Question

What is the standard deviation of the data set? 30, 30, 42, 18, 42, 24

Answer 1

The data set given are: 30, 30, 42, 18, 42, 24

The formula for the mean of the data set is:

`\begin{gathered} \bar{x}=(\Sigma x)/(n) \\ x\text{ represents the data} \\ n\text{ (the number of data) = 6} \end{gathered}`

The formula for standard deviation (S.D) is:

`\begin{gathered} S.D\text{ = }\sqrt[]{\frac{\Sigma|x-\bar{x}|^2}{n}} \\ \text{ where, }\bar{x}=\text{ mean, n= number of data} \end{gathered}`

Step 1: Find the mean

`\begin{gathered} \bar{x}=(30+30+42+18+42+24)/(6) \\ \bar{x}=(186)/(6)=31 \end{gathered}`

step 2: Find the summation of the absolute value deviation of the data from the mean

`\begin{gathered} \Sigma|x-\bar{x}|^2=|(30-31)^2+(30-31)^2+(42-31)^2+(18-31)^2+ \\ (42-31)^2+(24-31)^2| \end{gathered}`
`\begin{gathered} \Sigma|x-\bar{x}|^2=1+1+11^2+13^2+11^2+7^2 \\ =1+1+121+169+121+49 \\ =462 \end{gathered}`

step 3: Substitute the value obtained in step 2 into the standard deviation formula

`\begin{gathered} S.D\text{ = }\sqrt[]{\frac{\Sigma|x-\bar{x}|^2}{n}}=\sqrt[]{(462)/(6)}=\sqrt[]{77} \\ =8.775 \end{gathered}`

Therefore, the standard deviation of the data set