# How do I calculate the variance and standard deviation of 102, 104, 106, 108, 110?

Sep 28, 2017

Variance $= 8$
Standard Deviation $= \sqrt{8} \approx 2.828$

#### Explanation:

To find the variance, you must find "the average of the squared distances from the mean." What this means is that to find the variance, you individually subtract the mean from each data point, square the result, then add all the answers together and divide them by how many answer there were.

To do this we must first find the mean. We do this by adding all the numbers of the data set together, then dividing them by how many numbers there were, so:
$\frac{102 + 104 + 106 + 108 + 110}{5}$
$= \frac{530}{5}$
$= 106$ is the mean .

Next we find the variance as mentioned above. The process is as follows: Select a value from the data set $\left(102\right)$, subtract the mean from it $\left(102 - 106 = - 4\right)$, then square the result $\left(- {4}^{2} = 16\right)$. We do this for the rest of the numbers, to get
${\left(104 - 106\right)}^{2} = {\left(- 2\right)}^{2} = 4$,
${\left(106 - 106\right)}^{2} = {0}^{2} = 0 ,$
${\left(108 - 106\right)}^{2} - {\left(2\right)}^{2} = 4 ,$
${\left(110 - 106\right)}^{2} = {\left(4\right)}^{2} = 16$.

The last thing to do is to add these results together, then divide them by how many there are, like so:
$\frac{16 + 4 + 0 + 4 + 16}{5}$
$= \frac{40}{5}$
$= 8$, the variance.

To find the standard deviation, we simply take the square root of the variance, so:
$\sqrt{8} \approx 2.828$, therefore the standard deviation is roughly equal to $2.828$.

I hope I helped!