Question

listed below are the numbers of hurricanes that occurred in each year in a certain region. the data are listed in order by year. find the range, variance, and standard deviation for the given sample data. include appropriate units in the results. what important feature of the data is not revealed by any of the measures of variation?
15 18 18 1 20 14 3
6 7 6 2 7 6 17
the range of the sample data is 19.0 hurricanes.
(round to one decimal place as needed.)
the standard deviation of the sample data is 6.7 hurricanes
(round to one decimal place as needed.)
the variance of the sample data is □
(round to one decimal place as needed.)

Explanation:

Step1: Recall the formula for variance

The variance \(s^{2}\) of a sample is given by \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), where \(n\) is the sample size, \(x_{i}\) are the data points, and \(\bar{x}\) is the sample mean.
First, find the sample mean \(\bar{x}\).
The data set is \(x=\{15,18,18,1,20,14,3,6,7,6,2,7,17\}\).
\(n = 13\)
\(\bar{x}=\frac{15 + 18+18+1+20+14+3+6+7+6+2+7+17}{13}=\frac{154}{13}\approx11.8\)

Step2: Calculate \((x_{i}-\bar{x})^{2}\) for each \(x_{i}\)

\((15 - 11.8)^{2}=(3.2)^{2}=10.24\)
\((18 - 11.8)^{2}=(6.2)^{2}=38.44\) (twice)
\((1 - 11.8)^{2}=(- 10.8)^{2}=116.64\)
\((20 - 11.8)^{2}=(8.2)^{2}=67.24\)
\((14 - 11.8)^{2}=(2.2)^{2}=4.84\)
\((3 - 11.8)^{2}=(-8.8)^{2}=77.44\)
\((6 - 11.8)^{2}=(-5.8)^{2}=33.64\) (twice)
\((2 - 11.8)^{2}=(-9.8)^{2}=96.04\)
\((7 - 11.8)^{2}=(-4.8)^{2}=23.04\) (twice)
\((17 - 11.8)^{2}=(5.2)^{2}=27.04\)

Step3: Sum up \((x_{i}-\bar{x})^{2}\)

\(\sum_{i = 1}^{13}(x_{i}-\bar{x})^{2}=10.24+38.44\times2 + 116.64+67.24+4.84+77.44+33.64\times2+96.04+23.04\times2+27.04\)
\(=10.24 + 76.88+116.64+67.24+4.84+77.44+67.28+96.04+46.08+27.04\)
\(=690.72\)

Step4: Calculate the variance

\(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}=\frac{690.72}{13 - 1}=\frac{690.72}{12}=57.6\)

Answer:

The variance of the sample data is \(57.6\) hurricanes².