Question

Watch help videoFor the following set of data, find the number of data within 2population standard deviations of the mean.54, 103, 62, 102, 60, 60, 54, 58

Answer 1

To find the mean, add the given data and then divide the sum by the number of data.

`\begin{gathered} \bar{x}=(54+103+62+102+60+60+54+58)/(8) \\ \bar{x}=(553)/(8) \\ \bar{x}=69.125 \end{gathered}`

Find the standard deviation by

(1) Subtracting the mean from each data.

(2) Square each difference.

(3) Find the mean of the squares.

(4) Determine the square root of the mean of the squares.

Thus, the standard deviation is as follows.

`\begin{gathered} \sigma=\sqrt[]{((54-69.125)^2+(103-69.125)^2+\cdots+(54-69.125)^2+(58-69.125)^2)/(8)} \\ \sigma=\sqrt[]{((-15.125)^2+(33.875)^2+\cdots+(-15.125)^2+(-11.125)^2)/(8)} \\ \sigma\approx\sqrt[]{(228.765625+1147.515625+\cdots+228.765625+123.765625)/(8)} \\ \sigma\approx\sqrt[]{(3026.875)/(8)} \\ \sigma\approx19.45 \end{gathered}`

Therefore, to obtain the number of data within 2 standard deviations, add and subtract the mean by the standard deviation. Thus, we obtain the following.

`\begin{gathered} \bar{x}+2\sigma=69.125+2(19.45) \\ =69.125+38.9 \\ =108.025 \end{gathered}`
`\begin{gathered} \bar{x}-2\sigma=69.125-2(19.45) \\ =69.125-38.9 \\ =30.225 \end{gathered}`

Since the range would be from 30.225 to 108.025, we can say that all of the given data are within 2 standard deviations of the mean. Thus, there are 8 data.