Question

Find the variance of the sample of observations \(2,5,7,9,12\).

Short answer

The variance of the sample of observations \(2, 5, 7, 9, 12\) is \(11.6\).

Step-by-step solution

Calculate the mean of the sample.

First, we need to find the mean of the sample. The mean is the average of all the observations. To find the mean, add all the observations and divide the sum by the number of observations. Mean = \(\frac{sum \ of \ observations}{number \ of \ observations}\) The sum of the observations: \(2 + 5 + 7 + 9 + 12 = 35\) Number of observations: \(5\) Mean = \(\frac{35}{5} = 7\)

Find the differences between each observation and the mean.

Now, we need to find the differences between each observation and the mean calculated in Step 1. Difference 1: \(2 - 7 = -5\) Difference 2: \(5 -7 = -2\) Difference 3: \(7- 7 = 0\) Difference 4: \(9 - 7 = 2\) Difference 5: \(12 - 7 = 5\)

Square the differences.

In this step, we need to square each of the differences found in Step 2. Squared Difference 1: \((-5)^2 = 25\) Squared Difference 2: \((-2)^2 = 4\) Squared Difference 3: \((0)^2 = 0\) Squared Difference 4: \((2)^2 = 4\) Squared Difference 5: \((5)^2 = 25\)

Find the mean of the squared differences.

To find the mean of the squared differences, add all the squared differences and divide the sum by the number of squared differences. Mean of Squared Differences = \(\frac{sum \ of \ squared \ differences}{number \ of \ squared \ differences}\) The sum of squared differences: \(25 + 4 + 0 + 4 + 25 = 58\) Number of squared differences: \(5\) Mean of Squared Differences = \(\frac{58}{5} = 11.6\)

The variance is the result obtained in step 4.

Finally, the variance of the sample of observations is the result obtained in step 4, which is 11.6. So, the variance of the sample of observations \(2,5,7,9,12\) is \(11.6\).