Z-scores and Correlations

Lecture 6, Psych 350 - R. Chris Fraleyhttp://www.yourpersonality.net/psych350/fall2012/

Announcements

• No lab on Wed this week• No lecture next week• Email TA about zero-acquaintance data;

working with it on Friday

Answering Descriptive Questions in Multivariate Research

• When we are studying more than one variable, we are typically asking one (or more) of the following two questions:– How does a person’s score on the first variable compare to

his or her score on a second variable?– How do scores on one variable vary as a function of scores

on a second variable?

Making Sense of Scores

• Let’s work with this first issue for a moment. • Let’s assume we have Marc’s scores on his

first two Psych 350 exams.• Marc has a score of 50 on his first exam and

a score of 50 on his second exam.• On which exam did Marc do best?

GRADE

0 20 40 60 80 100

05

1015 •In one case, Marc’s exam

score is 10 points above the mean

•In the other case, Marc’s exam score is 10 points below the mean

•In an important sense, we must interpret Marc’s grade relative to the average performance of the class

Exam1 Exam2

Mean Exam2 = 60

Example 1

Mean Exam1 = 40

•Both distributions have the same mean (40), but different standard deviations (10 vs. 20).

•In one case, Marc is performing better than almost 95% of the class. In the other, he is performing better than approximately 68% of the class.

•Thus, how we evaluate Marc’s performance depends on how much spread or variability there is in the exam scores.GRADE

0 20 40 60 80 100

05

1015

2025

30Example 2

Exam2

Exam1

Standard Scores

• In short, what we would like to do is express Marc’s score for any one exam with respect to (a) how far he is from the average score in the class and (b) the variability of the exam scores.

– how far a person is from the mean: • (X – M)

– variability in scores: • SD

Standard Scores

• Standardized scores, or z-scores, provide a way to express how far a person is from the mean, relative to the variation of the scores.

• (1) Subtract the person’s score from the mean. (2) Divide that difference by the standard deviation.

** This tells us how far a person is from the mean, in the metric of standard deviation units **

Z = (X – M)/SD

Marc’s z-score on Exam1:

z = (50 - 40)/10 = 1

(one SD above the mean)

Marc’s z-score on Exam2

z = (50 - 60)/10 = -1

(one SD below the mean)

Example 1

GRADE

0 20 40 60 80 100

05

1015

Exam1 Exam2

Mean Exam2 = 60

SD = 10

Mean Exam1 = 40

SD = 10

An example where the means are identical, but the two sets of scores have different spreads

Marc’s Exam1 Z-score

Z = (50-40)/5 = 2

Marc’s Exam2 Z-score

Z = (50-40)/20 = .5GRADE

0 20 40 60 80 100

05

1015

2025

30

Example 2

Exam2

SD = 20

Exam1

SD = 5

Some Useful Properties of Standard Scores

(1) The mean of a set of z-scores is always zero

Why? If we subtract a constant, C, from each score, the mean of the scores will be off by that amount (M – C). If we subtract the mean from each score, then mean will be off by an amount equal to the mean (M – M = 0).

50 60 70 80403020

0 1 2 3-1-2-3

x

z

M = 50

SD = 10

if x = 60,

110

10

10

5060

(2) The SD of a set of standardized scores is always 1

Why? SD/SD = 1

(3) The distribution of a set of standardized scores has the same shape as the unstandardized (raw) scores

UNSTANDARDIZED

0.4 0.6 0.8 1.0

02

46

STANDARDIZED

-6 -4 -2 0 2

0.0

0.1

0.2

0.3

0.4

0.5

SCORE

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4

A “Normal” Distribution

The “normalization” (mis)interpretation

Some Useful Properties of Standard Scores

(4) Standard scores can be used to compute easily centile scores: the proportion of people with scores less than or equal to a particular score.

SCORE

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4

34% 34%

14%14%

2%2%

50%

The area under a normal curve

Some Useful Properties of Standard Scores

(5) Z-scores provide a way to “standardize” different metrics (i.e., metrics that differ in variation or meaning). Different variables expressed as z-scores can be interpreted on the same metric (the z-score metric). (Each score comes from a distribution with the same mean [zero] and the same standard deviation [1].)

Person Heart Rate

Complaints Z-score (Heart Rate)

Z-score (Complaints)

Average

A 80 2 (80-100)/20 = -1 (2-2.5)/.5 = -1 -1

B 80 3 (80-100)/20 = -1 (3-2.5)/.5 = 1 0

C 120 2 (120-100)/20 = 1 (2-2.5)/.5 = -1 0

D 120 3 (120-100)/20 = 1 (3-2.5)/.5 = 1 1

Average 100 2.5 0 0 0

SD 20 .5 1 1 1

Correlations in Personality Research

• Many research questions that are addressed in personality psychology are concerned with the relationship between two or more variables.

Some examples

• How does dating/marital satisfaction vary as a function of personality traits, such as emotional stability?

• Are people who are relatively sociable as children also likely to be relatively sociable as adults?

• What is the relationship between individual differences in violent video game playing and aggressive behavior in adolescents?

Graphic presentation

• Many of the relationships we’ll focus on in this course are of the linear variety.

• The relationship between two variables can be represented as a line.

violent video game playing

aggr

essi

ve b

ehav

i or

• Linear relationships can be negative or positive.

violent game playing

aggr

essi

ve b

ehav

i or

aggr

essi

ve b

ehav

i or

violent game playing

• How do we determine whether there is a positive or negative relationship between two variables?

Scatter plotsag

gres

sive

beh

avi o

r

One way of determining the form of the relationship between two variables is to create a scatter plot or a scatter graph.

The form of the relationship (i.e., whether it is positive or negative) can often be seen by inspecting the graph.

violent game playing

Use one variable as the x-axis (the horizontal axis) and the other as the y-axis (the vertical axis).

Plot each person in this two dimensional space as a set of (x, y) coordinates.

How to create a scatter plot

How to create a scatter plot in SPSS

How to create a scatter plot in SPSS

• Select the two variables of interest.

• Click the “ok” button.

positive relationship negative relationship no relationship

Quantifying the relationship

• How can we quantify the linear relationship between two variables?

• One way to do so is with a commonly used statistic called the correlation coefficient (often denoted as r).

Some useful properties of the correlation coefficient

(1) Correlation coefficients range between –1 and + 1.

Note: In this respect, r is useful in the same way that z-scores are useful: they both use a standardized metric.

Some useful properties of the correlation coefficient

(2) The value of the correlation conveys information about the form of the relationship between the two variables.– When r > 0, the relationship between the two variables is

positive.– When r < 0, the relationship between the two variables is

negative--an inverse relationship (higher scores on x correspond to lower scores on y).

– When r = 0, there is no relationship between the two variables.

r = .80 r = -.80 r = 0

Some useful properties of the correlation coefficient

(3) The correlation coefficient can be interpreted as the slope of the line that maps the relationship between two standardized variables.

slope as rise over run

x

y

-2 -1 0 1 2

-2-1

01

23

run

rise

moving from 0 to 1 on x

takes you up .5 on y

r = .50

How do you compute a correlation coefficient?

• First, transform each variable to a standardized form (i.e., z-scores).

• Multiply each person’s z-scores together.• Finally, average those products across

people.

rN

zz YX

Example

Person Violent game playing (z-scores): Zx

Aggressive behavior (z-scores): Zy

Adair 1 1 1

Antoine 1 1 1

Colby -1 -1 1

Trotter -1 -1 1

Average 0 0 1 =

Matching z-scores via products

Why products? Important Note on 2 x 2

Important Note on 2 x 2

x

y

-2 -1 0 1 2

-2-1

01

23

Computing Correlations in SPSS

• Go to the “Analyze” menu.

• Select “Correlate”

• Select “Bivariate…”

Computing Correlations in SPSS

• Select the variables you want to correlate

• Shoot them over to the right-most window

• Click on the “Ok” button.

Magnitude of correlations

• When is a correlation “big” versus “small?”• Cohen’s standards:

– .1 small– .3 medium– > .5 large

What are typical correlations in personality psychology?

Typical sample sizes and effect sizes in studies conducted in personality psychology.

Mdn M SD Range

N 120 179 159 15 – 508

r .21 .24 .17 0 – .96

Note. The absolute value of r was used in the calculations reported here. Data are based on articles published in the 2004 volumes of JPSP:PPID and JP.

A selection of effect sizes from various domains of research

Variables r

Effect of sugar consumption on the behavior and cognitive process of children

.00

Chemotherapy and surviving breast cancer .03

Coronary artery bypass surgery for stable heart disease and survival at 5 years

.08

Combat exposure in Vietnam and subsequent PTSD within 18 years

.11

Self-disclosure and likeability .14

Post-high school grades and job performance .16

Psychotherapy and subsequent well-being .32

Social conformity under the Asch line judgment task .42

Attachment security of parent and quality of offspring attachment .47

Gender and height for U.S. Adults .67

Note. Table adapted from Table 1 of Meyer et al. (2001).

Magnitude of correlations

• “real world” correlations are rarely get larger than .30.

• Why is this the case? – Any one variable can be influenced by a hundred other

variables. To the degree to which a variable is multi-determined, the correlation between it and any one variable must be small.

Qualify

• For the purposes of this class, I want you to describe the correlation: What is it numerically? And, qualitatively speaking, is it “zero or close to zero” (< .1), “small” (.1 to .29), “medium” (.30 to .49), or “large” (> .50).