Question

Consider a sample with data values of 8, 10, 7, 19, and 16. Compute the variance. (to 1 decimal) Compute the standard deviation. (to 2 decimals)

Answer 1

Answer 2

Step 1: Calculate the Mean (Average)

First, find the mean of the data set by adding all the values and dividing by the number of data points.

$$\text{Mean}=\frac{8+10+7+19+16}{5}=\frac{60}{5}=12$$

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Step 2: Compute the Squared Differences from the Mean

Subtract the mean from each data value and square the result.

$$\begin{array}{rl}{\displaystyle (8-12{)}^2}& {\displaystyle =(-4{)}^2=16}\\ {\displaystyle (10-12{)}^2}& {\displaystyle =(-2{)}^2=4}\\ {\displaystyle (7-12{)}^2}& {\displaystyle =(-5{)}^2=25}\\ {\displaystyle (19-12{)}^2}& {\displaystyle =7^2=49}\\ {\displaystyle (16-12{)}^2}& {\displaystyle =4^2=16}\end{array}$$

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Step 3: Calculate the Variance

Add the squared differences and divide by the number of data points minus 1 (since this is a sample).

$$\text{Variance}=\frac{16+4+25+49+16}{5-1}=\frac{110}{4}=27.5$$

Variance (to 1 decimal): 27.5

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Step 4: Compute the Standard Deviation

Take the square root of the variance to find the standard deviation.

$$\text{Standard Deviation}=\sqrt{27.5}\approx 5.244$$

Standard Deviation (to 2 decimals): 5.24

Answer is :-

• Variance: 27.5

• Standard Deviation: 5.24