While working with statistical problems, we come across with standard deviation quite frequently. It is the measure of dispersion of the statistical data. Dispersion or variation is the property of the data to spread over a field. Dispersion measures the deviation of the data from its average or mean position. The degree of dispersion or variation is calculated by the means of measures of variation.
There are many different measures of variation most common of those are listed below :
Among those, SD is the most frequently used measure of dispersion. It is denoted by $\sigma$, pronounced as sigma.
The measures of dispersion are the statistical tool which measures the variability or the dispersion of a data. The measures of central tendency, such as  mean, median and mode, are the central values of the values. Therefore, these are known as first order averages. On the other hand, the measures of variation or dispersion mentioned above are averages of deviations derived from the average values. Therefore, they are known as second order averages.
Related Calculators  
Calculate Standard Deviation Calculator  Relative Standard Deviation Calculator 
Confidence Interval Standard Deviation Calculator  Normal Distribution Standard Deviation Calculator 
A measure of dispersion in statistics. It gives an idea of how the individual data in a data set is dispersed from the mean. Also known as the root mean square deviation. The symbol used is $\sigma$.
1. The minimum positive value is 0. i.e. it cannot be negative.
2. When the items in a series are more dispersed from the mean, then the standard deviation is also large.
Demerits
To find, we first find the arithmetic mean of the
values. Then we find the deviation of each item from the mean. Find
the squares of the deviations and add them. Then divide the sum by the
number of items in the series and take the square root.
Steps to Calculate the Standard Deviation:
Step 1: Calculate the arithmetic mean.
Step 2: Find the deviation of each item from the mean.
Step 3: Square these deviations and add them.
Step 4: We get $\sum (x  \bar{x})^{2}$.
Step 5: Divide this sum by the total number of items.
Step 6: Take the square root of the result of step 5.
Step 1: Calculate 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 
Step 2: Find the sum of $(X  M)^2$
=> $(X  M)^2$ = 4 + 1 + 0 + 1 + 4 = 10
Step 3: N = 5, the total number of values.
=> N  1 = 5  1 = 4.
Step 4: Now find the standard deviation by deviation formula.
=> S = $\sqrt{\frac{\sum (X  M)^2}{N  1}}$
= $\sqrt{\frac{10}{4}}$
=> S = 1.58113.
We first find the arithmetic mean of
the values. Then we find the deviation of each item from the mean.
Find the squares of the deviations and add them. Then divide the sum by
the number of items in the series and take the square root.
The calculations are shown in the table below
X 
$(x_{1}  \bar{x})$ 
$(x_{1}  \bar{x})^{2}$ 
2 
2  4 =  2 
4 
3 
3  4 =  1 
1 
4 
4  4 = 0 
0 
5 
5  4 = 1 
1 
6 
6  4 = 2 
4 


10 
Mean $\bar{x}$ = $\frac{2 + 3 + 4 + 5 + 6}{5}$ = 4
Standard deviation $\sigma$ = $\sqrt{\frac{\sum (x_{1}  \bar{x})^{2}}{n}}$
= $\sqrt{\frac{10}{4}}$
= $\sqrt{2.5}$
=1.58
=> $\sigma$ = 1.58
Here you can see that all items are taken into consideration. Also we can see that it give more importance to bigger values since we are squaring the values.
X 
1 
2 
3 
4 
f  10 
4 
3  3 
Step 1: The calculations are shown in the table below
x 
f 
$(x_{1}\bar{x})$ 
$(x_{1}\bar{x})^{2}$ 
$f(x_{1}\bar{x})^{2}$ 
1 
10 
11.95 = 0.95 
0.9025  9.025 
2 
4 
21.95 = 0.05 
0.0025 
0.01 
3 
3 
31.95 = 1.05 
1.1025  3.3075 
4 
3 
41.95 = 2.05 
4.2025  12.6075 

20 

Total 
24.95 
Step 2:
Mean, $\bar{x}$ = $\frac{\sum fx}{\sum f}$
= $\frac{1 * 10 + 2 * 4 + 3 * 3 + 4 * 3}{20}$
= $\frac{10 + 8 + 9 + 12}{20}$
= $\frac{39}{20}$
= 1.95
=> Mean = 1.95
Step 3:
Now,
Standard deviation, $\sigma$ = $\sqrt{\frac{\sum f(x_{1}\bar{x})^{2}}{N}}$
= $\sqrt{\frac{24.95}{20}}$
= 1.12
More topics in Standard Deviation  
Standard Deviation Examples  
Related Topics  
Math Help Online  Online Math Tutor 