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# Standard Deviation Calculator

Enter Data Set Values (Separated by Comma)

Table of Content

 1 What Is Standard Deviation? 2 Formula Of Standard Deviation: 3 How To Find Standard Deviation (Step-by-Step): 4 Why can variance not be negative? 5 What is the range of the standard deviation? 6 Does standard deviation have units?

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Our standard deviation calculator allows you to find the standard deviation, variance, mean, coefficient of variance, and margin of error for a data set. Provide your data set values and the tool will show the calculations needed.

## What Is Standard Deviation In Statistics?

“It is the square root of the variance of any statistical data set”

## Standard Deviation Formula:

Our standard deviation calculator utilises the following formulas to calculate standard deviation:

### Population Standard Deviation:

$$σ = \sqrt{\dfrac{1}{N} \sum_{i=1}^N\left(x_{i} – μ\right)^2}$$
Where:
xi = Individual Value
μ = Average Mean Value
N = Total Number Of Values

### Sample Standard Deviation:

$$s = \sqrt{\dfrac{1}{N – 1} \sum_{i=1}^N\left(x_{i} – \bar{x}\right)^2}$$
Where:
xi= Each Single Value In the Data Set
x = sample mean
N = total sample size

## How To Find Standard Deviation? (Step-by-Step):

Let’s resolve an example to make your concept broader. Just stay focused!

### Example # 01:

How to find the standard deviation for the following sample data set below:

$$3, 4, 9, 7, 2, 5$$

#### Solution:

Step 1(Calculate Mean Value):

$$\bar{x} = {\dfrac{3 + 4 + 9 + 7 + 2 + 5}{6}}$$

$$\bar{x} = {30}{6}$$

$$\bar{x} = 5$$

Step 2 (Calculate The Value Of $$\left(x_{i} – \bar{x}\right)$$:

$$x_1-\bar{x} = 3 – 5 = -2$$

$$x_2-\bar{x} = 4 – 5 = -1$$

$$x_3-\bar{x} = 9 – 5 = 4$$

$$x_4-\bar{x} = 7 – 5 = 2$$

$$x_5-\bar{x} = 2 – 5 = -3$$

$$x_6-\bar{x} = 5 – 5 = 0$$

Now we have:

$$(x_1-µ)^2 = (-2)^2 = 4$$

$$(x_2-µ)^2 = (-1)^2 = 1$$

$$(x_3-µ)^2 = (-4)^2 = 16$$

$$(x_4-µ)^2 = (2)^2 = 4$$

$$(x_5-µ)^2 = (-3)^2 = 9$$

$$(x_6-µ)^2 = (0)^2 = 0$$

The variance and standard deviation calculator automatically determine all of these values without creating any difficulty for you.

Step 3 (Calculate Sample Standard Deviation):

Now we know that:

$$s = \sqrt{\dfrac{1}{N – 1} \sum_{i=1}^N\left(x_{i} – \bar{x}\right)^2}$$

$$s = \sqrt {\dfrac { 4+1+16+4+9+0}{ 6-1}}$$

$$s = \sqrt {\dfrac { 34 }{5}}$$

$$s = \sqrt {6.8}$$

$$s = 2.60$$

## How Does Standard Deviation Calculator Work?

The simple interface of the STD calculator makes your calculations super fast and 100% accurate! What you need to do includes:

Inputs:

• Select either sample or population data set
• Enter the values for the dataset
• Tap Calculate

Outputs:

• Standard deviation, variance, and mean of the dataset
• Total count & sum of data values
• Coefficient of Variance
• Standard Error of Mean (SE)
• Sum of Squares of the numbers
• Frequency table for the given dataset
• Step-by-Step calculation

## References:

From the source of Wikipedia : General understanding and basic examples

From the site of scribbr.com : Formulas for population and sample standard deviation