# Question #d4b35

Jun 10, 2017

c), the variance is 8.24

#### Explanation:

The variance of a set of data, ${\sigma}^{2}$, is calculated by the expression:

${\sigma}^{2} = \frac{\sum {x}^{2}}{n} - {\left(\frac{\sum x}{n}\right)}^{2}$

Where $x$ represents the data and $n$ represents the number of items of data. So for the data set: 6, 8, 1, 9, 4:

${\sigma}^{2} = \frac{{6}^{2} + {8}^{2} + {1}^{2} + {9}^{2} + {4}^{2}}{5} - {\left(\frac{6 + 8 + 1 + 9 + 4}{5}\right)}^{2}$
$= \frac{198}{5} - {\left(\frac{28}{5}\right)}^{2} = \frac{206}{25} = 8.24$.

Jun 11, 2017

C) $8.2$

#### Explanation:

Manual calculation:

Normally, to calculate standard deviation, you find the mean of the data set, find the sum square of the difference between every data, and find the mean of the sum. Then you would square root it again to get the standard deviation. However, since we're finding the variance, we don't need to square root as the last step.

First, find the mean by adding all the values together and dividing by the number of values in the data set:

$\frac{6 + 8 + 1 + 9 + 4}{5} = 5.6$

Now find the square of the differences:

${\left(6 - 5.6\right)}^{2} = .16$
${\left(8 - 5.6\right)}^{2} = 5.76$
${\left(1 - 5.6\right)}^{2} = 21.16$
${\left(9 - 5.6\right)}^{2} = 11.56$
${\left(4 - 5.6\right)}^{2} = 2.56$

Find the mean of these values:

$\frac{.16 + 5.76 + 21.16 + 11.56 + 2.56}{5} = 8.24$

Graphing calculator:

Go to stat -> edit, and plug in your numbers. Then go to stat, calc, and go to 1-var stats. Plug in your list and take note of $\sigma x$.

In our case, $\sigma x \approx 2.87$

That is the standard deviation. To convert this to a variance, we square it:

${2.87}^{2} = 8.2369$

It's not as accurate because we used two decimal places for $\sigma x$ but it is close to $8.2$ so we take that answer.