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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?. Example 1. In one case, Dave’s exam score is 10 points above the mean

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today s question
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?
example 1
Example 1
  • 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 Calculus = 60

Mean Statistics = 40

example 2
Example 2
  • 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 scores

Statistics

Calculus

standard scores
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 scores5
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
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 **

example 17
Example 1

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 28
Example 2

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

Statistics

Calculus

thee properties of standard scores
Thee Properties of Standard Scores
  • 1. The mean of a set of z-scores is always zero
properties of standard scores
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
Three Properties of Standard Scores
  • 2. The SD of a set of standardized scores is always 1
why is the sd of z scores always equal to 1 0
Why is the SD of z-scores always equal to 1.0?

M = 50

SD = 10

if x = 60,

x

20

30

40

50

60

70

80

z

-3

-2

-1

0

1

2

3

three properties of standard scores13
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
two advantages of standard scores
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.

two advantages of standard scores17
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
Two Disadvantages of Standard Scores
  • 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 scores19
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.