Today’s Question

• Example: Dave gets a 50 on his Statistics midterm and an 50 on his Calculus midterm. Did he do equally well on these two exams?

• Big question: How can we compare a person’s score on different variables?

GRADE

0 20 40 60 80 100

05

1015 •In one case, Dave’s exam

score is 10 points above the mean

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

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

Statistics Calculus

Mean Statistics = 40

Mean Calculus

= 60

Example 1

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

•In one case, Dave 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 Dave’s performance depends on how much variability there is in the exam scoresGRADE

0 20 40 60 80 100

05

1015

2025

30Example 2

Calculus

Statistics

Standard Scores

• In short, we would like to be able to express a person’s score with respect to both (a) the mean of the group and (b) the variability of the scores– how far a person is from the mean– variability

Standard Scores

• In short, we would like to be able to express a person’s score with respect to both (a) the mean of the group and (b) the variability of the scores– how far a person is from the mean = X - M– variability = SD

Standard (Z) Scores

• In short, we would like to be able to express a person’s score with respect to both (a) the mean of the group and (b) the variability of the scores– how far a person is from the mean = X - M– variability = SD

Standard score or

** How far a person is from the mean, in the metric of standard deviation units **

SD

MXZ ii

)(

GRADE

0 20 40 60 80 100

05

1015

Dave in Statistics:

(50 - 40)/10 = 1

(one SD above the mean)

Dave in Calculus

(50 - 60)/10 = -1

(one SD below the mean)

Statistics Calculus

Mean

Statistics = 40 Mean

Calculus = 60

Example 1

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

Dave’s Stats Z-score

(50-40)/5 = 2

Dave’s Calc Z-score

(50-40)/20 = .5

GRADE

0 20 40 60 80 100

05

1015

2025

30

Calculus

Statistics

Example 2

Thee Properties of Standard Scores

• 1. The mean of a set of z-scores is always zero

Properties of Standard Scores

• Why?

• The mean has been subtracted from each score. Therefore, following the definition of the mean as a balancing point, the sum (and, accordingly, the average) of all the deviation scores must be zero.

Three Properties of Standard Scores

• 2. The SD of a set of standardized scores is always 1

50 60 70 80403020

0 1 2 3-1-2-3

x

z

M = 50

SD = 10if x = 60,

110

10

10

5060

Why is the SD of z-scores always equal to 1.0?

Three Properties of Standard Scores

• 3. The distribution of a set of standardized scores has the same shape as the unstandardized scores– beware of the “normalization” misinterpretation

The shape is the same (but the scaling or metric is different)

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

Two Advantages of Standard Scores

1. We can use standard scores to find centile scores: the proportion of people with scores less than or equal to a particular score. Centile scores are intuitive ways of summarizing a person’s location in a larger set of scores.

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

Two Advantages of Standard Scores

2. Standard scores provides a way to standardize or equate different metrics. We can now interpret Dave’s scores in Statistics and Calculus 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].)

Two Disadvantages of Standard Scores

1. Because a person’s score is expressed relative to the group (X - M), the same person can have different z-scores when assessed in different samples

Example: If Dave had taken his Calculus exam in a class in which everyone knew math well his z-score would be well below the mean. If the class didn’t know math very well, however, Dave would be above the mean. Dave’s score depends on everyone else’s scores.

Two Disadvantages of Standard Scores

2. If the absolute score is meaningful or of psychological interest, it will be obscured by transforming it to a relative metric.

time

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

7

6

5

4

3

2

1

X

time

A

B

C

D

E

1.42

.71

0

-.71

-1.42

A

B

C

D

E

A

B

C

D

E

A

B

C

D

E

z

In metric of original scores, everyone is increasing over time

If we standardize within time slices, everyone appears the same over time

SD = 1.41

Developmental Example