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
standard deviation formula
Population Standard Deviation
population standard deviation formula
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.
XM(X-M)(X-M)2
13-24
23-11
3300
4311
5324
Step 2: Find the sum of (X-M)2

4+1+0+1+4 = 10

Step 3:

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

Step 4: Now find Standard Deviation using the formula

√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
XX2
11
24
39
416
525
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.

Step 3: Now find the population standard deviation using the formula.

√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

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

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