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? 20 18 14 3 13 18 9 12 19 19 12 20 1 17 the range of the sample data is (round to one decimal place as needed.)

Explanation:

Step1: Find the maximum and minimum values

The data set is \(20,18,14,3,13,18,9,12,19,19,12,20,1,17\). The maximum value \(x_{max}=20\) and the minimum value \(x_{min}=1\).

Step2: Calculate the range

The formula for the range \(R\) of a data - set is \(R = x_{max}-x_{min}\). Substituting the values, we get \(R=20 - 1=19\).

Step3: Calculate the mean \(\bar{x}\)

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 14\) and \(\sum_{i=1}^{14}x_{i}=20 + 18+14 + 3+13+18+9+12+19+19+12+20+1+17=195\). So, \(\bar{x}=\frac{195}{14}\approx13.9286\).

Step4: Calculate the squared - differences \((x_{i}-\bar{x})^2\)

For example, when \(x_1 = 20\), \((x_1-\bar{x})^2=(20 - 13.9286)^2=(6.0714)^2 = 36.861\). Do this for all \(x_i\) values and sum them up: \(\sum_{i = 1}^{14}(x_{i}-\bar{x})^2=(20 - 13.9286)^2+(18 - 13.9286)^2+\cdots+(17 - 13.9286)^2=379.2143\).

Step5: Calculate the variance \(s^{2}\)

The formula for the sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}{n - 1}\). Substituting \(n = 14\) and \(\sum_{i = 1}^{14}(x_{i}-\bar{x})^2=379.2143\), we get \(s^{2}=\frac{379.2143}{13}\approx29.1703\) (in number of hurricanes\(^2\)).

Step6: Calculate the standard deviation \(s\)

The standard deviation \(s=\sqrt{s^{2}}\). So, \(s=\sqrt{29.1703}\approx5.4\) (number of hurricanes).

The important feature not revealed by the measures of variation is the pattern of data over time (e.g., trends, seasonality).

Answer:

The range of the sample data is \(19.0\) number of hurricanes.
The variance of the sample data is approximately \(29.2\) number of hurricanes\(^2\).
The standard deviation of the sample data is approximately \(5.4\) number of hurricanes.
The important feature not revealed is the pattern of data over time.