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 (round to one decimal place as needed.) the standard deviation of the sample data is 6.1 hurricanes. (round to one decimal place as needed.) the variance of the sample data is 37.2 hurricanes². (round to one decimal place as needed.) what important feature of the data is not revealed through the different measures of variation? a. the measures of variation reveal no information about the scale of the data. b. the measures of variation do not reveal the difference between the largest number of hurricanes and the smallest number of hurricanes in the data. c. the measures of variation reveal nothing about the pattern over time. d. the measures of variation reveal nothing about how the numbers of hurricanes are spread.

Explanation:

Step1: Find the range

Range = Max - Min. The maximum value in the data set is 20 and the minimum is 1. So, Range = 20 - 1 = 19 hurricanes.

Step2: Recall the formula for sample variance $s^{2}$

The formula for the sample variance of a data - set $x_1,x_2,\cdots,x_n$ is $s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$, where $\bar{x}$ is the sample mean and $n$ is the sample size. First, find the sample mean $\bar{x}=\frac{20 + 18+14 + 3+13+18+9+12+19+19+12+20+1+17}{14}=\frac{195}{14}\approx13.9$. Then calculate $(x_i-\bar{x})^2$ for each $x_i$, sum them up and divide by $n - 1=13$. After calculation, $s^{2}\approx37.2$ hurricanes$^{2}$.

Step3: Recall the formula for sample standard deviation $s$

The sample standard deviation $s=\sqrt{s^{2}}$. Since $s^{2}\approx37.2$, then $s=\sqrt{37.2}\approx6.1$ hurricanes.

Step4: Analyze the features not revealed by measures of variation

The range, variance, and standard - deviation only measure the spread of the data values. They do not take into account the order of the data over time. For example, they cannot tell us if there is a trend (increasing or decreasing) in the number of hurricanes over the years.

Answer:

The range of the sample data is 19 hurricanes.
The variance of the sample data is 37.2 hurricanes$^{2}$.
The standard deviation of the sample data is 6.1 hurricanes.
C. The measures of variation reveal nothing about the pattern over time.