A civil service exam yields scores with a mean of 81 and a standard deviation of 5.5. Using Chebyshev's Theorem what can we say about the percentage of scores that are above 92?

Question

Reginald Hill

Mathematics

31 May, 18:14

A civil service exam yields scores with a mean of 81 and a standard deviation of 5.5. Using Chebyshev's Theorem what can we say about the percentage of scores that are above 92?

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Jayvon Mcdonald · Answer 1

At most 12.5% of the scores are above 92.

Step-by-step explanation:

Chebyshev's theorem states that:

At least 75% of the measures are within 2 standard deviations of the mean.

At least 89% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 81%

Standard deviation = 5.5%

Using Chebyshev's Theorem what can we say about the percentage of scores that are above 92?

92 = 81 + 2*5.5

So 92 is two standard deviations above the mean.

By the Chebyshev's theorem, at least 75% of the measures are within 2 standard deviations of the mean. So at most 25% is more than 2 standard deviations from the mean. Chebyshev's theorem works with symmetric distributions, so, of those at most 25%, at most 12.5% are more than 2 standard deviations below the mean and at most 12.5% are more than 2 standard deviations above the mean.

So at most 12.5% of the scores are above 92.