Practicing with the Central Limit Theorem

Predicting Test Scores

You are  amiddles chool principal and your 100 eighth-graders are about to take a national standardized test. The test is designed so that the mean score is μ=400 with a standard deviation of σ=70.  Assume the scores are normally distributed.

  1. What is the likelihood that one of your eighth-graders, selected at random, will score below 375 on the exam?
  2. Your performance as a principal depends on how well your entire group of eighth-graders scores on the exam. What is the likelihood that your group of 100 eighth-graders will have a mean score of 375?


  1. In dealing with an individual score, we use the method of standard scores (z-scores). Given the mean of 400 and a standard deviation of 70, a score of 375 has a standard score of

z =   data value – mean/ standard deviation

=  375 – 400/ 70

=  -0.36

Since a z-score of -0.36 corresponds to about the 36th percentile, then 36% of the students are expected to score below 375.  So there is a 0.36 chance that a randomly selected student will score below 375.  Remember: We need to know that the scores have a normal distribution for us to make this calculation because z-scores only apply to normal distributions.

2. The question about the mean of a group of students must be handled with the Central Limit Theorem. According to this theorem, if we take random samples of size n =100 students and compute the mean test score for the group, the distribution of means is approximately normal. Moreover, the mean of this distribution is μ = 400 and its standard deviation is σ/√n or 70/√100 = 7. With these values for the mean and standard deviation, the standard score for a mean test score of 375 is

z =  375 – 400 / 7 =  -3.57

Since  z-score of -3.5 corresponds to the 0.02th percentile, and the standard score in this case is even lower, we can say that fewer than 0,02% of all random samples of 100 students will have a mean score of less than 375. Therefore the chance that a randomly selected group of 100 students will have a mean score below 375 is less than 0.0002, or about 1 in 5,000. Notice that this calculation regarding the group mean did not depend upon the individual scores having a normal distribution.

The likelihood of an individual scoring below 375 is more than 1 in 3 (36%), but the likelihood of a group of 100 students having a mean score below 375 is less than 1 in 5,000 (0.02%). In other words, there is much more variation in the scores of individuals than in the means of groups of individuals.

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