis of Variance 365 C H A P t E R 13 Repeated-Measures Analysis of Variance 413 C H A P t E R 14 Two-Factor Analysis of Variance (Independent Measures) 447 CHAPtER 15 Correlation 485 C H A P t E R 16 Introduction to Regression 529 C H A P t E R 17 The Chi-Square Statistic: Tests for Goodness of Fit and Independence 559 C H A P t E R 18 The Binomial Test 603 iii Co n tEn t S CHAPtER 1 Introduction to Statistics PREVIEW 1 2 1.1 Statistics, Science, and Observations 2 1.2 Data Structures, Research Methods, and Statistics 10 1.3 Variables and Measurement 18 1.4 Statistical Notation 25 Summary 29 Focus on Problem Solving 30 Demonstration 1.1 30 Problems 31 CHAPtER 2 Frequency Distributions PREVIEW 33 34 2.1 Frequency Distributions and Frequency Distribution Tables 35 2.2 Grouped Frequency Distribution Tables 38 2.3 Frequency Distribution Graphs 42 2.4 Percentiles, Percentile Ranks, and Interpolation 49 2.5 Stem and Leaf Displays 56 Summary 58 Focus on Problem Solving 59 Demonstration 2.1 60 Demonstration 2.2 61 Problems 62 v vi CONTENTS CHAPtER 3 Central Tendency PREVIEW 67 68 3.1 Overview 68 3.2 The Mean 70 3.3 The Median 79 3.4 The Mode 83 3.5 Selecting a Measure of Central Tendency 86 3.6 Central Tendency and the Shape of the Distribution 92 Summary 94 Focus on Problem Solving 95 Demonstration 3.1 96 Problems 96 CHAPtER 4 Variability PREVIEW 99 100 4.1 Introduction to Variability 101 4.2 Defining Standard Deviation and Variance 103 4.3 Measuring Variance and Standard Deviation for a Population 108 4.4 Measuring Standard Deviation and Variance for a Sample 111 4.5 Sample Variance as an Unbiased Statistic 117 4.6 More about Variance and Standard Deviation 119 Summary 125 Focus on Problem Solving 127 Demonstration 4.1 128 Problems 128 z-Scores: Location of Scores C H A P t E R 5 and Standardized Distributions PREVIEW 132 5.1 Introduction to z-Scores 133 5.2 z-Scores and Locations in a Distribution 135 5.3 Other Relationships Between z, X, 𝛍, and 𝛔 138 131 CONTENTS vii 5.4 Using z-Scores to Standardize a Distribution 141 5.5 Other Standardized Distributions Based on z-Scores 145 5.6 Computing z-Scores for Samples 148 5.7 Looking Ahead to Inferential Statistics 150 Summary 153 Focus on Problem Solving 154 Demonstration 5.1 155 Demonstration 5.2 155 Problems 156 CHAPtER 6 Probability PREVIEW 159 160 6.1 Introduction to Probability 160 6.2 Probability and the Normal Distribution 165 6.3 Probabilities and Proportions for Scores from a Normal Distribution 172 6.4 Probability and the Binomial Distribution 179 6.5 Looking Ahead to Inferential Statistics 184 Summary 186 Focus on Problem Solving 187 Demonstration 6.1 188 Demonstration 6.2 188 Problems 189 Probability and Samples: The Distribution C H A P t E R 7 of Sample Means PREVIEW 194 7.1 Samples, Populations, and the Distribution of Sample Means 194 7.2 The Distribution of Sample Means for any Population and any Sample Size 199 7.3 Probability and the Distribution of Sample Means 206 7.4 More about Standard Error 210 7.5 Looking Ahead to Inferential Statistics 215 193 viii CONTENTS Summary 219 Focus on Problem Solving 219 Demonstration 7.1 220 Problems 221 CHAPtER 8 Introduction to Hypothesis Testing PREVIEW 223 224 8.1 The Logic of Hypothesis Testing 225 8.2 Uncertainty and Errors in Hypothesis Testing 236 8.3 More about Hypothesis Tests 240 8.4 Directional (One-Tailed) Hypothesis Tests 245 8.5 Concerns about Hypothesis Testing: Measuring Effect Size 250 8.6 Statistical Power 254 Summary 260 Focus on Problem Solving 261 Demonstration 8.1 262 Demonstration 8.2 263 Problems 263 CHAPtER 9 Introduction to the t Statistic PREVIEW 268 9.1 The t Statistic: An Alternative to z 268 9.2 Hypothesis Tests with the t Statistic 274 9.3 Measuring Effect Size for the t Statistic 279 9.4 Directional Hypotheses and One-Tailed Tests 288 Summary 291 Focus on Problem Solving 293 Demonstration 9.1 293 Demonstration 9.2 294 Problems 295 267 CONTENTS CHAPtER 10 The t Test for Two Independent Samples PREVIEW ix 299 300 10.1 Introduction to the Independent-Measures Design 300 10.2 The Null Hypothesis and the Independent-Measures t Statistic 302 10.3 Hypothesis Tests with the Independent-Measures t Statistic 310 10.4 Effect Size and Confidence Intervals for the Independent-Measures t 316 10.5 The Role of Sample Variance and Sample Size in the Independent-Measures t Test Summary 322 325 Focus on Problem Solving 327 Demonstration 10.1 328 Demonstration 10.2 329 Problems 329 CHAPtER 11 The t Test for Two Related Samples PREVIEW 335 336 11.1 Introduction to Repeated-Measures Designs 336 11.2 The t Statistic for a Repeated-Measures Research Design 339 11.3 Hypothesis Tests for the Repeated-Measures Design 343 11.4 Effect Size and Confidence Intervals for the Repeated-Measures t 347 11.5 Comparing Repeated- and Independent-Measures Designs 352 Summary 355 Focus on Problem Solving 358 Demonstration 11.1 358 Demonstration 11.2 359 Problems 360 CHAPtER 12 Introduction to Analysis of Variance PREVIEW 366 12.1 Introduction (An Overview of Analysis of Variance) 366 12.2 The Logic of Analysis of Variance 372 12.3 ANOVA Notation and Formulas 375 365 x CONTENTS 12.4 Examples of Hypothesis Testing and Effect Size with ANOVA 383 12.5 Post Hoc Tests 393 12.6 More about ANOVA 397 Summary 403 Focus on Problem Solving 406 Demonstration 12.1 406 Demonstration 12.2 408 Problems 408 CHAPtER 13 Repeated-Measures Analysis of Variance PREVIEW 413 414 13.1 Overview of the Repeated-Measures ANOVA 415 13.2 Hypothesis Testing and Effect Size with the Repeated-Measures ANOVA 420 13.3 More about the Repeated-Measures Design 429 Summary 436 Focus on Problem Solving 438 Demonstration 13.1 439 Demonstration 13.2 440 Problems 441 Two-Factor Analysis of Variance C H A P t E R 14 (Independent Measures) PREVIEW 447 448 14.1 An Overview of the Two-Factor, Independent-Measures, ANOVA: Main Effects and Interactions 448 14.2 An Example of the Two-Factor ANOVA and Effect Size 458 14.3 More about the Two-Factor ANOVA 467 Summary 473 Focus on Problem Solving 475 Demonstration 14.1 476 Demonstration 14.2 478 Problems 479 CONTENTS CHAPtER 15 Correlation PREVIEW xi 485 486 15.1 Introduction 487 15.2 The Pearson Correlation 489 15.3 Using and Interpreting the Pearson Correlation 495 15.4 Hypothesis Tests with the Pearson Correlation 506 15.5 Alternatives to the Pearson Correlation 510 Summary 520 Focus on Problem Solving 522 Demonstration 15.1 523 Problems 524 CHAPtER 16 Introduction to Regression PREVIEW 529 530 16.1 Introduction to Linear Equations and Regression 530 16.2 The Standard Error of Estimate and Analysis of Regression: The Significance of the Regression Equation 538 16.3 Introduction to Multiple Regression with Two Predictor Variables 544 Summary 552 Linear and Multiple Regression 554 Focus on Problem Solving 554 Demonstration 16.1 555 Problems 556 The Chi-Square Statistic: Tests for Goodness C H A P t E R 17 of Fit and Independence PREVIEW 560 17.1 Introduction to Chi-Square: The Test for Goodness of Fit 561 17.2 An Example of the Chi-Square Test for Goodness of Fit 567 17.3 The Chi-Square Test for Independence 573 17.4 Effect Size and Assumptions for the Chi-Square Tests 17.5 Special Applications of the Chi-Square Tests 587 582 559 xii CONTENTS Summary 591 Focus on Problem Solving 595 Demonstration 17.1 595 Demonstration 17.2 597 Problems 597 CHAPtER 18 The Binomial Test PREVIEW 603 604 18.1 Introduction to the Binomial Test 604 18.2 An Example of the Binomial Test 608 18.3 More about the Binomial Test: Relationship with Chi-Square and the Sign Test Summary 612 617 Focus on Problem Solving 619 Demonstration 18.1 619 Problems 620 A PPE N D IX E S A Basic Mathematics Review 625 A.1 A.2 A.3 A.4 A.5 Symbols and Notation 627 Proportions: Fractions, Decimals, and Percentages 629 Negative Numbers 635 Basic Algebra: Solving Equations 637 Exponents and Square Roots 640 B Statistical Tables 647 C Solutions for Odd-Numbered Problems in the Text 663 D General Instructions for Using SPSS 683 E Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests 687 Statistics Organizer: Finding the Right Statistics for Your Data References 717 Name Index 723 Subject Index 725 701 PREFACE M any students in the behavioral sciences view the required statistics course as an intimidating obstacle that has been placed in the middle of an otherwise interesting curriculum. They want to learn about human behavior—not about math and science. As a result, the statistics course is seen as irrelevant to their education and career goals. However, as long as the behavioral sciences are founded in science, knowledge of statistics will be necessary. Statistical procedures provide researchers with objective and systematic methods for describing and interpreting their research results. Scientific research is the system that we use to gather information, and statistics are the tools that we use to distill the information into sensible and justified conclusions. The goal of this book is not only to teach the methods of statistics, but also to convey the basic principles of objectivity and logic that are essential for science and valuable for decision making in everyday life. Those of you who are familiar with previous editions of Statistics for the Behavioral Sciences will notice that some changes have been made. These changes are summarized in the section entitled “To the Instructor.” In revising this text, our students have been foremost in our minds. Over the years, they have provided honest and useful feedback. Their hard work and perseverance has made our writing and teaching most rewarding. We sincerely thank them. Students who are using this edition should please read the section of the preface entitled “To the Student.” The book chapters are organized in the sequence that we use for our own statistics courses. We begin with descriptive statistics, and then examine a variety of statistical procedures focused on sample means and variance before moving on to correlational methods and nonparametric statistics. Information about modifying this sequence is presented in the To The Instructor section for individuals who prefer a different organization. Each chapter contains numerous examples, many based on actual research studies, learning checks, a summary and list of key terms, and a set of 20–30 problems. Ancillaries Ancillaries for this edition include the following. ■■ MindTap® Psychology: MindTap® Psychology for Gravetter/Wallnau’s Statistics for The Behavioral Sciences, 10th Edition is the digital learning solution that helps instructors engage and transform today’s students into critical thinkers. Through paths of dynamic assignments and applications that you can personalize, real-time course analytics, and an accessible reader, MindTap helps you turn cookie cutter into cutting edge, apathy into engagement, and memorizers into higher-level thinkers. As an instructor using MindTap you have at your fingertips the right content and unique set of tools curated specifically for your course, such as video tutorials that walk students through various concepts and interactive problem tutorials that provide students opportunities to practice what they have learned, all in an interface designed to improve workflow and save time when planning lessons and course structure. The control to build and personalize your course is all yours, focusing on the most relevant xiii xiv PREFACE ■■ ■■ ■■ material while also lowering costs for your students. Stay connected and informed in your course through real time student tracking that provides the opportunity to adjust the course as needed based on analytics of interactivity in the course. Online Instructor’s Manual: The manual includes learning objectives, key terms, a detailed chapter outline, a chapter summary, lesson plans, discussion topics, student activities, “What If” scenarios, media tools, a sample syllabus and an expanded test bank. The learning objectives are correlated with the discussion topics, student activities, and media tools. Online PowerPoints: Helping you make your lectures more engaging while effectively reaching your visually oriented students, these handy Microsoft PowerPoint® slides outline the chapters of the main text in a classroom-ready presentation. The PowerPoint® slides are updated to reflect the content and organization of the new edition of the text. Cengage Learning Testing, powered by Cognero®: Cengage Learning Testing, Powered by Cognero®, is a flexible online system that allows you to author, edit, and manage test bank content. You can create multiple test versions in an instant and deliver tests from your LMS in your classroom. Acknowledgments It takes a lot of good, hard-working people to produce a book. Our friends at Cengage have made enormous contributions to this textbook. We thank: Jon-David Hague, Product Director; Timothy Matray, Product Team Director; Jasmin Tokatlian, Content Development Manager; Kimiya Hojjat, Product Assistant; and Vernon Boes, Art Director. Special thanks go to Stefanie Chase, our Content Developer and to Lynn Lustberg who led us through production at MPS. Reviewers play a very important role in the development of a manuscript. Accordingly, we offer our appreciation to the following colleagues for their assistance: Patricia Case, University of Toledo; Kevin David, Northeastern State University; Adia Garrett, University of Maryland, Baltimore County; Carrie E. Hall, Miami University; Deletha Hardin, University of Tampa; Angela Heads, Prairie View A&M University; Roberto Heredia, Texas A&M International University; Alisha Janowski, University of Central Florida; Matthew Mulvaney, The College at Brockport (SUNY); Nicholas Von Glahn, California State Polytechnic University, Pomona; and Ronald Yockey, Fresno State University. To the Instructor Those of you familiar with the previous edition of Statistics for the Behavioral Sciences will notice a number of changes in the 10th edition. Throughout this book, research examples have been updated, real world examples have been added, and the end-of-chapter problems have been extensively revised. Major revisions for this edition include the following: 1. Each section of every chapter begins with a list of Learning Objectives for that specific section. 2. Each section ends with a Learning Check consisting of multiple-choice questions with at least one question for each Learning Objective. PREFACE xv 3. The former Chapter 19, Choosing the Right Statistics, has been eliminated and an abridged version is now an Appendix replacing the Statistics Organizer, which appeared in earlier editions. Other examples of specific and noteworthy revisions include the following. Chapter 1 The section on data structures and research methods parallels the new Appendix, Choosing the Right Statistics. Chapter 2 The chapter opens with a new Preview to introduce the concept and purpose of frequency distributions. Chapter 3 Minor editing clarifies and simplifies the discussion the median. Chapter 4 The chapter opens with a new Preview to introduce the topic of Central Tendency. The sections on standard deviation and variance have been edited to increase emphasis on concepts rather than calculations. The section discussion relationships between z, X, μ, and σ has been expanded and includes a new demonstration example. Chapter 5 Chapter 6 The chapter opens with a new Preview to introduce the topic of Probability. The section, Looking Ahead to Inferential Statistics, has been substantially shortened and simplified. Chapter 7 The former Box explaining difference between standard deviation and standard error was deleted and the content incorporated into Section 7.4 with editing to emphasize that the standard error is the primary new element introduced in the chapter. The final section, Looking Ahead to Inferential Statistics, was simplified and shortened to be consistent with the changes in Chapter 6. Chapter 8 A redundant example was deleted which shortened and streamlined the remaining material so that most of the chapter is focused on the same research example. Chapter 9 The chapter opens with a new Preview to introduce the t statistic and explain why a new test statistic is needed. The section introducing Confidence Intervals was edited to clarify the origin of the confidence interval equation and to emphasize that the interval is constructed at the sample mean. Chapter 10 The chapter opens with a new Preview introducing the independent-measures t statistic. The section presenting the estimated standard error of (M1 – M2) has been simplified and shortened. Chapter 11 The chapter opens with a new Preview introducing the repeated-measures t statistic. The section discussing hypothesis testing has been separated from the section on effect size and confidence intervals to be consistent with the other two chapters on t tests. The section comparing independent- and repeated-measures designs has been expanded. Chapter 12 The chapter opens with a new Preview introducing ANOVA and explaining why a new hypothesis testing procedure is necessary. Sections in the chapter have been reorganized to allow flow directly from hypothesis tests and effect size to post tests. xvi PREFACE Chapter 13 Substantially expanded the section discussing factors that influence the outcome of a repeated-measures hypothesis test and associated measures of effect size. Chapter 14 The chapter opens with a new Preview presenting a two-factor research example and introducing the associated ANOVA. Sections have been reorganized so that simple main effects and the idea of using a second factor to reduce variance from individual differences are now presented as extra material related to the two-factor ANOVA. Chapter 15 The chapter opens with a new Preview presenting a correlational research study and the concept of a correlation. A new section introduces the t statistic for evaluating the significance of a correlation and the section on partial correlations has been simplified and shortened. Chapter 16 The chapter opens with a new Preview introducing the concept of regression and its purpose. A new section demonstrates the equivalence of testing the significance of a correlation and testing the significance of a regression equation with one predictor variable. The section on residuals for the multiple-regression equation has been edited to simplify and shorten. Chapter 17 A new chapter Preview presents an experimental study with data consisting of frequencies, which are not compatible with computing means and variances. Chi-square tests are introduced as a solution to this problem. A new section introduces Cohen’s w as a means of measuring effect size for both chi-square tests. Chapter 18 Substantial editing clarifies the section explaining how the real limits for each score can influence the conclusion from a binomial test. The former Chapter 19 covering the task of matching statistical methods to specific types of data has been substantially shortened and converted into an Appendix. ■■Matching the Text to Your Syllabus The book chapters are organized in the sequence that we use for our own statistics courses. However, different instructors may prefer different organizations and probably will choose to omit or deemphasize specific topics. We have tried to make separate chapters, and even sections of chapters, completely self-contained, so they can be deleted or reorganized to fit the syllabus for nearly any instructor. Some common examples are as follows. ■■ ■■ ■■ It is common for instructors to choose between emphasizing analysis of variance (Chapters 12, 13, and 14) or emphasizing correlation/regression (Chapters 15 and 16). It is rare for a one-semester course to complete coverage of both topics. Although we choose to complete all the hypothesis tests for means and mean differences before introducing correlation (Chapter 15), many instructors prefer to place correlation much earlier in the sequence of course topics. To accommodate this, Sections 15.1, 15.2, and 15.3 present the calculation and interpretation of the Pearson correlation and can be introduced immediately following Chapter 4 (variability). Other sections of Chapter 15 refer to hypothesis testing and should be delayed until the process of hypothesis testing (Chapter 8) has been introduced. It is also possible for instructors to present the chi-square tests (Chapter 17) much earlier in the sequence of course topics. Chapter 17, which presents hypothesis tests for proportions, can be presented immediately after Chapter 8, which introduces the process of hypothesis testing. If this is done, we also recommend that the Pearson correlation (Sections 15.1, 15.2, and 15.3) be presented early to provide a foundation for the chi-square test for independence. PREFACE xvii To the Student A primary goal of this book is to make the task of learning statistics as easy and painless as possible. Among other things, you will notice that the book provides you with a number of opportunities to practice the techniques you will be learning in the form of Learning Checks, Examples, Demonstrations, and end-of-chapter problems. We encourage you to take advantage of these opportunities. Read the text rather than just memorizing the formulas. We have taken care to present each statistical procedure in a conceptual context that explains why the procedure was developed and when it should be used. If you read this material and gain an understanding of the basic concepts underlying a statistical formula, you will find that learning the formula and how to use it will be much easier. In the “Study Hints,” that follow, we provide advice that we give our own students. Ask your instructor for advice as well; we are sure that other instructors will have ideas of their own. Over the years, the students in our classes and other students using our book have given us valuable feedback. If you have any suggestions or comments about this book, you can write to either Professor Emeritus Frederick Gravetter or Professor Emeritus Larry Wallnau at the Department of Psychology, SUNY College at Brockport, 350 New Campus Drive, Brockport, New York 14420. You can also contact Professor Emeritus Gravetter directly at fgravett@brockport.edu. ■■Study Hints You may find some of these tips helpful, as our own students have reported. ■■ ■■ ■■ ■■ ■■ The key to success in a statistics course is to keep up with the material. Each new topic builds on previous topics. If you have learned the previous material, then the new topic is just one small step forward. Without the proper background, however, the new topic can be a complete mystery. If you find that you are falling behind, get help immediately. You will learn (and remember) much more if you study for short periods several times per week rather than try to condense all of your studying into one long session. For example, it is far more effective to study half an hour every night than to have a single 3½-hour study session once a week. We cannot even work on writing this book without frequent rest breaks. Do some work before class. Keep a little ahead of the instructor by reading the appropriate sections before they are presented in class. Although you may not fully understand what you read, you will have a general idea of the topic, which will make the lecture easier to follow. Also, you can identify material that is particularly confusing and then be sure the topic is clarified in class. Pay attention and think during class. Although this advice seems obvious, often it is not practiced. Many students spend so much time trying to write down every example presented or every word spoken by the instructor that they do not actually understand and process what is being said. Check with your instructor—there may not be a need to copy every example presented in class, especially if there are many examples like it in the text. Sometimes, we tell our students to put their pens and pencils down for a moment and just listen. Test yourself regularly. Do not wait until the end of the chapter or the end of the week to check your knowledge. After each lecture, work some of the end-of-chapter problems and do the Learning Checks. Review the Demonstration Problems, and be sure you can define the Key Terms. If you are having trouble, get your questions answered immediately—reread the section, go to your instructor, or ask questions in class. By doing so, you will be able to move ahead to new material. xviii PREFACE ■■ ■■ Do not kid yourself! Avoid denial. Many students observe their instructor solve problems in class and think to themselves, “This looks easy, I understand it.” Do you really understand it? Can you really do the problem on your own without having to leaf through the pages of a chapter? Although there is nothing wrong with using examples in the text as models for solving problems, you should try working a problem with your book closed to test your level of mastery. We realize that many students are embarrassed to ask for help. It is our biggest challenge as instructors. You must find a way to overcome this aversion. Perhaps contacting the instructor directly would be a good starting point, if asking questions in class is too anxiety-provoking. You could be pleasantly surprised to find that your instructor does not yell, scold, or bite! Also, your instructor might know of another student who can offer assistance. Peer tutoring can be very helpful. Frederick J Gravetter Larry B. Wallnau A B o U t tH E AU tH o R S Frederick Gravetter Frederick J Gravetter is Professor Emeritus of Psychology at the State University of New York College at Brockport. While teaching at Brockport, Dr. Gravetter specialized in statistics, experimental design, and cognitive psychology. He received his bachelor’s degree in mathematics from M.I.T. and his Ph.D in psychology from Duke University. In addition to publishing this textbook and several research articles, Dr. Gravetter co-authored Research Methods for the Behavioral Science and Essentials of Statistics for the Behavioral Sciences. Larry B. Wallnau Larry B. WaLLnau is Professor Emeritus of Psychology at the State University of New York College at Brockport. While teaching at Brockport, he published numerous research articles in biopsychology. With Dr. Gravetter, he co-authored Essentials of Statistics for the Behavioral Sciences. Dr. Wallnau also has provided editorial consulting for numerous publishers and journals. He has taken up running and has competed in 5K races in New York and Connecticut. He takes great pleasure in adopting neglected and rescued dogs. xix CH A P T ER Introduction to Statistics 1 © Deborah Batt PREVIEW 1.1 Statistics, Science, and Observations 1.2 Data Structures, Research Methods, and Statistics 1.3 Variables and Measurement 1.4 Statistical Notation Summary Focus on Problem Solving Demonstration 1.1 Problems 1 PREVIEW Before we begin our discussion of statistics, we ask you to read the following paragraph taken from the philosophy of Wrong Shui (Candappa, 2000). The Journey to Enlightenment In Wrong Shui, life is seen as a cosmic journey, a struggle to overcome unseen and unexpected obstacles at the end of which the traveler will find illumination and enlightenment. Replicate this quest in your home by moving light switches away from doors and over to the far side of each room.* Why did we begin a statistics book with a bit of twisted philosophy? In part, we simply wanted to lighten the mood with a bit of humor—starting a statistics course is typically not viewed as one of life’s joyous moments. In addition, the paragraph is an excellent counterexample for the purpose of this book. Specifically, our goal is to do everything possible to prevent you from stumbling around in the dark by providing lots of help and illumination as you journey through the world of statistics. To accomplish this, we begin each section of the book with clearly stated learning objectives and end each section with a brief quiz to test your mastery of the new material. We also introduce each new statistical procedure by explaining the purpose it is intended to serve. If you understand why a new procedure is needed, you will find it much easier to learn. 1.1 The objectives for this first chapter are to provide an introduction to the topic of statistics and to give you some background for the rest of the book. We discuss the role of statistics within the general field of scientific inquiry, and we introduce some of the vocabulary and notation that are necessary for the statistical methods that follow. As you read through the following chapters, keep in mind that the general topic of statistics follows a well-organized, logically developed progression that leads from basic concepts and definitions to increasingly sophisticated techniques. Thus, each new topic serves as a foundation for the material that follows. The content of the first nine chapters, for example, provides an essential background and context for the statistical methods presented in Chapter 10. If you turn directly to Chapter 10 without reading the first nine chapters, you will find the material confusing and incomprehensible. However, if you learn and use the background material, you will have a good frame of reference for understanding and incorporating new concepts as they are presented. *Candappa, R. (2000). The little book of wrong shui. Kansas City: Andrews McMeel Publishing. Reprinted by permission. Statistics, Science, and Observations LEARNING OBJECTIVEs 1. Define the terms population, sample, parameter, and statistic, and describe the relationships between them. 2. Define descriptive and inferential statistics and describe how these two general categories of statistics are used in a typical research study. 3. Describe the concept of sampling error and explain how this concept creates the fundamental problem that inferential statistics must address. ■■Definitions of Statistics By one definition, statistics consist of facts and figures such as the average annual snowfall in Denver or Derrick Jeter’s lifetime batting average. These statistics are usually informative and time-saving because they condense large quantities of information into a few simple figures. Later in this chapter we return to the notion of calculating statistics (facts and figures) but, for now, we concentrate on a much broader definition of statistics. Specifically, we use the term statistics to refer to a general field of mathematics. In this case, we are using the term statistics as a shortened version of statistical procedures. For example, you are probably using this book for a statistics course in which you will learn about the statistical techniques that are used to summarize and evaluate research results in the behavioral sciences. 2 SEctIon 1.1 | Statistics, Science, and Observations 3 Research in the behavioral sciences (and other fields) involves gathering information. To determine, for example, whether college students learn better by reading material on printed pages or on a computer screen, you would need to gather information about students’ study habits and their academic performance. When researchers finish the task of gathering information, they typically find themselves with pages and pages of measurements such as preferences, personality scores, opinions, and so on. In this book, we present the statistics that researchers use to analyze and interpret the information that they gather. Specifically, statistics serve two general purposes: 1. Statistics are used to organize and summarize the information so that the researcher can see what happened in the research study and can communicate the results to others. 2. Statistics help the researcher to answer the questions that initiated the research by determining exactly what general conclusions are justified based on the specific results that were obtained. DEFInItIon The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information. Statistical procedures help ensure that the information or observations are presented and interpreted in an accurate and informative way. In somewhat grandiose terms, statistics help researchers bring order out of chaos. In addition, statistics provide researchers with a set of standardized techniques that are recognized and understood throughout the scientific community. Thus, the statistical methods used by one researcher will be familiar to other researchers, who can accurately interpret the statistical analyses with a full understanding of how the analysis was done and what the results signify. ■■Populations and Samples Research in the behavioral sciences typically begins with a general question about a specific group (or groups) of individuals. For example, a researcher may want to know what factors are associated with academic dishonesty among college students. Or a researcher may want to examine the amount of time spent in the bathroom for men compared to women. In the first example, the researcher is interested in the group of college students. In the second example, the researcher wants to compare the group of men with the group of women. In statistical terminology, the entire group that a researcher wishes to study is called a population. DEFInItIon A population is the set of all the individuals of interest in a particular study. As you can well imagine, a population can be quite large—for example, the entire set of women on the planet Earth. A researcher might be more specific, limiting the population for study to women who are registered voters in the United States. Perhaps the investigator would like to study the population consisting of women who are heads of state. Populations can obviously vary in size from extremely large to very small, depending on how the investigator defines the population. The population being studied should always be identified by the researcher. In addition, the population need not consist of people—it could be a population of rats, corporations, parts produced in a factory, or anything else an investigator wants to study. In practice, populations are typically very large, such as the population of college sophomores in the United States or the population of small businesses. Because populations tend to be very large, it usually is impossible for a researcher to examine every individual in the population of interest. Therefore, researchers typically select 4 chaPtER 1 | Introduction to Statistics a smaller, more manageable group from the population and limit their studies to the individuals in the selected group. In statistical terms, a set of individuals selected from a population is called a sample. A sample is intended to be representative of its population, and a sample should always be identified in terms of the population from which it was selected. A sample is a set of individuals selected from a population, usually intended to represent the population in a research study. DEFInItIon Just as we saw with populations, samples can vary in size. For example, one study might examine a sample of only 10 students in a graduate program and another study might use a sample of more than 10,000 people who take a specific cholesterol medication. So far we have talked about a sample being selected from a population. However, this is actually only half of the full relationship between a sample and its population. Specifically, when a researcher finishes examining the sample, the goal is to generalize the results back to the entire population. Remember that the research started with a general question about the population. To answer the question, a researcher studies a sample and then generalizes the results from the sample to the population. The full relationship between a sample and a population is shown in Figure 1.1. ■■Variables and Data Typically, researchers are interested in specific characteristics of the individuals in the population (or in the sample), or they are interested in outside factors that may influence the individuals. For example, a researcher may be interested in the influence of the weather on people’s moods. As the weather changes, do people’s moods also change? Something that can change or have different values is called a variable. DEFInItIon A variable is a characteristic or condition that changes or has different values for different individuals. THE POPULATION All of the individuals of interest The results from the sample are generalized to the population F I G U R E 1.1 The relationship between a population and a sample. The sample is selected from the population THE SAMPLE The individuals selected to participate in the research study SEctIon 1.1 | Statistics, Science, and Observations 5 Once again, variables can be characteristics that differ from one individual to another, such as height, weight, gender, or personality. Also, variables can be environmental conditions that change such as temperature, time of day, or the size of the room in which the research is being conducted. To demonstrate changes in variables, it is necessary to make measurements of the variables being examined. The measurement obtained for each individual is called a datum, or more commonly, a score or raw score. The complete set of scores is called the data set or simply the data. DEFInItIon Data (plural) are measurements or observations. A data set is a collection of measurements or observations. A datum (singular) is a single measurement or observation and is commonly called a score or raw score. Before we move on, we should make one more point about samples, populations, and data. Earlier, we defined populations and samples in terms of individuals. For example, we discussed a population of graduate students and a sample of cholesterol patients. Be forewarned, however, that we will also refer to populations or samples of scores. Because research typically involves measuring each individual to obtain a score, every sample (or population) of individuals produces a corresponding sample (or population) of scores. ■■Parameters and Statistics When describing data it is necessary to distinguish whether the data come from a population or a sample. A characteristic that describes a population—for example, the average score for the population—is called a parameter. A characteristic that describes a sample is called a statistic. Thus, the average score for a sample is an example of a statistic. Typically, the research process begins with a question about a population parameter. However, the actual data come from a sample and are used to compute sample statistics. DEFInItIon A parameter is a value, usually a numerical value, that describes a population. A parameter is usually derived from measurements of the individuals in the population. A statistic is a value, usually a numerical value, that describes a sample. A statistic is usually derived from measurements of the individuals in the sample. Every population parameter has a corresponding sample statistic, and most research studies involve using statistics from samples as the basis for answering questions about population parameters. As a result, much of this book is concerned with the relationship between sample statistics and the corresponding population parameters. In Chapter 7, for example, we examine the relationship between the mean obtained for a sample and the mean for the population from which the sample was obtained. ■■Descriptive and Inferential Statistical Methods Although researchers have developed a variety of different statistical procedures to organize and interpret data, these different procedures can be classified into two general categories. The first category, descriptive statistics, consists of statistical procedures that are used to simplify and summarize data. DEFInItIon Descriptive statistics are statistical procedures used to summarize, organize, and simplify data. 6 chaPtER 1 | Introduction to Statistics Descriptive statistics are techniques that take raw scores and organize or summarize them in a form that is more manageable. Often the scores are organized in a table or a graph so that it is possible to see the entire set of scores. Another common technique is to summarize a set of scores by computing an average. Note that even if the data set has hundreds of scores, the average provides a single descriptive value for the entire set. The second general category of statistical techniques is called inferential statistics. Inferential statistics are methods that use sample data to make general statements about a population. DEFInItIon Inferential statistics consist of techniques that allow us to study samples and then make generalizations about the populations from which they were selected. Because populations are typically very large, it usually is not possible to measure everyone in the population. Therefore, a sample is selected to represent the population. By analyzing the results from the sample, we hope to make general statements about the population. Typically, researchers use sample statistics as the basis for drawing conclusions about population parameters. One problem with using samples, however, is that a sample provides only limited information about the population. Although samples are generally representative of their populations, a sample is not expected to give a perfectly accurate picture of the whole population. There usually is some discrepancy between a sample statistic and the corresponding population parameter. This discrepancy is called sampling error, and it creates the fundamental problem inferential statistics must always address. DEFInItIon Sampling error is the naturally occurring discrepancy, or error, that exists between a sample statistic and the corresponding population parameter. The concept of sampling error is illustrated in Figure 1.2. The figure shows a population of 1,000 college students and 2 samples, each with 5 students who were selected from the population. Notice that each sample contains different individuals who have different characteristics. Because the characteristics of each sample depend on the specific people in the sample, statistics will vary from one sample to another. For example, the five students in sample 1 have an average age of 19.8 years and the students in sample 2 have an average age of 20.4 years. It is also very unlikely that the statistics obtained for a sample will be identical to the parameters for the entire population. In Figure 1.2, for example, neither sample has statistics that are exactly the same as the population parameters. You should also realize that Figure 1.2 shows only two of the hundreds of possible samples. Each sample would contain different individuals and would produce different statistics. This is the basic concept of sampling error: sample statistics vary from one sample to another and typically are different from the corresponding population parameters. One common example of sampling error is the error associated with a sample proportion. For example, in newspaper articles reporting results from political polls, you frequently find statements such as this: Candidate Brown leads the poll with 51% of the vote. Candidate Jones has 42% approval, and the remaining 7% are undecided. This poll was taken from a sample of registered voters and has a margin of error of plus-or-minus 4 percentage points. The “margin of error” is the sampling error. In this case, the percentages that are reported were obtained from a sample and are being generalized to the whole population. As always, you do not expect the statistics from a sample to be perfect. There always will be some “margin of error” when sample statistics are used to represent population parameters. SEctIon 1.1 | Statistics, Science, and Observations 7 F I G U R E 1. 2 A demonstration of sampling error. Two samples are selected from the same population. Notice that the sample statistics are different from one sample to another and all the sample statistics are different from the corresponding population parameters. The natural differences that exist, by chance, between a sample statistic and population parameter are called sampling error. Population of 1000 college students Population Parameters Average Age 5 21.3 years Average IQ 5 112.5 65% Female, 35% Male Sample #1 Sample #2 Eric Jessica Laura Karen Brian Tom Kristen Sara Andrew John Sample Statistics Average Age 5 19.8 Average IQ 5 104.6 60% Female, 40% Male Sample Statistics Average Age 5 20.4 Average IQ 5 114.2 40% Female, 60% Male As a further demonstration of sampling error, imagine that your statistics class is separated into two groups by drawing a line from front to back through the middle of the room. Now imagine that you compute the average age (or height, or IQ) for each group. Will the two groups have exactly the same average? Almost certainly they will not. No matter what you chose to measure, you will probably find some difference between the two groups. However, the difference you obtain does not necessarily mean that there is a systematic difference between the two groups. For example, if the average age for students on the right-hand side of the room is higher than the average for students on the left, it is unlikely that some mysterious force has caused the older people to gravitate to the right side of the room. Instead, the difference is probably the result of random factors such as chance. The unpredictable, unsystematic differences that exist from one sample to another are an example of sampling error. ■■Statistics in the Context of Research The following example shows the general stages of a research study and demonstrates how descriptive statistics and inferential statistics are used to organize and interpret the data. At the end of the example, note how sampling error can affect the interpretation of experimental results, and consider why inferential statistical methods are needed to deal with this problem. 8 chaPtER 1 | Introduction to Statistics ExamplE 1.1 Figure 1.3 shows an overview of a general research situation and demonstrates the roles that descriptive and inferential statistics play. The purpose of the research study is to address a question that we posed earlier: Do college students learn better by studying text on printed pages or on a computer screen? Two samples are selected from the population of college students. The students in sample A are given printed pages of text to study for 30 minutes and the students in sample B study the same text on a computer screen. Next, all of the students are given a multiple-choice test to evaluate their knowledge of the material. At this point, the researcher has two sets of data: the scores for sample A and the scores for sample B (see the figure). Now is the time to begin using statistics. First, descriptive statistics are used to simplify the pages of data. For example, the researcher could draw a graph showing the scores for each sample or compute the average score for each sample. Note that descriptive methods provide a simplified, organized Step 1 Experiment: Compare two studying methods Data Test scores for the students in each sample Step 2 Descriptive statistics: Organize and simplify Population of College Students Sample A Read from printed pages 25 27 30 19 29 26 21 28 23 26 20 25 28 27 24 26 22 30 Average Score = 26 Step 3 Inferential statistics: Interpret results F i g u r E 1. 3 The role of statistics in experimental research. Sample B Read from computer screen 20 20 23 25 22 18 22 17 28 19 24 25 30 27 23 21 22 19 Average Score = 22 The sample data show a 4-point difference between the two methods of studying. However, there are two ways to interpret the results. 1. There actually is no difference between the two studying methods, and the sample difference is due to chance (sampling error). 2. There really is a difference between the two methods, and the sample data accurately reflect this difference. The goal of inferential statistics is to help researchers decide between the two interpretations. SEctIon 1.1 | Statistics, Science, and Observations 9 description of the scores. In this example, the students who studied printed pages had an average score of 26 on the test, and the students who studied text on the computer averaged 22. Once the researcher has described the results, the next step is to interpret the outcome. This is the role of inferential statistics. In this example, the researcher has found a difference of 4 points between the two samples (sample A averaged 26 and sample B averaged 22). The problem for inferential statistics is to differentiate between the following two interpretations: 1. There is no real difference between the printed page and a computer screen, and the 4-point difference between the samples is just an example of sampling error (like the samples in Figure 1.2). 2. There really is a difference between the printed page and a computer screen, and the 4-point difference between the samples was caused by the different methods of studying. In simple English, does the 4-point difference between samples provide convincing evidence of a difference between the two studying methods, or is the 4-point difference just chance? The purpose of inferential statistics is to answer this question. ■ lE arn in g Ch ECk 1. A researcher is interested in the sleeping habits of American college students. A group of 50 students is interviewed and the researcher finds that these students sleep an average of 6.7 hours per day. For this study, the average of 6.7 hours is an example of a(n) . a. parameter b. statistic c. population d. sample 2. A researcher is curious about the average IQ of registered voters in the state of Florida. The entire group of registered voters in the state is an example of a . a. sample b. statistic c. population d. parameter 3. Statistical techniques that summarize, organize, and simplify data are classified as . a. population statistics b. sample statistics c. descriptive statistics d. inferential statistics 4. In general, statistical techniques are used to summarize the data from a research study and statistical techniques are used to determine what conclusions are justified by the results. a. inferential, descriptive b. descriptive, inferential c. sample, population d. population, sample 10 chaPtER 1 | Introduction to Statistics 5. IQ tests are standardized so that the average score is 100 for the entire group of people who take the test each year. However, if you selected a group of 20 people who took the test and computed their average IQ score you probably would not get 100. What statistical concept explains the difference between your mean and the mean for the entire group? a. statistical error b. inferential error c. descriptive error d. sampling error an s wE r s 1. B, 2. C, 3. C, 4. B, 5. D 1.2 Data Structures, Research Methods, and Statistics LEARNING OBJECTIVEs 4. Differentiate correlational, experimental, and nonexperimental research and describe the data structures associated with each. 5. Define independent, dependent, and quasi-independent variables and recognize examples of each. ■■Individual Variables: Descriptive Research Some research studies are conducted simply to describe individual variables as they exist naturally. For example, a college official may conduct a survey to describe the eating, sleeping, and study habits of a group of college students. When the results consist of numerical scores, such as the number of hours spent studying each day, they are typically described by the statistical techniques that are presented in Chapters 3 and 4. Non-numerical scores are typically described by computing the proportion or percentage in each category. For example, a recent newspaper article reported that 34.9% of Americans are obese, which is roughly 35 pounds over a healthy weight. ■■Relationships Between Variables Most research, however, is intended to examine relationships between two or more variables. For example, is there a relationship between the amount of violence in the video games played by children and the amount of aggressive behavior they display? Is there a relationship between the quality of breakfast and academic performance for elementary school children? Is there a relationship between the number of hours of sleep and grade point average for college students? To establish the existence of a relationship, researchers must make observations—that is, measurements of the two variables. The resulting measurements can be classified into two distinct data structures that also help to classify different research methods and different statistical techniques. In the following section we identify and discuss these two data structures. I. One Group with Two Variables Measured for Each Individual: The Correlational Method One method for examining the relationship between variables is to observe the two variables as they...