# Item Discrimination Definition Essay

Susan Matlock-Hetzel

Texas A&M University, January 1997

When norm-referenced tests are developed for instructional purposes, to assess the effects of educational programs, or for educational research purposes, it can be very important to conduct item and test analyses. These analyses evaluate the quality of the items and of the test as a whole. Such analyses can also be employed to revise and improve both items and the test as a whole. However, some best practices in item and test analysis are too infrequently used in actual practice. The purpose of the present paper is to summarize the recommendations for item and test analysis practices, as these are reported in commonly-used measurement textbooks (Crocker & Algina, 1986; Gronlund & Linn, 1990; Pedhazur & Schemlkin, 1991; Sax, 1989; Thorndike, Cunningham, Thorndike, & Hagen, 1991).

Paper presented at the annual meeting of the Southwest Educational Research Association, Austin, January, 1997.Making fair and systematic evaluations of others' performance can be a challenging task. Judgments cannot be made solely on the basis of intuition, haphazard guessing, or custom (Sax, 1989). Teachers, employers, and others in evaluative positions use a variety of tools to assist them in their evaluations. Tests are tools that are frequently used to facilitate the evaluation process. When norm-referenced tests are developed for instructional purposes, to assess the effects of educational programs, or for educational research purposes, it can be very important to conduct item and test analyses.

Test analysis examines how the test items perform as a set. Item analysis "investigates the performance of items considered individually either in relation to some external criterion or in relation to the remaining items on the test" (Thompson & Levitov, 1985, p. 163). These analyses evaluate the quality of items and of the test as a whole. Such analyses can also be employed to revise and improve both items and the test as a whole.

However, some best practices in item and test analysis are too infrequently used in actual practice. The purpose of the present paper is to summarize the recommendations for item and test analysis practices, as these are reported in commonly-used measurement textbooks (Crocker & Algina, 1986; Gronlund & Linn, 1990; Pedhazur & Schemlkin, 1991; Sax, 1989; Thorndike, Cunningham, Thorndike, & Hagen, 1991). These tools include item difficulty, item discrimination, and item distractors.

Item Difficulty

Item difficulty is simply the percentage of students taking the test who answered the item correctly. The larger the percentage getting an item right, the easier the item. The higher the difficulty index, the easier the item is understood to be (Wood, 1960). To compute the item difficulty, divide the number of people answering the item correctly by the total number of people answering item. The proportion for the item is usually denoted as __p__ and is called item difficulty (Crocker & Algina, 1986). An item answered correctly by 85% of the examinees would have an item difficulty, or __p__ value, of .85, whereas an item answered correctly by 50% of the examinees would have a lower item difficulty, or __p__ value, of .50.

A __p__ value is basically a behavioral measure. Rather than defining difficulty in terms of some intrinsic characteristic of the item, difficulty is defined in terms of the relative frequency with which those taking the test choose the correct response (Thorndike et al, 1991). For instance, in the example below, which item is more difficult?

- Who was Boliver Scagnasty?
- Who was Martin Luther King?

One cannot determine which item is more difficult simply by reading the questions. One can recognize the name in the second question more readily than that in the first. But saying that the first question is more difficult than the second, simply because the name in the second question is easily recognized, would be to compute the difficulty of the item using an intrinsic characteristic. This method determines the difficulty of the item in a much more subjective manner than that of a __p__ value.

Another implication of a __p__ value is that the difficulty is a characteristic of both the item and the sample taking the test. For example, an English test item that is very difficult for an elementary student will be very easy for a high school student. A __p__ value also provides a common measure of the difficulty of test items that measure completely different domains. It is very difficult to determine whether answering a history question involves knowledge that is more obscure, complex, or specialized than that needed to answer a math problem. When __p__ values are used to define difficulty, it is very simple to determine whether an item on a history test is more difficult than a specific item on a math test taken by the same group of students.

To make this more concrete, take into consideration the following examples. When the correct answer is not chosen (__p__ = 0), there are no individual differences in the "score" on that item. As shown in Table 1, the correct answer C was not chosen by either the upper group or the lower group. (The upper group and lower group will be explained later.) The same is true when everyone taking the test chooses the correct response as is seen in Table 2. An item with a __p__ value of .0 or a __p__ value of 1.0 does not contribute to measuring individual differences, and this is almost certain to be useless. Item difficulty has a profound effect on both the variability of test scores and the precision with which test scores discriminate among different groups of examinees (Thorndike et al, 1991). When all of the test items are extremely difficult, the great majority of the test scores will be very low. When all items are extremely easy, most test scores will be extremely high. In either case, test scores will show very little variability. Thus, extreme __p__ values directly restrict the variability of test scores.

Table 1

Minimum Item Difficulty Example Illustrating No Individual Differences

Note. * denotes correct response

Item difficulty: (0 + 0)/30 = .00p

Discrimination Index: (0 - 0)/15 = .00

Table 2

Maximum Item Difficulty Example Illustrating No Individual Differences

Note. * denotes correct response

Item difficulty: (15 + 15)/30 = 1.00p

Discrimination Index: (15-15)/15 = .00

In discussing the procedure for determining the minimum and maximum score on a test, Thompson and Levitov (1985) stated that

For example, many teachers may think that the minimum score on a test consisting of 100 items with four alternatives each is 0, when in actuality the theoretical floor on such a test is 25. This is the score that would be most likely if a student answered every item by guessing (e.g., without even being given the test booklet containing the items).

Similarly, the ideal percentage of correct answers on a four-choice multiple-choice test is not 70-90%. According to Thompson and Levitov (1985), the ideal difficulty for such an item would be halfway between the percentage of pure guess (25%) and 100%, (25% + {(100% - 25%)/2}. Therefore, for a test with 100 items with four alternatives each, the ideal mean percentage of correct items, for the purpose of maximizing score reliability, is roughly 63%. Tables 3, 4, and 5 show examples of items with p values of roughly 63%.

Table 3

Maximum Item Difficulty Example Illustrating Individual Differences

Note. * denotes correct response

Item difficulty: (13 + 5)/30 = .60p

Discrimination Index: (13-5)/15 = .53

Table 4

Maximum Item Difficulty Example Illustrating Individual Differences

Note. * denotes correct response

Item difficulty: (11 + 7)/30 = .60p

Discrimination Index: (11-7)/15 = .267

Table 5

Maximum Item Difficulty Example Illustrating Individual Differences

Note. * denotes correct response

Item difficulty: (11 + 7)/30 = .60p

Discrimination Index: (7 - 11)/15 = .267

If the test and a single item measure the same thing, one would expect people who do well on the test to answer that item correctly, and those who do poorly to answer the item incorrectly. A good item discriminates between those who do well on the test and those who do poorly. Two indices can be computed to determine the discriminating power of an item, the item discrimination index, __D__, and discrimination coefficients.

Item Discrimination Index, D

The method of extreme groups can be applied to compute a very simple measure of the discriminating power of a test item. If a test is given to a large group of people, the discriminating power of an item can be measured by comparing the number of people with high test scores who answered that item correctly with the number of people with low scores who answered the same item correctly. If a particular item is doing a good job of discriminating between those who score high and those who score low, more people in the top-scoring group will have answered the item correctly.

In computing the discrimination index, __D__, first score each student's test and rank order the test scores. Next, the 27% of the students at the top and the 27% at the bottom are separated for the analysis. Wiersma and Jurs (1990) stated that "27% is used because it has shown that this value will maximize differences in normal distributions while providing enough cases for analysis" (p. 145). There need to be as many students as possible in each group to promote stability, at the same time it is desirable to have the two groups be as different as possible to make the discriminations clearer. According to Kelly (as cited in Popham, 1981) the use of 27% maximizes these two characteristics. Nunnally (1972) suggested using 25%.

The discrimination index, __D__, is the number of people in the upper group who answered the item correctly __minus__ the number of people in the lower group who answered the item correctly, __divided__ by the number of people in the largest of the two groups. Wood (1960) stated that

The higher the discrimination index, the better the item because such a value indicates that the item discriminates in favor of the upper group, which should get more items correct, as shown in Table 6. An item that everyone gets correct or that everyone gets incorrect, as shown in Tables 1 and 2, will have a discrimination index equal to zero. Table 7 illustrates that if more students in the lower group get an item correct than in the upper group, the item will have a negative __D__ value and is probably flawed.

Table 6

Positive Item Discrimination Index D

Note. * denotes correct response

74 students took the test

27% = 20(N)

Item difficulty: (15 + 3)/40 = .45p

Discrimination Index: (15 - 3)/20 = .60

Table 7

Negative Item Discrimination Index D

Note. * denotes correct response

Item difficulty: (0 + 15)/30 = .50p

Discrimination Index: (0 - 15)/15 = -1.0

A negative discrimination index is most likely to occur with an item covers complex material written in such a way that it is possible to select the correct response without any real understanding of what is being assessed. A poor student may make a guess, select that response, and come up with the correct answer. Good students may be suspicious of a question that looks too easy, may take the harder path to solving the problem, read too much into the question, and may end up being less successful than those who guess. As a rule of thumb, in terms of discrimination index, .40 and greater are very good items, .30 to .39 are reasonably good but possibly subject to improvement, .20 to .29 are marginal items and need some revision, below .19 are considered poor items and need major revision or should be eliminated (Ebel & Frisbie, 1986).

Discrimination Coefficients

Two indicators of the item's discrimination effectiveness are point biserial correlation and biserial correlation coefficient. The choice of correlation depends upon what kind of question we want to answer. The advantage of using discrimination coefficients over the discrimination index (D) is that every person taking the test is used to compute the discrimination coefficients and only 54% (27% upper + 27% lower) are used to compute the discrimination index, __D__.

__Point biserial.__ The point biserial (__r__) correlation is used to find out if the right people are getting the items right, and how much predictive power the item has and how it would contribute to predictions. Henrysson (1971) suggests that the __r__ tells more about the predictive validity of the total test than does the biserial __r__, in that it tends to favor items of average difficulty. It is further suggested that the __r__ is a combined measure of item-criterion relationship and of difficulty level.

__Biserial correlation.__ Biserial correlation coefficients (__r__) are computed to determine whether the attribute or attributes measured by the criterion are also measured by the item and the extent to which the item measures them. The __r__ gives an estimate of the well-known Pearson product-moment correlation between the criterion score and the hypothesized item continuum when the item is dichotomized into right and wrong (Henrysson, 1971). Ebel and Frisbie (1986) state that the __r__ simply describes the relationship between __scores on a test item (e.g., "0" or "1") and scores (e.g., "0", "1",..."50") on the total test for all examinees__.

Distractors

Analyzing the distractors (e.i., incorrect alternatives) is useful in determining the relative usefulness of the decoys in each item. Items should be modified if students consistently fail to select certain multiple choice alternatives. The alternatives are probably totally implausible and therefore of little use as decoys in multiple choice items. A discrimination index or discrimination coefficient should be obtained for each option in order to determine each distractor's usefulness (Millman & Greene, 1993). Whereas the discrimination value of the correct answer should be positive, the discrimination values for the distractors should be lower and, preferably, negative. Distractors should be carefully examined when items show large positive __D__ values. When one or more of the distractors looks extremely plausible to the informed reader and when recognition of the correct response depends on some extremely subtle point, it is possible that examinees will be penalized for partial knowledge.

Thompson and Levitov (1985) suggested computing reliability estimates for a test scores to determine an item's usefulness to the test as a whole. The authors stated, "The total test reliability is reported first and then each item is removed from the test and the reliability for the test less that item is calculated" (Thompson & Levitov, 1985, p.167). From this the test developer deletes the indicated items so that the test scores have the greatest possible reliability.

Developing the perfect test is the unattainable goal for anyone in an evaluative position. Even when guidelines for constructing fair and systematic tests are followed, a plethora of factors may enter into a student's perception of the test items. Looking at an item's difficulty and discrimination will assist the test developer in determining what is wrong with individual items. Item and test analysis provide empirical data about how individual items and whole tests are performing in real test situations.

Crocker, L., & Algina, J. (1986). __Introduction to classical and modern test theory.__ New York: Holt, Rinehart and Winston.

Ebel, R.L., & Frisbie, D.A. (1986). __Essentials of educational measurement.__ Englewood Cliffs, NJ: Prentice-Hall.

Gronlund, N.E., & Linn, R.L. (1990). __Measurement and evaluation in teaching__ (6th ed.). New York: MacMillan.

Henrysson, S. (1971). Gathering, analyzing, and using data on test items. In R.L. Thorndike (Ed.), __Educational Measurement__ (p. 141). Washington DC: American Council on Education.

Millman, J., & Greene, J. (1993). The specification and development of tests of achievement and ability. In R.L. Linn (Ed.), __Educational measurement__ (pp. 335-366). Phoenix, AZ: Oryx Press.

Nunnally, J.C. (1972). __Educational measurement and evaluation__ (2nd ed.). New York: McGraw-Hill.

Pedhazur, E.J., & Schmelkin, L.P. (1991). __Measurement, design, and analysis: An integrated approach.__ Hillsdale, NJ: Erlbaum.

Popham, W.J. (1981). __Modern educational measurement__. Englewood Cliff, NJ: Prentice-Hall.

Sax, G. (1989). __Principles of educational and psychological measurement and evaluation__ (3rd ed.). Belmont, CA: Wadsworth.

Thompson, B., & Levitov, J.E. (1985). Using microcomputers to score and evaluate test items. __Collegiate Microcomputer__, __3__, 163-168.

Thorndike, R.M., Cunningham, G.K., Thorndike, R.L., & Hagen, E.P. (1991). __Measurement and evaluation in psychology and education__ (5th ed.). New York: MacMillan.

Wiersma, W. & Jurs, S.G. (1990). __Educational measurement and testing__ (2nd ed.). Boston, MA: Allyn and Bacon.

Wood, D.A. (1960). __Test construction: Development and interpretation of achievement tests.__ Columbus, OH: Charles E. Merrill Books, Inc.

The single best measure of the effectiveness of an item is its ability to separate students who vary in their degree of knowledge of the material tested and their ability to use it. If one group of students has mastered the material and the other group had not, a larger portion of the former group should be expected to correctly answer a test item. Item discrimination is the difference between the percentage correct for these two groups.

Item discrimination can be calculated by ranking the students according to total score and then selecting the top 27 percent and the lowest 27 percent in terms of total score. For each item, the percentage of students in the upper and lower groups answering correctly is calculated. The difference is one measure of **item discrimination (IDis)**. The formula is:

IDis = (Upper Group Percent Correct) – (Lower Group Percent Correct)

Item #1 in the attached report would have an IDis of

100% - 62.5% = 37.5% (or .375 as a decimal).

The maximum item discrimination difference is 100 percent. This would occur if all those in the upper group answered correctly and all those in the lower group answered incorrectly. Zero discrimination occurs when equal numbers in both groups answer correctly. Negative discrimination, a highly undesirable condition, occurs when more students in the lower group then the upper group answer correctly. Item #6 on the attached report has a negative IDis.

The following levels may be used as a guideline for acceptable items.

Negative IDis | Unacceptable – check item for error |

0% - 24% | Usually unacceptable – might be approved |

25% - 39% | Good item |

40% - 100% | Excellent item |

## Comments