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# Standard Deviation Calculator

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Standard deviation estimates the distribution of a data set from its average. Standard deviation is calculated by taking the square root of the variance i.e., $\sigma ^{2}$. Plug in the set of numbers into the standard deviation calculator to get the standard deviation value along with its mean and variance value.

## Steps

Step 1 : Read the given problem and note the given set of numbers.

Step 2 : First calculate the mean($\mu$) of the given set of number.

Step 3 : Subtract each term by their mean and square the result. Then, get the sum of the terms.

Step 4 : Determine the standard deviation by the formula given,

$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2}$

where,
$\sigma$ = standard deviation,
$x_{i}$ = each set of data,
$\mu$ = mean of the data set,
N = number of data values.

## Problems

Based on standard deviation, some of the problems are given below.

### Solved Examples

Question 1: Determine the standard deviation of the given set :
2, 4, 7, 9, 5, 12.
Solution:

Step 1 : Given set of numbers : 2, 4, 7, 9, 5, 12.

Step 2 : Mean ($\mu$) =  $\frac{2 + 4 + 7 + 9 + 5 + 12}{6}$

Mean ($\mu$) = $\frac{39}{6}$ = 6.5

Step 3 : Finding sunm of ((x_i-\mu )^2).
$(2 - 6.5)^2$ = 20.25;
$(4 - 6.5)^2$ = 6.25;
$(7 - 6.5)^2$ = 0.25;
$(9 - 6.5)^2$ = 6.25;
$(5 - 6.5)^2$ = 2.25;
$(12 - 6.5)^2$ = 30.25;

$\sum_{i=1}^{6}(x_i-\mu )^2$ = 20.25 + 6.25 + 0.25 + 6.25 + 2.25 + 30.25 = 65.5.

Step 4 : $\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2}$

$\sigma = \sqrt{\frac{1}{6}(65.5)}$

$\sigma = \sqrt{10.91}$

$\sigma$ = 3.30

Therefore, the standard deviation of the given set is 3.30.

Question 2: Determine the standard deviation of the given set :
6, 5, 4, 3, 2, 1.
Solution:

Step 1 : Given set of numbers : 6, 5, 4, 3, 2, 1.

Step 2 : Mean ($\mu$) =  $\frac{6 + 5 + 4 + 3 + 2 + 1}{6}$

Mean ($\mu$) = $\frac{39}{6}$ = 3.5

Step 3 : Finding sum of ((x_i-\mu )^2).
$(6 - 3.5)^2$ = 6.25;
$(5 - 3.5)^2$ = 2.25;
$(4 - 3.5)^2$ = 0.25;
$(3 - 3.5)^2$ = 0.25;
$(2 - 3.5)^2$ = 2.25;
$(1 - 3.5)^2$ = 6.25;

$\sum_{i=1}^{6}(x_i-\mu )^2$ = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 =  17.5

Step 4 : $\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2}$

$\sigma = \sqrt{\frac{1}{6}(17.5)}$

$\sigma = \sqrt{2.91}$

$\sigma$ = 1.70

Therefore, the standard deviation of the given set is 1.70.