Examples
Adapted from the forthcoming CyberStats introductory statistics course.

Example 2 - Cutoff Scores for Admissions to a School's "Gifted" Program.

In the United States, many school districts have "gifted" programs for particularly bright students. Some school districts use IQ tests to screen students for entrance to the gifted program.  Most schools don't use an IQ test as the sole criterion for entrance, but will place into the program any student whose IQ test score is higher than a specified cutoff.  The cutoff IQ is a specified percentile of the distribution of IQ scores, for instance, the 95th percentile.

The Stanford-Binet IQ test is calibrated so that, for the general population, the mean score is 100 with a standard deviation equal to 16.  The distribution of scores is approximated by a normal curve.  What IQ score is the cutoff score for the gifted program in a district that uses the 95th percentile of the Stanford-Binet test as the criterion?

The Cutoff

The following graph describes the problem.  The proportion less than the cutoff is 0.95.  What's the cutoff?

95th percentile

The answer is about 126.

Interactivity

Try verifying this.  The steps in the solution are:
  1. Use the calculator below to determine what Z score is the 95th percentile of the standard normal curve. The given information is Prob<Z.
  2. Calculate the IQ that has this Z score.  The Z score is the number of standard deviations that the 95th percentile is from the mean score.  The mean is 100, the standard deviation is 16.

Z Score

Prob<Z

Some Cautions
The calculations are based on a model for IQ scores. 

  • If the model is wrong, the school district is using a borderline that is not the desired percentile. 
  • Are the assumed mean and standard deviation correct? 
  • Is the distribution really normal? 

The answers to these questions affect the correctness of the cutoff score.

Example 1
Example 2

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