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?
13 1 8 7 2 10 10
10 20 20 8 17 17 4
the range of the sample data is 19.0
(round to one decimal place as needer)

Explanation:

Step1: Calculate the mean

First, find the sum of the data values: \(13 + 1+8 + 7+2 + 10+10+10+20+20+8+17+17+4=157\).
There are \(n = 14\) data values.
The mean \(\bar{x}=\frac{157}{14}\approx11.214\).

Step2: Calculate the variance

Use the formula \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\).
\((13-11.214)^{2}=3.183\), \((1 - 11.214)^{2}=104.326\), \((8-11.214)^{2}=10.326\), \((7-11.214)^{2}=17.758\), \((2-11.214)^{2}=84.878\), \((10-11.214)^{2}=1.474\) (three times for the three \(10\)s), \((20-11.214)^{2}=77.194\) (two times for the two \(20\)s), \((8-11.214)^{2}=10.326\), \((17-11.214)^{2}=33.474\) (two times for the two \(17\)s), \((4-11.214)^{2}=52.042\).
Sum of \((x_{i}-\bar{x})^{2}=3.183+104.326 + 10.326+17.758+84.878+1.474\times3+77.194\times2+10.326+33.474\times2+52.042\)
\(=3.183+104.326+10.326+17.758+84.878 + 4.422+154.388+10.326+66.948+52.042\)
\(=508.699\)
Variance \(s^{2}=\frac{508.699}{14 - 1}=\frac{508.699}{13}\approx39.131\) (units: \(hurricanes^{2}\))

Step3: Calculate the standard deviation

Standard deviation \(s=\sqrt{s^{2}}=\sqrt{39.131}\approx6.3\) (units: \(hurricanes\))

Answer:

The range of the sample data is \(19.0\) \(hurricanes\).
The variance of the sample data is approximately \(39.1\) \(hurricanes^{2}\).
The standard deviation of the sample data is approximately \(6.3\) \(hurricanes\).
An important feature not revealed by the measures of variation is the pattern of the data over time (since the data is ordered by year).