Question Difficulty Analyzer

Classical item analysis describes a test question with two numbers. The difficulty index, conventionally written p, is the proportion answering correctly - so despite its name a high p means an easy item. The discrimination index measures whether the students who did well overall were also the ones who got this item right; in point-biserial form it is the correlation between item score and total score and ranges from minus 1 to plus 1. The two are linked through item variance p times (1 minus p), which peaks at 0.25 when p equals 0.5. That is the mathematical reason mid-difficulty items carry the most information: an item everyone or no one answers correctly has zero variance and cannot discriminate at all. Practitioner thresholds are well established. For norm-referenced testing, items with p between 0.30 and 0.70 are retained, discrimination at or above 0.40 is rated excellent, 0.30 to 0.39 good, 0.20 to 0.29 marginal, and anything below 0.20 or negative is flagged for rewrite - a negative value usually means a miskeyed answer or a distractor that attracts strong students. Because guessing inflates p on selected-response items, the corrected index (p minus g) divided by (1 minus g) is used, where g is 0.25 for four options and 0.20 for five. Caveats: both statistics are sample dependent, so an item is only easy relative to the group tested; estimates are unstable below roughly 100 examinees; and on criterion-referenced mastery tests high p values are the goal, not a defect.

Item analysis emerged from the mental testing movement of the early twentieth century. The percentage-passing measure of difficulty was already in use in the Army Alpha and Beta group tests administered to roughly 1.7 million US recruits from 1917 under Robert Yerkes, where items were ordered by the proportion of soldiers answering correctly. During the 1930s the statistical machinery matured: Truman Kelley published work in 1939 showing that upper and lower groups of about 27 percent of examinees maximize the stability of a difference-based discrimination index, a rule of thumb still embedded in item analysis software. Reliability coefficients from Charles Spearman, and later the Kuder-Richardson formulas of 1937 and Lee Cronbach's alpha in 1951, gave test builders a way to connect item statistics to whole-test quality. The canonical synthesis of this framework is Harold Gulliksen's Theory of Mental Tests of 1950, which laid out classical test theory in the form still taught today. Its central weakness - that p and discrimination describe an item only in relation to the particular sample that took it - motivated the development of item response theory. Georg Rasch, a Danish statistician, published his one-parameter logistic model in 1960, and Frederic Lord and Melvin Novick's 1968 volume with Allan Birnbaum introduced the two- and three-parameter models, the third parameter being an explicit pseudo-guessing asymptote. Modern large-scale programs calibrate with IRT, but classical indices remain the standard tool for classroom and small-sample assessment because they need no specialized software.