Standard deviation calculator

solution

You entered the following data set:

$$\begin{array}{cccc}5.14&5.15&5.17&5.20\\5.20&5.18&5.15&5.15\\5.16&5.16&&\end{array} $$

The standard deviation of the data set is:

$$ \sigma = 0.02 $$

explanation

To find standard deviation we use the following formula

$$ \sigma = \sqrt{ \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n} } $$

We will compute this formula in 4 steps.

Step 1: Find the mean. $ \left( \overline{X} \right)$

In this example: $ \overline{X} = 5.166 $. (use this calculator for step by step explanation on how to find mean).

Step 2: Create the following table.

data	data-mean	(data - mean)2
5.14	-0.026000000000001	0.00067600000000004
5.15	-0.016	0.000256
5.17	0.0039999999999996	1.5999999999996E-5
5.20	0.034	0.001156
5.20	0.034	0.001156
5.18	0.013999999999999	0.00019599999999998
5.15	-0.016	0.000256
5.15	-0.016	0.000256
5.16	-0.0060000000000002	3.6000000000003E-5
5.16	-0.0060000000000002	3.6000000000003E-5

Step 3: Find the sum of numbers in the last column to get.

$$ \sum{\left(x_i - \overline{X}\right)^2} = 0.004 $$

Step 4: Calculate $ \sigma $ using the above formula.

$$ \sigma = \sqrt{ \frac{ \sum{\left(x_i - \overline{X}\right)^2 }}{n} } = \sqrt{ \frac{ 0.004 }{ 10} } \approx 0.02$$