Standard deviation prices

Question

In December​ 2018, the average price of regular unleaded gasoline excluding taxes in the United States was ​$3.06 per gallon. Assume that the standard deviation price per gallon is ​$0.09 per gallon and use​ Chebyshev's Inequality to answer the following.

​(a) What minimum percentage of gasoline stations had prices within 2 standard deviations of the​ mean?

​(b) What minimum percentage of gasoline stations had prices within 2.5 standard deviations of the​ mean? What are the gasoline prices that are within 2.5 standard deviations of the​ mean?

​(c) What is the minimum percentage of gasoline stations that had prices between ​$2.70 and ​$3.42?

Answer 1

Chebyshev's inequality is a neat way to determine the percentage of values that fall within (or outside of) a certain number of standard deviations from the mean of any probability distribution, which is equal to 1/k^2 (where k is the number of standard deviations away from the mean). We use this when we either don't know the distribution or don't have a nice z-table or empirical rule applicable to the distribution we do have. In other words, it gives ballpark figures

a) 1/(2)^2 = 1/4 of gas stations have prices above 2 standard deviations of the mean thus 1 - 1/4 - 3/4 = 75% of gas stations have prices within 2 standard deviations of the mean.

b) 1/(2.5)^2 = 1/6.25 = 0.16 or 16% of gas stations have prices above 2 standard deviations of the mean, thus 1 - 0.16 = 0.84 or 84% of gas stations have prices within 2.5 standard deviations of the mean. 2.5 * $0.09 = $0.225 and so the minimum and maximum values within 2.5 standard deviations are $2.84 and $3.29 rounded to two decimal places respectively.

c) $2.70 and $3.42 correspond to 4 standard deviations above and below the mean respectively. This means that 1/(4)^2 = 1/16 = 0.0625 or 6.25% percent of gas stations have prices 4 standard deviations from the mean and 93.75% have prices within 4 standard deviations of the mean.

I hope that helps