# The following data show the number of hours of sleep attained during a recent night for a sample of 20 workers: 6,5,10,5,6,9,9,5,9,5,8,7,8,6,9,8,9,6,10,8. What is the mean? What is the variance? What is the standard deviation?

Sep 27, 2017

Mean = 7.4
Standard Deviation $\approx 1.715$
Variance = 2.94

#### Explanation:

The mean is the sum of all the data points divided by number of data points. In this case, we have
$\frac{5 + 5 + 5 + 5 + 6 + 6 + 6 + 6 + 7 + 8 + 8 + 8 + 8 + 9 + 9 + 9 + 9 + 9 + 10 + 10}{20}$
$= \frac{148}{20}$

$= 7.4$

The variance is "the average of the squared distances from the mean." http://www.mathsisfun.com/data/standard-deviation.html
What this means is you subtract every data point from the mean, square the answers, then add them all together and divide them by the number of data points. In this question, it looks like this:
$4 \left(5 - 7.4\right)$
$= 4 {\left(- 2.4\right)}^{2}$
$= 4 \left(5.76\right)$
$= 23.04$

We add a 4 in front of the brackets because there are four 5's in this data set. We then do this to the rest of the numbers:
$4 {\left(6 - 7.4\right)}^{2} = 7.84$
$1 {\left(7 - 7.4\right)}^{2} = 0.16$
$4 {\left(8 - 7.4\right)}^{2} = 1.44$
$5 {\left(9 - 7.4\right)}^{2} = 12.8$
$2 {\left(10 - 7.4\right)}^{2} = 13.52$

The last step is to add them all together and then divide them by how many there are, which looks like this:
$\frac{23.04 + 7.84 + 0.16 + 1.44 + 12.8 + 13.52}{20}$
$= \frac{58.8}{20}$

$= 2.94$, therefore the variation is 2.94

The standard deviation is easy, it is simply the square root of the variation, which is
$\sqrt{2.94} \approx 1.715$.