To study the usefulness of the Empirical Rule, look at a distribution that is heavily skewed to the right, as shown in the accompanying figure.A histogram has a horizontal axis labeled "n= 25" with values from 0 to 10 and a vertical axis labeled "Frequency" with values from 0 to 10. The histogram has 11 bars. Each bar is associated with a label and an approximate value as listed below.

• 0: 10
• 1: 5
• 2: 3
• 3: 2
• 4: 1
• 5: 1
• 6: 0
• 7: 0
• 8: 1
• 9: 1
• 10: 1

Calculate

x

and s for the data shown. (NOTE: There are 10 zeros, 5 ones, and so on. Round your answer for s to three decimal places.)

x =

s =

Find the intervals

x s, x 2s,

and

x 3s

and locate them on the frequency distribution. (Round your answers to three decimal places.)

	Interval
x s	to 5.068
x 2s	3.777 to
x 3s	6.725 to

Calculate the proportion of the

n = 25

measurements that fall into each of the three intervals. Compare with Tchebysheff's Theorem and the Empirical Rule. Note that, although the proportion that falls into the interval

x s

does not agree closely with the Empirical Rule, the proportions that fall into the intervals

x 2s

and

x 3s

agree very well. Many times this is true, even for non-mound-shaped distributions of data. (Round your Tchebysheff values to the nearest integer.)

	Actual Percentage	Tchebysheff	Empirical
x s	%	at least 0%	%
x 2s	%	at least %	%
x 3s	%	at least %	99.7%