Transcript Today’s Question • Example: Dave gets a 50 on his Statistics midterm and an 50 on his Calculus midterm.

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?
15
Example 1
Statistics
Calculus
•In one case, Dave’s exam
score is 10 points above
the mean
10
•In the other case, Dave’s
exam score is 10 points
below the mean
0
5
•In an important sense, we
must interpret Dave’s grade
relative to the average
performance of the class
0
2 0
4 0
6 0
8 0
1 0 0
G R AD E
Mean Statistics
= 40
Mean Calculus
= 60
0 5 10 15 20 25 30
Example 2
•Both distributions have the
same mean (40), but
different standard deviations
(10 vs. 20)
Statistics
•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.
Calculus
0
•Thus, how we evaluate
Dave’s performance
depends on how much
2 0
4 0
6 0
8 0
1 0variability
0
there is in the
G R AD E exam 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 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
(Xi  M )
Zi 
SD
** How far a person is from the mean, in the metric of
standard deviation units **
Example 1
15
Dave in Statistics:
Statistics
Calculus
(50 - 40)/10 = 1
10
(one SD above the
mean)
5
Dave in Calculus
(50 - 60)/10 = -1
0
(one SD below the
mean)
0
2 0
4 0
6 0
8 0
1 0 0
Mean
Statistics = 40
G R AD E
Mean
Calculus = 60
0 5 10 15 20 25 30
Example 2
An example where the
means are identical, but
the two sets of scores
have different spreads
Statistics
Dave’s Stats Z-score
(50-40)/5 = 2
Calculus
Dave’s Calc Z-score
(50-40)/20 = .5
0
2 0
4 0
6 0
8 0
1 0 0
G R AD E
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
if x = 60,
SD = 10
60  50 10

1
10
10
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
The shape is the same
(but the scaling or metric is different)
STANDARDIZED
0
0 .0
0 .1
2
0 .2
0 .3
4
0 .4
6
0 .5
UNSTANDARDIZED
0.4
0.6
0.8
1.0
-6
-4
-2
0
2
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.
0. 0.1 0.2 0.3 0.4
The area under a normal curve
50%
34%
34%
14%
14%
2%
- 4
2%
- 2
0
SC
2
4
O R
E
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.