## Exercises - Chebyshev's Theorem

1. What amount of data does Chebyshev's Theorem guarantee is within three standard deviations from the mean?

$k=3$ in the formula and $k^2 = 9$, so $1 - 1/9 = 8/9$. Thus $8/9$ of the data is guaranteed to be within three standard deviations of the mean.

2. Given the following grades on a test:

86, 92, 100, 93, 89, 95, 79, 98, 68, 62, 71, 75, 88, 86, 93, 81, 100, 86, 96, 52

what percentage of scores lie within one standard deviation from the mean? two standard deviations? Are these results consistent with what Chebyshev's Theorem concludes?

13 out of 20 lie between 71.33 and 97.67, so 65% lie within one standard deviation of the mean; 19 out of 20, or 95% lie within two standard deviations of the mean. These are, of course, consistent with what Chebyshev's Theorem concludes -- namely, that at least 0% lies within one standard deviation of the mean (trivially true), and that $(1-1/2^2) = 75$% lies within two standard deviations of the mean (and note that 95% > 75%).

3. Given this sample of freshman GPA scores:

2.2, 2.9, 3.5, 4.0, 3.9, 3.5, 2.9, 2.8, 3.1, 3.5, 3.8, 4.0,
3.8, 2.4, 3.9, 3.4, 2.8, 2.4, 1.8, 3.6, 3.1, 2.9, 3.8, 4.0


what percentage of scores is within one standard deviation of the mean? two standard deviations? three standard deviations? Are these results consistent with the conclusions of Cheybshev's Theorem?

62.5%, 95.8%, 100% Yes, of course these are consistent with the conclusions of Chebyshev's Theorem which indicate these values must be at least 0%, 75%, and approximately 88.8%, respectively. In each case, the proportion seen in the sample exceeds the bound Chebyshev's theorem establishes.