# How to Calculate Standard Deviation, Variance - Definition, Formula, Example

## How to Calculate Standard Deviation, Variance - Tutorial

##### Standard Deviation Definition:

Standard deviation is a statistical measure of spread or variability.The standard deviation is the root mean square (RMS) deviation of the values from their arithmetic mean.

##### Variance Definition:

The square of the standard deviation. A measure of the degree of spread among a set of values; a measure of the tendency of individual values to vary from the mean value.

#### Formula:

##### Standard Deviation ##### Population Standard Deviation ###### where

Σ = Sum of X = Individual score M = Mean of all scores N = Sample size (Number of scores) Variance : Variance = s2

##### Standard Deviation Method1 Example:

To find the Standard deviation of 1,2,3,4,5.

###### Step 1: Calculate the mean and deviation.
 X M (X-M) (X-M)2 1 3 -2 4 2 3 -1 1 3 3 0 0 4 3 1 1 5 3 2 4

4+1+0+1+4 = 10

###### Step 3:

N = 5, the total number of values. Find N-1. 5-1 = 4

√10/√4 = 1.58113

##### Standard Deviation Method2 Example:

To find the Standard deviation of 1,2,3,4,5.

###### Step 1: First, square each of the scores
 X X2 1 1 2 4 3 9 4 16 5 25
###### Step2: Use the formula

s = square root of[(sum of Xsquared -((sum of X)*(sum of X)/N))/(N-1)] = square root of[(55-((15)*(15)/5))/(5-1)] = square root of[(55-(225/5))/4] = square root of[(55-45)/4] = square root of[10/4] = square root of[2.5] s = 1.58113

##### Population Standard Deviation Example:

To find the Population Standard deviation of 1,2,3,4,5. Perform the steps 1 and 2 as seen in above example.

√10/√5 = 1.414

##### Variance Example:

To find the Variance of 1,2,3,4,5. After finding the standard deviation square the values. (1.58113)2 = 2.4999 Same for Population standard deviation. (1.414)2 = 2

#### Related Calculator:

This tutorial will help you dynamically to find the standard deviation problems.