Answer:
The variance is [tex]var = 1197.7[/tex].
The standard deviation is [tex]\sigma = 34.6[/tex].
Step-by-step explanation:
We have the following data set:
[tex]\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}[/tex]
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:
[tex]var = \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1}[/tex]
Step 1: Find the mean [tex]\left( \overline{X} \right)[/tex].
The mean of a data set is the sum of the terms divided by the total number of terms.
[tex]\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}[/tex]
Step 2: Create the below table.
Step 3: Find the sum of numbers in the last column.
[tex]\sum{\left(x_i - \overline{X}\right)^2} = 22755.75[/tex]
Step 4: Calculate variance using the above formula.
[tex]var = \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1} = \frac{ 22755.75 }{ 20 - 1 } \approx 1197.7[/tex]
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
[tex]\sigma = \sqrt{ \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1} }[/tex]
[tex]\sigma = \sqrt{ \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n-1} } = \sqrt{ \frac{ 22755.75 }{ 20 - 1} } \approx 34.6[/tex]
