In Statistics, Standard Deviation is the measure of variability or spread of scores within a set of data. It is calculated as the root mean square deviation of the values from their arithmetic mean
Standard Deviation Equation or Formula for Standard Deviation
 |
| Standard Deviation |
1. Sample data (selection from a bigger population)
2. Population data (from a group of sample)
The standard deviation formula or the standard deviation equation is given as,
Population Standard Deviation, s = square root of [sigma(xi-x(bar))^2/(n)]
Sample Standard Deviation, s = square root of [sigma(xi-x(bar))^2/(n-1)]
[ s is the standard deviation, xi = all the data items of a sample data (i is 1 to n), x(bar) = mean of the sample data, n=number of data items]
How to do Standard Deviation:
To find the Standard Deviation the steps involved are:
• First we need to count the number of data items in the given sample, this gives us ‘n’
• The mean of the data is calculated, x(bar)
• Subtract the mean from each data item and square the difference, [(x-x(bar)]2 tabulate this values
• Find the sum of the above tabulated values, sigma [(x-x(bar))^2]
• Divide the sum with the number of data items (n); sigma [(x-x(bar))^2]/n
• Once all this done, find the square root; square root {sigma [(x-x(bar))^2]/n}
Using the formula square root of {sigma [(x-x(bar))^2]/n} gives us the standard deviation of the given sample data
Consider a sample data, 5, 8, 11, 13, 18, 7, 14, 12. Let us find the standard deviation of the given sample data using a Standard Deviation Chart
There are eight scores in the given sample data, so n=8
The mean =(5+8+11+13+18+7+14+12)/8= 88/8 = 11
Data Values (x) [x-x(bar)]2
5 (-6)^2=36
8 (-3)^2=9
11 (0)^2=0
13 (2)^2=4
18 (7)^2=49
7 (-4)^2=16
14 (3)^2=9
12 (1)^2=1
Sigma [x-x(bar)]^2= 36+9+0+4+49+16+9+1= 124
Sigma [x-x(bar)]2/n = 124/8 = 15.5
Standard Deviation =Square root{ Sigma [x-x(bar)]2/n} = square root {15.5} = 3.94
So, the standard deviation of the given sample data is 3.94
Standard Deviation Graph
The standard deviation is a statistic which tells us how the various samples or data items of a sample data are distributed around the mean in a set of data. In a normal distribution, the graph of the standard deviation is in the shape of a bell curve. When the samples are mostly together then the graph we get is a steep bell curve; which shows that the standard deviation is small. When the samples are spread apart then the bell curve we get is relatively flat; which shows that the standard deviation is relatively large.
