Chapter 6
Foundations of Educational Measurement
Part 1
Jeffrey Oescher
Descriptive Statistics

Two things you
need to know


Four types of
descriptive
statistics
How to interpret
descriptive
statistical results
Descriptive Statistics
Tell me something about the following scores
33 31 31 31 32 29 30 30 32 30 32 30 28 29
30 29 31 31 28 28 29 30 27 30 29 30 30
Descriptive Statistics



The scores vary
from 27 – 33.
The typical score
appears to be
somewhere
around 30.
There seems to be
about as many
high scores as low
scores.
Descriptive Statistics
9
8
7
6
5
4
27
28
29
30
31
3
32
2
33
1
0
Descriptive Statistics

Four statistics we’ll
study




Central tendency
Variability
Relationship
Standard scores
Descriptive Statistics

Central Tendency



Mean: the arithmetic average
Median: the score above and below
which one-half of all of the scores in
the distribution lie
Mode: the most frequently occurring
score(s)
Descriptive Statistics

Variation



Standard deviation: the average
deviation of all scores around the
mean
Variance: the average “squared”
deviation of all scores around the mean
Range: the difference between the
highest and lowest scores in a
distribution
Descriptive Statistics

Relationships

Tell me something about the
relationship between the following
variables
GRE scores and graduate school
performance
 Misbehavior in class and achievement
 Students’ height and their academic
performance

Descriptive Statistics

Relationships

Pearson correlation coefficient

Magnitude: 0 to 1



Weak 0.0 – 0.3
Moderate 0.4 – 0.7
Strong 0.8 – 1.0
Direction: positive (+) or negative (-)
 Notation: rxy


Other coefficients
Descriptive Statistics

Examples of correlations






Weak positive +0.15
Weak negative –0.23
Moderate positive +0.42
Moderate negative –0.51
Strong positive +0.84
Strong negative –0.84
Interpreting Descriptive Statistics
Table 1
1
Scale Scores for the Experimental and Control Groups
Scale
DSP
OSI
TSM
SCSI
1
Experimental Group
N
Mean
SD
15
117.30
43.00
15
348.20
29.37
15
208.70
40.80
15
40.00
10.38
N
15
15
15
15
Control Group
Mean
SD
132.87
25.26
362.07
36.31
224.90
43.60
48.47
9.56
Scale scores reflect levels of stress, so lower scores are more desirable.
1.
2.
3.
4.
5.
6.
Which group had the higher scores on the DSP?
Which group had the better scores on the DSP?
Which group had more variation in their OSI scores?
Which group had less variation in their TSM scores?
Which group had more students on the SCSI?
REMEMBER THE DIFFERENCE BETWEEN A NORM-REFERENCED
AND CRITERION-REFERENCED INTERPRETATION
Interpreting Descriptive Statistics

How would you describe the following
relationships?








+0.83
+0.65
+0.21
-0.83
-0.54
-0.03
Which relationship is stronger, +0.83 or
0.83?
The following link provides the tables discussed
above as well as the answers to the questions on
the slides.

Interpreting Descriptive Statistics
Descriptive Statistics

Properties of the normal curve



Bell-shaped curve
Symmetric around the mean
Standard deviation units and proportions
under the curve
Approximately 68% of all scores fall between
±1 standard deviation of the mean
 Approximately 97% of all scores fall between
±2 standard deviations of the mean

Descriptive Statistics

Standard scores

How good is an IQ of 130?

The mean of IQ scores is 100 with a standard
deviation of 15




A z-score for an IQ of 130 is +2.00
This score is +2.00 standard deviations above the mean
This is an excellent score as it is better than about 96%
of all other scores
How good is a SAT score of 400?

The mean for each section of the SAT is 500 with a
standard deviation of 100



A z-score for a SAT score of 400 is –1.00
This score is –1.00 standard deviation below the mean
This is not a good score as it is better than only 16% of
all other scores
Descriptive Statistics

Z-scores



A z-score represents the number of standard
deviation units a specific score differs from the
mean
Z = [ x – X ] / sx
No matter what the scale of your test
or the nature of the scores, you can
always convert a raw score to a z-score
Descriptive Statistics

Standard scores




CEEB scores
Normal Curve Equivalents (NCE)
Stanines
Figure 7.11 on page 169 is a good
reference for the relationships
between several standard scores
Descriptive Statistics

Standard Scores

Which score represents the best effort, 50 or 100?
 What are the z-scores for each score if the first
test has a mean of 40 and a standard deviation of
10 while the second has a mean of 125 and a
standard deviation of 25?
 What if the first test has a mean of 35 and a
standard deviation of 15 while the second has a
mean of 95 with a standard deviation of 5?