Question

the graph shows the distribution of hand lengths, in centimeters, of the members of an orchestra. what is the variance of the data? 1 2 4 5 hand lengths (cm) 14 15 16 17 18 19 20 21 22 23 24 x

Explanation:

1. Recall the property of a normal - distribution graph:

• For a normal - distribution (bell - shaped curve), if the data is approximately normally distributed, and we assume that the empirical rule (68 - 95 - 99.7 rule) applies. The empirical rule states that about 68% of the data lies within 1 standard deviation (\(\sigma\)) of the mean, about 95% of the data lies within 2 standard deviations of the mean, and about 99.7% of the data lies within 3 standard deviations of the mean.
• If we assume that the interval from the peak of the normal curve to the inflection point (where the curve changes from concave - up to concave - down) represents 1 standard deviation. Looking at the graph, if we assume that the mean is around \(x = 19\) and the inflection points are around \(x=17\) and \(x = 21\).
• The distance from the mean to the inflection point gives the standard deviation. So, \(\sigma=\vert19 - 17\vert=\vert21 - 19\vert = 2\).

1. Recall the formula for variance:

• The variance (\(\sigma^{2}\)) is related to the standard deviation (\(\sigma\)) by the formula \(\sigma^{2}=\sigma\times\sigma\).
• Since \(\sigma = 2\), then the variance \(\sigma^{2}=2\times2=4\).

Answer:

C. 4