# What are the variance and standard deviation of {18, -9, -57, 30, 18, 5, 700, 7, 2, 1}?

Jan 13, 2016

Assuming we are dealing with the entire population and not just a sample:
Variance ${\sigma}^{2} = 44 , 383.45$
Standard Deviation $\sigma = 210.6738$

#### Explanation:

Most scientific calculators or spreadsheets will allow you to determine these values directly.

If you need to do it in a more methodical way:

1. Determine the sum of the given data values.
2. Calculate the mean by dividing the sum by the number of data entries.
3. For each data value calculate its deviation from the mean by subtracting the data value from the mean.
4. For each data value's deviation from the mean calculate the squared deviation from the mean by squaring the deviation.
5. Determine the sum of the squared deviations
6. Divide the sum of the squared deviations by the number of original data values to get the population variance
7. Determine the square root of the population variance to get the population standard deviation

If you want the sample variance and sample standard deviation:
in step 6. divide by 1 less than the number of original data values.

Here it is as a detailed spreadsheet image: Note: I would normally simply used the functions
$\textcolor{w h i t e}{\text{XXX}}$VARP(B2:B11)
and
$\textcolor{w h i t e}{\text{XXX}}$STDEVP(B2:B11)

Jan 13, 2016

Variance = 44383.45
Standard deviation$\approx$210.674

#### Explanation:

$\sum X = 18 - 9 - 57 + 30 + 18 + 5 + 700 + 7 + 2 + 1$

$= 715$

$\sum {X}^{2} = {18}^{2} + {9}^{2} + {57}^{2} + {30}^{2} + {18}^{2} + {5}^{2} + {700}^{2} + {7}^{2} + {2}^{2} + {1}^{2} = 494957$

The mean is given by

$\mu = \frac{\sum X}{N} = \frac{715}{10} = 71.5$

The variance is given by

${\sigma}^{2} = \frac{1}{N} \left(\sum {X}^{2} - {\left(\sum X\right)}^{2} / N\right) = 44383.45$

The standard deviation is given by

$\sigma \approx 210.674$