# How do you find the standard deviation of 3, 7, 4, 6, and 5?

Dec 19, 2016

The standard deviation is $\sigma = \sqrt{2} \approx 1.41$

#### Explanation:

The standard deviation of a data set is given as:

## $\sigma = \sqrt{\frac{1}{N} \times {\Sigma}_{i = 1}^{N} {\left({x}_{i} - \overline{x}\right)}^{2}}$

where:
$N$- number of data;
$\overline{x}$ - mean value;
${x}_{i}$ - data in data set

To calculate the deviation we can use the following algorythm:

1. Calculate mean $\overline{x}$
2. Calculate ${\left({x}_{i} - \overline{x}\right)}^{2}$ for all $i$
3. Add all values calculated in $2$.
4. Divide the sum by $N$
5. Get square root to calculate the deviation.

Here we have:

1. $\overline{x} = \frac{3 + 4 + 5 + 6 + 7}{5} = \frac{25}{5} = 5$
2. ${\left(3 - 5\right)}^{2} = {\left(- 2\right)}^{2} = 4$
${\left(4 - 5\right)}^{2} = {\left(- 1\right)}^{2} = 1$
${\left(5 - 5\right)}^{2} = 0$
${\left(6 - 5\right)}^{2} = {1}^{2} = 1$
${\left(7 - 5\right)}^{2} = {2}^{2} = 4$
3. Sum is $4 + 1 + 0 + 1 + 4 = 10$
4. $10 \div 5 = 2$
5. $\sigma = \sqrt{2} \approx 1.41$