Question

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Hurricanes Listed below are the numbers of Atlantic hurricanes that occurred in each year. The data are listed in order by year, starting with the year 2000. What important feature of the data is not revealed by any of the measures of variation?

8 9 8 7 9 15 5 6 8 4 12 7 8 2

Short answer

The measures of variation calculated are as follows:

• The sample range is equal to 13.0 hurricanes.
• The sample variance is equal to 10.2${\text{hurricanes}}^2$ .
• The sample standard deviation is equal to 3.2 hurricanes.

As the sample has values corresponding to a series of times (14 consecutive years), the values of the measures of variation do not show any sign of gradual change in the pattern of the data or detect any trend in the data.

This is an important feature that the above values do not reveal.

Step-by-step solution

Given information

A time-series data showing the number of hurricanes that have occurred starting from the year 2000 is given.

8, 9, 8, 7, 9, 15, 5, 6, 8, 4, 12, 7, 8, 2

The number of hurricanes (n) is 14.

Measures of variation

Measures of variation/dispersion provide a fair idea of how distant a set of values is from its mean.

The sample rangedeals with the upper limit and the lower limit of the values of the sample by evaluating the difference between them.

.$$\begin{gathered} \text{Range}=\text{Maximum}\text{Value}-\text{Minimum}\text{Value} \\ =15-2 \\ =13.0 \end{gathered}$$

Therefore, the range is equal to 13.0.

Thesample variance$\left(s^2\right)$andsample standard deviation$\left(s\right)$ give an estimate of the disparity in the values of the sample.

The mathematical expressions to calculate the quantities are given below:

$$\begin{gathered} s^2=\frac{{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}{n-1} \\ s=\sqrt{s^2} \end{gathered}$$

Here,

x represents the observations in the sample, and

$\overline{x}$is the sample mean.

The sample mean is calculated as

$$\begin{gathered} \overline{x}=\frac{{\sum }_{1=1}^n{x}_i}{n} \\ =\frac{8+9+...+2}{14} \\ =\frac{108}{14} \\ \approx 7.7 \end{gathered}$$

Thus, the sample mean is 7.7 hurricanes.

The variance of the sample is calculated as

$$\begin{gathered} s^2=\frac{{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}{n-1} \\ =\frac{{\left(8-7.7\right)}^2+{\left(9-7.7\right)}^2+...+{\left(2-7.7\right)}^2}{14-1} \\ =\frac{132.86}{13} \\ =10.2 \end{gathered}$$

Therefore, the sample variance is equal to 10.2 ${\text{hurricanes}}^2$.

The sample standard deviation is calculated as

$$\begin{gathered} s=\sqrt{s^2} \\ =\sqrt{10.22} \\ \approx 3.2 \end{gathered}$$

Therefore, the sample standard deviation is equal to 3.2 hurricanes.

Interpretation

Here, the data values are given for a sequence of years from 2000 to 2013. Like every time-series data, this data might also have a certain pattern that it follows.

The values of the measures of variation do not give an idea of any trend or cycle that the number of hurricanes has followed through the years. This is something that these values do not reveal.