# Standard Deviation Calculator

Our free online Standard Deviation Calculator (σ) in order to calculate not only the Standard Deviation of a given set of numbers, but the variance and mean also.

Please provide numbers separated by comma (e.g: 7,1,8,5), space (e.g: 7 1 8 5) or line break and press the "Calculate" button.

## Standard Deviation

Standard deviation is used when it comes to statistics and probability theory. It is used in order to measure both variability and diversity and will show the precision of your data.

When you calculate standard deviation, you are calculating the square root of its variance. A low standard deviation will indicate that the entered data points are most likely to be closer to the mean, compared to if there is a high standard deviation which is an indication that the entered data points are most likely further away from the mean.

Calculating both the variance and the standard deviation depends on not only the set of numbers you are using to calculate them but whether the set of numbers is a population or a sample.

## Variance and Standard Deviation of a Population

When calculating the standard deviation of a population you are calculating the variation of data within a population. The standard deviation is usually an unknown constant. Standard deviation (σ) and Variance (σ^{2}) of the population are given as:

- Our Standard Deviation Calculator is appropriate for carrying out any mathematical calculations that consist of the formulae and algorithms found below.

- σ = Standard Deviation of the Population
- σ
^{2}= Variance of the Population - x
_{1}, ..., x_{N}= the Population Data Set - μ = Mean of the Population Data Set
- N = Size of the Population Data Set

## The Variance and Standard Deviation of a Sample

When calculating the standard deviation of a sample, you are calculating an estimate of the standard deviation of a population. Variance (s^{2}) and the Standard Deviation (s) of the sample are calculated using the following formulae.

- s = Standard Deviation of the Sample
- s
^{2}= Variance of the Sample - x
_{1}, ..., x_{N}= the Sample Data Set - x̄ = Mean of the Sample Data Set
- N = Size of the Sample Data Set

## The Coefficient of the Variation

When standard deviation is expressed as a percentage of the mean value, it is known as the Coefficient of Variation.

In order to calculate the Coefficient of Variation, you use the following formula:

*Coefficient of Variation = Standard Deviation (σ) / Mean Value (μ)*

In order to calculate the mean, you use the following formula:

The formula found above will depict the Coefficient of Variation's meaning which can then be applied to not only the population, but the sample of a certain distribution. You can also express the coefficient of variation as a percentage by multiplying your result from the above formula by 100. This is a useful way to measure variability when multiple groups are given and are in different measurement units.

## Standard Deviation Calculation Examples

Consider the set X of numbers 1, 2, 3, 4, 5

**Step 1 :**

- Mean = Sum of X values / N (Number of Values)
- = (1 + 2 + 3 + 4 + 5) / 5
- = 15 / 5
- = 3

**Step 2 :**

- In order to find the variance,
- Subtract the mean from each of the values,
- 1 - 3 = -2
- 2 - 3 = -1
- 3 - 3 = 0
- 4 - 3 = 1
- 5 - 3 = 2
- Now square all of the answers that you had gotten from subtraction.
- (-2)² = 4
- (-1)² = 1
- (0)² = 0
- (1)² = 1
- (2)² = 4
- Add all of the Squared numbers,
- 4 + 1 + 0 + 1 + 4 = 10
- Divide the sum of squares by (n-1)
- 10 / (5 - 1) = 10 / 4 = 2.5
- Therefore, Variance = 2.5

**Step 3 :**

- To find the standard deviation, find the square root of variance,
- √2.5 = 1.581
- Therefore, Standard deviation is 1.581
- To find minimum and maximum standard deviation,
- Minimum SD = Mean − SD
- = 3 - 1.581
- = 1.419
- Maximum SD = Mean + SD
- =3 + 1.581
- = 4.581

**Step 4 :**

- To find the population standard deviation,
- Divide the sum of squares found in step 2 by n
- 10 / 5 = 2
- Find the square root of 2, √2 = 1.414