Question

The following data are the distances between a sample of 20 retail stores and a large distribution center. The distances are in miles.
29; 35; 39; 40; 58; 67; 68; 69; 76; 80; 88; 95; 96; 96; 99; 106; 112; 127; 145; 150
Find the variance and standard deviation (round each to the nearest tenth).

Answer 1

Answer:

The variance is $var = 1197.7$.

The standard deviation is $\sigma = 34.6$.

Step-by-step explanation:

We have the following data set:

$\begin{array}{cccccccc}29&35&39&40&58&67&68&69\\76&80&88&95&96&96&99&106\\112&127&145&150&&&&\end{array}$

The variance measures how far a data set is spread out. It is mathematically defined as the average of the squared differences from the mean.

To find variance we use the following formula:

$var = \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1}$

Step 1: Find the mean $\left( \overline{X} \right)$.

The mean of a data set is the sum of the terms divided by the total number of terms.

$\begin{aligned}Sum ~ of ~ terms~&=~29~+~35~+~39~+~40~+~58~+~\cdots +~150~=~1675\\ Number ~ of ~ terms &= 20 \\ \overline{X} & = \frac{Sum ~ of ~ terms}{Number ~ of ~ terms} \\ \overline{X} & = \frac{ 1675 }{ 20 } \\ \overline{X} &= \frac{ 335 }{ 4 } =83.75 \end{aligned}$

Step 2: Create the below table.

Step 3: Find the sum of numbers in the last column.

$\sum{\left(x_i - \overline{X}\right)^2} = 22755.75$

Step 4: Calculate variance using the above formula.

$var = \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1} = \frac{ 22755.75 }{ 20 - 1 } \approx 1197.7$

The standard deviation measures how close the set of data is to the mean value of the data set.

To find standard deviation we use the following formula

$\sigma = \sqrt{ \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1} }$

$\sigma = \sqrt{ \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1} } = \sqrt{ \frac{ 22755.75 }{ 20 - 1} } \approx 34.6$