The **Standard Deviation Calculator** can be used to calculate both the *Population Standard Deviation*, as well as the *Sample Standard Deviation*. Each value should be separated by a comma. Example is included below.

## How to use the Standard Deviation Calculator

Let's review a simple example to demonstrate how to use the Standard Deviation Calculator, where:

- The data points to be entered into the calculator are:
**2,7,15,4,8**

To get the Standard Deviation, simply type/copy the above data points into the entry box, and then click on the **Calculate Standard Deviation **button:

You'll then get the the Population Standard Deviation of **4.445**, and the Sample Standard Deviation of **4.970**:

## How to Manually Calculate the Standard Deviation

You can use the following formula to calculate the **Population Standard Deviation**:

[Σ(xi - μ)^{2}]^{0.5}

Population Standard Deviation (σ) = (N)^{0.5}

Where:

- μ = Mean (average of all data points)
- xi = Value of each data point
- N = Total number of data points

Alternatively, you may use this formula to get the **Sample Standard Deviation**:

` [ Σ(xi - x̅)`^{2 }]^{0.5}

Sample Standard Deviation (s) = (n-1)^{0.5}

Where:

- x̅ = Sample Mean (average of all data points)
- xi = Value of each data point
- n = Sample size

## Calculation Example

Suppose that you have the following data points: 2,7,15,4,8.

Your goal is to calculate the:

- Population Standard Deviation (σ); and
- Sample Standard Deviation (s)

### Calculate the Population Standard Deviation

- μ = (2+7+15+4+8) / 5 =
**7.2** - Σ(xi - μ)
^{2}= (2-7.2)^{2}+ (7-7.2)^{2}+ (15-7.2)^{2}+ (4-7.2)^{2}+ (8-7.2)^{2}=**98.8** - N =
**5**

[Σ(xi - μ)^{2}]^{0.5}[98.8]^{0.5}Population Standard Deviation (σ) = (n)^{0.5}= (5)^{0.5}= 4.445

You'll then get the Population Standard Deviation of **4.445**.

### Calculate the Sample Standard Deviation

- x̅ = (2+7+15+4+8) / 5 =
**7.2** - Σ(xi - x̅)
^{2}= (2-7.2)^{2}+ (7-7.2)^{2}+ (15-7.2)^{2}+ (4-7.2)^{2}+ (8-7.2)^{2}=**98.8** - n =
**5**

[ Σ(xi - x̅)^{2 }]^{0.5}[98.8]^{0.5}Sample Standard Deviation (s) = (n-1)^{0.5}= (5-1)^{0.5}= 4.970

You'll now get the Sample Standard Deviation of **4.970**.