Bluman A.G. Elementary Statistics- A Step By Step Approach

Section 3–2 Measures of Variation 135Figure 3–3Chebyshev’s TheoremAt least88.89%At least75%X – 3s X – 2sX X + 2s X + 3sThis theorem can be applied to any distribution regardless of its shape (seeFigure 3–3).Examples 3–27 and 3–28 illustrate the application of Chebyshev’s theorem.Example 3–27Prices of HomesThe mean price of houses in a certain neighborhood is $50,000, and the standarddeviation is $10,000. Find the price range for which at least 75% of the houseswill sell.SolutionChebyshev’s theorem states that three-fourths, or 75%, of the data values will fall within2 standard deviations of the mean. Thus,and$50,000 2($10,000) $50,000 $20,000 $70,000$50,000 2($10,000) $50,000 $20,000 $30,000Hence, at least 75% of all homes sold in the area will have a price range from $30,000to $70,000.Chebyshev’s theorem can be used to find the minimum percentage of data values thatwill fall between any two given values. The procedure is shown in Example 3–28.Example 3–28Travel AllowancesA survey of local companies found that the mean amount of travel allowance for executiveswas $0.25 per mile. The standard deviation was $0.02. Using Chebyshev’s theorem, findthe minimum percentage of the data values that will fall between $0.20 and $0.30.3–33