Question

Lightning Deaths Listed below are the numbers of deaths from lightning strikes in the United States each year for a sequence of 14 recent and consecutive years. Find the values of the indicated statistics.

51 44 51 43 32 38 48 45 27 34 29 26 28 23

a. Mean

b. Median

c. Standard deviation

d. Variance

e. Range

f. What important feature of the data is not revealed from an examination of the statistics, and what tool would be helpful in revealing it?

Short answer

a. Mean: 37.1 deaths

b. Median: 36.0 deaths

c. Standard deviation: 9.8 deaths

d. Variance: 96.8 deaths

e. Range: 28.0 deaths

f. The change of values over the year is not revealed by the above values of statistics.

A time-series plot would correctly depict the change in the number of deaths due to lightning strikes over the years.

Step-by-step solution

Given information

Data are given on the number of deaths that have occurred due to lightning strikes for a series of 14 consecutive years.

Mean

a.

Let x be the number of deaths due to lightning strikes.

The mean number of deaths that have occurred due to lightning strikes is computed:

$$\begin{gathered} \overline{x}=\frac{51+44+........+23}{14} \\ =37.1 \end{gathered}$$

Thus, the mean value is equal to 37.1 deaths.

Median

b.

The data is arranged in ascending order as follows:

23	26	27	28	29	32	34
38	43	44	45	48	51	51

As the number of data values present is even (14), the following formula is used to compute the median value:

$$\begin{gathered} \text{Median}=\frac{{\left(\frac{n}{2}\right)}^{th}obs+{\left(\frac{n}{2}+1\right)}^{th}obs}{2} \\ =\frac{{\left(\frac{14}{2}\right)}^{th}obs+{\left(\frac{14}{2}+1\right)}^{th}obs}{2} \\ =\frac{7^{th}obs+8^{th}obs}{2} \\ =\frac{34+38}{2} \\ =36 \end{gathered}$$

Therefore, the median value is equal to 36.0 deaths.

Standard deviation

c.

The sample standard deviation of the number of deaths is computed below:

$$\begin{gathered} s=\sqrt{\frac{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}{n-1}} \\ =\sqrt{\frac{{\left(51-37.1\right)}^2+{\left(44-37.1\right)}^2+.......+{\left(23-37.1\right)}^2}{14-1}} \\ =9.8 \end{gathered}$$

Therefore, the value of the standard deviation is equal to 9.8 deaths.

Variance

d.

The sample variance of the number of deaths is computed below:

$$\begin{gathered} s^2=\frac{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}{n-1} \\ =\frac{{\left(51-37.2\right)}^2+{\left(44-37.2\right)}^2+.......+{\left(23-37.2\right)}^2}{14-1} \\ =96.8 \end{gathered}$$

Therefore, the sample variance is equal to 96.8 deaths.

Range

e.

The range of the number of deaths is computed below:

$$\begin{gathered} \text{Range}=\text{Maximum}\text{value}-\text{Minimum}\text{value} \\ =51-23 \\ =28.0 \end{gathered}$$

Therefore, the range is equal to 28.0 deaths.

Missing information

f.

The gradual change in the number of deaths due to lightning strikes over the years is not depicted by the computed statistics.

A time-series graph showing the trend of the values over 14 years would be an appropriate tool.