Transcript Five Questions about Test Scores

Lesson Fourteen
Interpreting Scores
Contents
Five Questions about Test Scores
1. The general pattern of the set of scores
 How do scores run or what do they look
like?
 2. What’s the typical level of performance?
What does the typical individual look like?
 3. How do scores spread out away from
the average value?
 4. What does an individual score on a test
mean?
 5. How are two sets of scores related?

General Pattern of Scores

What do scores look like?
– List all scores, but no sense of the scores.
– List scores from lowest to highest “order”,
out of chaos, but if too many scores, it’s
difficult to list.
– List scores with frequency marked reduce
the length of the distribution, but still have a
long table if N is big.
– Group the scores into intervals (i.e.,
grouped frequency distribution)  ideal: 10
~15 groups
– Convert grouped frequency distribution into
a graphical representation
General Pattern of Scores
Frequency distribution: a table that
shows how often each score has
occurred
 Grouped frequency distribution: normally
10 ~ 15 groups
 Graphic representation

– Histogram (bar graphs) (Bailey 94)
– Frequency polygon (Bailey 95)
Frequency Distribution

Score f
2
1
4
2
5
1
6
2
7
1
9
1
11
2
12
2
14
2
Grouped Frequency Distribution

Scores
f
46-49
1
42-45
1
38-41
5
34-37
2
:
:
14-17
5
10-13
4
6-9
4
2-5
4
Graphic Representation

Histogram

Bar graph

Frequency polygon
Measures of Central Tendency (1)

2. What’s the average level of
performance?  measures of central
tendency
– To identify the typical score (i.e., the “middle”
of the group)
Mean (x bar )
 Median (m)
 Mode (M)

Central Tendency (2)

Mean
– The sum of all scores divided by the total
number of the scores
– Arithmetic average
– The most commonly used
– Mean = X
N
Central Tendency (3)

Median
– The middle score that separates the scores into
half
– There is the same number of scores above and
below it (but the list of scores has to be in order,
e.g., from low/high to high/low)
– The 50th percentile (p50)
– The middle score
– e.g., a set of scores: 10, 5, 3  m = 5
– 20, 9, 7, 6, 1  m =?
– 15, 7, 6, 3  m = ?
– Location of the median: (N+1)/2
• e.g., N= 15, (15+1)/2 = 8  median = 8th score
Central Tendency (4)

Mode:
– the score that occurs most frequently
– E.g., 18-21= the modal group (in the
previous example)
– Possible to have bimodal (or even trimodal)
groups
Measures of Variability (1)
3. How spread out or how close
together are the scores? Or how
“widely” scores spread out around their
center?
 Measures of dispersion (or variability)

– Range
– Variance
– Standard deviation
Measures of Variability (2)

Range
– The difference between the highest and the
lowest scores
– The highest score – the lowest score (+ 1)
– Depending only on two extreme cases, so
not very dependable
Measures of Variability (3)
Variance (s2)
 A measure of dispersion around the
mean (Bailey 92)
 An indicator of the spread of scores in a
distribution
 S2 = (x-x bar)2/N - 1

Measures of Variability (4)
Standard deviation (s)
 The square root of variance
 Average of the differences of all score
from the mean
 e.g., two sets of scores

– 3, 4, 5  mean = 4
– 1, 4, 7  mean = 4
– 2nd set is more spread out  larger s

s>=0
Measures of Variability (5)

Under normal distribution, a constant
relationship exists between s and the
proportion of cases: (Bailey 105)
– between mean and either +1s or –1 s 
34.1% cases (between +- 1 s  68.2%)
– between mean and either +2 s or –2 s 
47.7% (i.e., between +- 2 s = 95.4%)
– between mean and either +3 s or –3 s 
49.9% (i.e., between +- 3 s = 99.8%)
Interpretation of Individual Scores


4. What does an individual score on a test
mean? What does it say about individual
performance? How does this person stand
relative to the group?
Percentile
– The score exceeds a specific fraction of the
scores

Standard deviation
– Provides a common unit of distance from the
mean of the group
– Individual score can be expressed as a distance
above or below the mean in the common unit
Measures of Relationship
5. To what extent are scores on two
different tests go together? How are
two sets of scores on different tests
related? What’s the relationship
between two measurements?
 Correlation coefficient (r): an index of
degree of relationship
 Correlation only tells you that two things
are related, but doesn’t mean causality.

Correlation

-1<= r <= 1
– r: “+”  rank people in the same order; exact agreement. “-”
 rank people in the reverse order; exact disagreement
– r: how strong the relationship is (the strength)

r= 1  perfect positive relationship
– High on one test, must be high on the other

r= 0  no relationship
– No pattern; random

r= -1  strong relationship, but in the opposite
direction
– High on one test, but low on the other
Correlation
3 settings that use correlation coefficients
 To determine how consistent a
measurement procedure is

– Info. on stability or reliability

To study the relationship between 2
measures
– To evaluate one as a predictor of the other

Relationships between variables
– To understand better how behavior is
organized