What are the percentages of the observations for A and B?

Question

A data set has a mean of 1,600 and a standard deviation of 120.

a. Using Chebyshev's theorem, what percentage of the observations fall between 1,240 and 1,960? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)

b. Using Chebyshev’s theorem, what percentage of the observations fall between 1,360 and 1,840? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)

Answer 1

Chebyshev's Theorem states that, for any distribution (normal or otherwise), the proportion of the data within k standard deviations of the mean is at least

1 - 1/k²

So, in this case, we're given a mean of 1600 and standard deviation of 120. For part (a), the range defined is between 1240 and 1960. This is 360 data units below and above the mean. Since the standard deviation is 120, it is within 3 standard deviations of the mean (360/120), so k = 3. Therefore, 1 - 1/3² = 1 - 1/9. This is 8/9, which is about 89%. However, for your answer, you should say at least 89% because we don't have enough information about the distribution to get a more exact answer.

For part (b), the range is from 1360 to 1840. This is 240 above and below, which is within 2 standard deviations of the mean. So, the proportion of the data in this range is at least 1 - 1/2², which is 3/4 - 75%. So at least 75% of the data is within this range.

Bonus fact: It can be helpful here to make a comparison to the normal distribution--specifically the empirical rule--to "verify" or show how "fuzzy" this theorem is. The empirical rule for the normal distribution says that about 99.7% of the data is within 3 standard deviations of the mean, and 95% is within 2 standard deviations. Both of these are consistent with Chebyshev's Theorem, but not particularly close. This gives a feel for how much of a difference knowing the distribution can make.