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Probability, Statistics and Queueing

A scored 80 on test X, and B scored 75 on test Y. Test X had a mean of 65 and a standard deviation of 10, and test Y had a mean of 60 and a standard deviation of 6. Which of the following correctly compares their relative positions within their own tests using the standardized value z = (score − mean) ÷ standard deviation?

  1. AA ranks higher, because A scored 5 points more than B
  2. BB ranks higher, because A's z is 1.5 and B's z is 2.5
  3. CThey are at the same position, because both scored 15 points above the mean
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Answer: B. B ranks higher, because A's z is 1.5 and B's z is 2.5

A's z = (80 − 65) ÷ 10 = 1.5, and B's z = (75 − 60) ÷ 6 = 2.5. Both are 15 points above the mean, but test Y has a smaller spread, so measured in units of the standard deviation, B is farther above the mean. Standardization makes it possible to compare relative positions even between data with different means and units.

  • A. Scores on tests with different means and standard deviations cannot be compared directly.
  • B. Correct. z is 1.5 for A and 2.5 for B, so B is farther above the mean.
  • C. Both are 15 points above the mean, but the standard deviations differ, so their distance relative to the spread is different.
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