The standard deviation formula that you will use to find the standard deviation (SD) is shown below.

- x represents a set of numbers. For example, x could be {5, 6, 14, 1, 6, 10}.

- The mean is the average of the set of numbers.

- The symbol Σ refers to an addition.

- n is the size of the list or set. For example, for the set above or {5, 6, 14, 1, 6, 10}, n = 6.

- σ is the symbol or the Greek letter sigma used for standard deviation.

In the formula, the expression inside the square root is called variance. Therefore, the standard deviation is the square root of the variance.

If you have collected information or data set from every single member of the population, then you will end calculating the population standard deviation using the formula below.

$$ Population \ standard \ deviation = σ = \sqrt {\frac{Σ(x - µ)^{2}} {N} } $$- x represents all the numbers in the population.

- µ, the Greek letter mu, is the population mean or the average of all numbers in the population.

- The symbol Σ refers to an addition.

- N is the size of the population.

- σ is the symbol used for standard deviation.

If you have collected information or dataset from some members of the population, then you will end calculating the sample standard deviation using the formula below.

$$ Sample \ standard \ deviation = s = \sqrt {\frac{Σ(x - \bar x)^{2}} {n - 1} } $$- x represents all the numbers in the sample.

- x̄ is the sample mean or the average of all numbers in the sample.

- The symbol Σ refers to an addition.

- n - 1 is the size of the sample.

- s is the symbol used for standard deviation.

Notice that the denominator of the variance is **n - 1** instead of N in the formula for sample standard deviation. This is an adjustment called Bessel's correction. This division by **n - 1** ensures that the sample standard deviation is a good **estimate** for the population standard deviation. It has been observed by statisticians that when we divide by n, it does not do a good job estimating the population standard deviation. You should use Bessel's correction only if you do not know the population mean and this is usually the case.

Σ(x - mean)^{2}
n

Using the standard deviation formula, I will now show you how to get the standard deviation step by step.

Let S = {4, 6, 8, 2, 5}

mean =

4 + 6 + 8 + 2 + 5
5

mean =

25
5

= 5
4 - 5 = -1

6 - 5 = 1

8 - 5 = 3

2 - 5 = -3

5 - 5 = 0

-1

1

3

-3

0

1 + 1 + 9 + 9 + 0
5

20
5

= 4
Standard deviation = σ =
√4
= 2

Using the short-cut formulas below to find the standard deviation is easier and it will reduce the computation time.

$$ Sample \ standard \ deviation = s = \sqrt {\frac{Σx^2 - \frac{(Σx)^2}{n} } {n - 1} } $$

Using the same dataset S = {4, 6, 8, 2, 5} and the short-cut formula, find SD.

Σx^{2} = 4^{2} + 6^{2} + 8^{2} + 2^{2} + 5^{2}

Σx^{2} = 16 + 36 + 64 + 4 + 25

**Σx ^{2} = 145**

Σx = 4 + 6 + 8 + 2 + 5

Σx = 25

(Σx)^{2} = 25^{2}

**(Σx) ^{2} = 625**