 # Difficulty Index

To determine the difficulty level of test items, a measure called the Difficulty Index is used. This measure calculates the proportion of students who answered the test item accurately.

By looking at each alternative (for multiple choice), we can also find out if there are answer choices that should be replaced. For example, let’s say you gave a multiple choice quiz and there were four answer choices (A, B, C, and D).

The following table illustrates how many students selected each answer choice for Question #1 and #2.

Question A B C D
#1 0 3 24 3
#2 12 3 3 2

Correct answer for #1 is C and for #2 is A.

For Question #1, we can see that A was not a very good distractor – no one selected that answer.

We can also compute the difficulty of the item by dividing the number of students who choose the correct answer (24) by the number of total students (30).

Using this formula, the difficulty of Question #1 (referred to as p) is equal to 24/30 or .80. A rough “rule-of-thumb” is that if the item difficulty is more than .75, it is an easy item; if the difficulty is below .25, it is a difficult item.

Given these parameters, this item could be regarded moderately easy – lots (80%) of students got it correct.

In contrast, Question #2 is much more difficult (12/30 = .40). In fact, on Question #2, more students selected an incorrect answer ( B ) than selected the correct answer (A). This item should be carefully analyzed to ensure that B is an appropriate distractor.

# Discrimination Index

Another measure, the Discrimination Index, refers to how well an assessment differentiates between high and low scorers. In other words, you should be able to expect that the high-performing students would select the correct answer for each question more often than the low-performing students. If this is true, then the assessment is said to have a positive discrimination index (between 0 and 1) – indicating that students who received a high total score chose the correct answer for a specific item more often than the students who had a lower overall score.

If, however, you find that more of the low-performing students got a specific item correct, then the item has a negative discrimination index (between -1 and 0).

This table displays the results of ten questions on a test. Note that the students are arranged with the top overall scorers at the top of the table.

Student Total Score (%) Q #1 Q #2 Q #3
Asif 90 1 0 1
Sam 90 1 0 1
Jill 80 0 0 1
Charlie 80 1 0 1
Sonya 70 1 0 1
Ruben 60 1 0 0
Clay 60 1 0 1
Kelley 50 1 1 0
Justin 50 1 1 0
Tonya 40 0 1 0

“1” indicates the answer was correct; “0” indicates it was incorrect.

Follow these steps to determine the Difficulty Index and the Discrimination Index.

After the students are arranged with the highest overall scores at the top, count the number of students in the upper and lower group who got each item correct. For Question #1, there were 4 students in the top half who got it correct, and 4 students in the bottom half.
Determine the Difficulty Index by dividing the number who got it correct by the total number of students. For Question #1, this would be 8/10 or p=.80.
Determine the Discrimination Index by subtracting the number of students in the lower group who got the item correct from the number of students in the upper group who got the item correct. Then, divide by the number of students in each group (in this case, there are five in each group). For Question #1, that means you would subtract 4 from 4, and divide by 5, which results in a Discrimination Index of 0.

Item # Correct (Upper Group) # Correct (Lower Group) Difficulty § Discrimination (D)
Q #1 4 4 .80 0
Q #2 0 3 .30 -0.6
Q #3 5 1 .60 0.8

We can see that Question #2 had a difficulty index of .30 (meaning it was quite difficult), and it also had a negative discrimination index of -0.6 (meaning that the low-performing students were more likely to get this item correct). This question should be carefully analyzed, and probably deleted or changed. Our “best” overall question is Question 3, which had a moderate difficulty level (.60), and discriminated extremely well (0.8).

## Important Change!

In the way how to calculate discrimination index, we made a little change.

As if the sample is less than 30, upper and lower groups are 50% of the sample, but if the sample is greater than 30, upper and lower groups are now taken as 28% of the sample.