# Comparing the Normal Ogive and Logistic Item Characteristic Curves

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In item response theory, the relationship between a latent ability () and the probability of a correct response () on a test item is modelled by an item characteristic curve. This Demonstration plots the item characteristic curve of a single dichotomous item under two different models: the normal ogive model and the logistic model. The parameters , , and represent item properties related to discrimination, difficulty, and guessing. The constant is used to scale the logistic curve. Notice that the two curves are nearly identical when .

Contributed by: Vincent Kieftenbeld (March 2011)

Open content licensed under CC BY-NC-SA

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## Details

The probability that a person with ability level gives a correct response () to an item with discrimination parameter , difficulty parameter , and pseudo-guessing parameter is modelled in the normal ogive model as

.

Alternatively, in the three-parameter logistic model,

.

The constant *D* is used to scale the logistic curve and represents the relationship between logits and probits. When , the models agree closely; that is, 1 logit is approximately equal to 1.7 probit. In fact, minimizes the maximum difference between the normal ogive and logistic curves.

Reference:

G. Camilli, "Origin of the Scaling Constant d=1.7 in Item Response Theory," *Journal of Educational and Behavioral Statistics*, 19(3), 1994 pp. 293–295.

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