Todays Question

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# Todays Question - PowerPoint PPT Presentation

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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Presentation Transcript
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
• 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
• 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
• 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 **

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

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.