1. Item & Test Statistics
Psych DeShon

2. Binary Item Statistics
Score an item or set of items as 0,1
1 represents “pass”, “agree”, or “endorse”
Notation:
Xj - jth item score
xij - ith person’s score on item j
j is the probability of a “1” on item j
nj is the number of people in the sample who receive a “1” on the jth item

3. Binary Item Statistics
Item difficulty = item mean
sample mean for item j
relative frequency of “1” (pass) in the sample
proportion of people with a “1” on the jth item
Estimates the “item difficulty” (j)
Range: 0 -> 1
1 means easy
0 means hard

4. Binary Item Statistics
Item Variance
Mean (“difficulty”) and variance are not independent
What does this imply about item discrimination or item information?

5. Binary Item Statistics
Covariance
Correlation

6. Quantitative Item Statistics
Assume a 5 point rating scale
frequency weighted approach X nc p  x
x2 5 7
.07 .1
0.5
2.5 4 17
.17 .2
0.8
3.2 3 41
.41 .4
1.2
3.6 2 25
.25
0.4 1 10
.10
0.1

7. Quantitative Item Statistics
Maximum variance of an item given the number of response options
used as the basis for the rwg statistic k 2 3 4 5 6 7 8 9 10
0.25
1.00
2.25
4.00
6.25
9.00
12.25
16.00
20.25

8. Quantitative Test Statistics
Forming test scores from quantitative items
Usually accomplished by simply adding the scores for each item
This is a unit weighted linear composite
Can directly compute the resulting test mean and SD
Variance is the sum of all elements in the var- covar matrix