Calculation:

Empirical rule:

If the distributions of a data set are approximately symmetric or bell shaped, then the standard deviations have the following features:

About 68% of the data falls within one standard deviation of the mean.

About 95% of the data falls within two standard deviations of the mean.

About 99.7% of the data falls within three standard deviations of the mean.

If the data entries lie more than two standard deviations from the mean, then the data entries are considered as unusual.

If the data entries lie more than three standard deviations from the mean, then the data entries are considered as very unusual.

The two standard deviations from the mean are as follows:

\(\displaystyle\overline{{{x}}}\pm{2}{s}={67}\pm{\left({2}\times{4}\right)}\)

\(\displaystyle={67}\pm{8}\)

\(\displaystyle={\left({67}-{8},{67}+{8}\right)}\)

\(\displaystyle={\left({59},{75}\right)}\)

From the data, the values 76,78,82 are above the range (59,75). This indicates that the values 76,78 and 82 are unusual, because these are more than two standard deviations from the mean.

The three standard deviations from the mean are as follows:

\(\displaystyle\overline{{{x}}}\pm{3}{s}={67}\pm{\left({3}\times{4}\right)}\)

\(\displaystyle={67}\pm{12}\)

\(\displaystyle={\left({67}-{12},{67}+{12}\right)}\)

\(\displaystyle={\left({55},{79}\right)}\)

From the data, the value 82 is above the range (55,79). This indicates that the value 82 is very unusual, because it is more than three standard deviations from the mean.

Empirical rule:

If the distributions of a data set are approximately symmetric or bell shaped, then the standard deviations have the following features:

About 68% of the data falls within one standard deviation of the mean.

About 95% of the data falls within two standard deviations of the mean.

About 99.7% of the data falls within three standard deviations of the mean.

If the data entries lie more than two standard deviations from the mean, then the data entries are considered as unusual.

If the data entries lie more than three standard deviations from the mean, then the data entries are considered as very unusual.

The two standard deviations from the mean are as follows:

\(\displaystyle\overline{{{x}}}\pm{2}{s}={67}\pm{\left({2}\times{4}\right)}\)

\(\displaystyle={67}\pm{8}\)

\(\displaystyle={\left({67}-{8},{67}+{8}\right)}\)

\(\displaystyle={\left({59},{75}\right)}\)

From the data, the values 76,78,82 are above the range (59,75). This indicates that the values 76,78 and 82 are unusual, because these are more than two standard deviations from the mean.

The three standard deviations from the mean are as follows:

\(\displaystyle\overline{{{x}}}\pm{3}{s}={67}\pm{\left({3}\times{4}\right)}\)

\(\displaystyle={67}\pm{12}\)

\(\displaystyle={\left({67}-{12},{67}+{12}\right)}\)

\(\displaystyle={\left({55},{79}\right)}\)

From the data, the value 82 is above the range (55,79). This indicates that the value 82 is very unusual, because it is more than three standard deviations from the mean.