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lys-0071 [83]
3 years ago
10

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).

Mathematics
1 answer:
zhenek [66]3 years ago
6 0

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

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Answer:

Step-by-step explanation:

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Hi,

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