Todays Question

1
Todays 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 persons score
  on different variables?
  

2
Example 1
In one case, Daves exam score is 10 points above
the mean In the other case, Daves exam score is
10 points below the mean In an important sense, w
e must interpret Daves grade relative to the
average performance of the class
Statistics
Calculus
Mean Calculus 60
Mean Statistics 40
3
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 Daves performance d
epends on how much variability there is in the
exam scores
Statistics
Calculus
4
Standard Scores

• In short, we would like to be able to express a
  persons 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

5
Standard Scores

• In short, we would like to be able to express a
  persons 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

6
Standard (Z) Scores

• In short, we would like to be able to express a
  persons 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

7
Example 1
Dave in Statistics (50 - 40)/10 1 (one SD abo
ve the mean) Dave in Calculus (50 - 60)/10 -
1
(one SD below the mean)
Statistics
Calculus
Mean Statistics 40
Mean Calculus 60
8
Example 2
An example where the means are identical, but the
two sets of scores have different spreads
Daves Stats Z-score (50-40)/5 2 Daves Ca
lc Z-score (50-40)/20 .5
Statistics
Calculus
9
Thee Properties of Standard Scores

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

10
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.

11
Three Properties of Standard Scores

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

12
Why is the SD of z-scores always equal to 1.0?
M 50 SD 10
if x 60,
50
60
70
80
40
30
20
x
0
1
2
3
-1
-2
-3
z
13
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

14
The shape is the same (but the scaling or metric
is different)
15
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
  persons location in a larger set of scores.

16
The area under a normal curve
50
34
34
14
14
2
2
17
Two Advantages of Standard Scores

• 2. Standard scores provides a way to standardize
  or equate different metrics. We can now
  interpret Daves 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.)

18
Two Disadvantages of Standard Scores

• Because a persons 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 didnt know math very well, however, Dave
  would be above the mean. Daves score depends on
  everyone elses scores.

19
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.