# Java Program to compute standard deviation,variance of population & sample with example.

In continuation to my earlier Java code to find all permutations of a String using recursion, let us write another useful java program to compute Variance and Standard Deviation. Before writing the java program, let us understand that what is Standard Deviation? and  how to calculate the variance and the standard deviation  with suitable  steps and example. Also let us see  that how standard deviation is applied in real life applications.   The standard deviation gives an idea of  how widely individual observations / values in a group are dispersed from the mean. (i.e. indicates that how much variation  is there from the average). A large standard deviation indicates that the data points are far from the mean and a small standard deviation indicates that they are  close to  the mean.  Volatility / Risk is measured using  standard deviations in real life applications such as trading business.  Investors can use standard deviation  to measure the risk. The formula for computing standard deviation is given below. Also it is important to distinguish between the standard deviation of a population and the standard deviation of a sample. They are computed differently.

The standard deviation of a population is calculated by the below formula.

std = sqrt [ sum (square( Xi – X )) / N ]

where Xi is the data item and X is the mean.

OR

The standard deviation of a sample is calculated by  different formula.

std = sqrt [ sum (square( Xi – X )) / (n-1) ]

where Xi is the data item and X is the mean.

OR

Let’s say we have a set of n numbers x1, x2, . . . , xn .

Here are the steps to compute Standard deviation

1. Compute the mean.(Average of the given numbers)

a) Compute the sum  of all numbers

b) Divide the sum by the number of observations (n)

2. Compute  the Variance

a) For each data value (Xi),  Calculate its deviation from the mean (Substract the mean from the data value) and   Square it (this removes the negative).

c) Sum the squared deviations.

d) Divide the sum by the  number of observations (n) (For sample standard deviation, divide by  n-1)

3. Compute the Standard Deviation

a) Compute the square root of the variance.

Consider a simple example. Let us calculate the standard deviation of  price variations. Prices of  stock  ABC is given below for  period of  one week.

Day 1 : Rs. 99/-

Day 2 : Rs. 100/-

Day 3 : Rs. 96/-

Day 4 : Rs. 100/-

Day 5 : Rs. 106/-

Day 6 : Rs. 102/-

Day 7 : Rs. 98/-

Calculate the average price, add up all prices and divide by the number of days:

(99+100+96+100 + 106 + 102 + 98  ) / 7 = Rs.  100.14

Let us  calculate the difference between each daily price and the average price  and square of that:

Periods     Price       Average     Difference           Square

Day 1 :    99    –        100.14       -1.14                  1.30

Day 2 :    100    –      100.14        -0.14                 0.02

Day 3 :    96      –      100.14       -4.14                 17.14

Day 4 :   100    –       100.14       -0.14                   0.02

Day 5 :   106    –       100.14         5.86                  34.34

Day 6 :   102    –       100.14        1.86                   3.46

Day 7 :    98    –        100.14         -2.14                4.58

Standard deviation = Squareroot( 1.30+ 0.02+17.14+ 0.02+34.34+2.46 + 4.58)/7)  =   2.94

In the above example,  average is 100.14 , and  1 standard deviation is 2.94.    So the price can deviate within plus or minus  of Rs. 2.94  from the average price of Rs. 100.14.  So 68.27% of the time, the price will  be within the range of  Rs. 97.2 and Rs. 103.08.

2 standard deviation is 2.94*2 = 5.88, So 95.45%  of the time , the price will  be within the range of  Rs. 94.26 and Rs. 106.02.

```
package com.javaonline;

//Calculate the standard deviation

public class Stdev

{

double getStdDev(String s, double...args)

{

return Math.sqrt(getVariance(s, args));

}

//Calculate the Variance

double getVariance(String str, double...args)

{

double mean = getMean( args);

double diff=0 , temp = 0, square=0 , variance=0.0;

int i=0;

System.out.println("Periods " + " Mean " + " Difference " + " Square " );

System.out.println("----------------------------------------------------------------------------------- " );

for(double data :args)

{

i=i+1;

diff=data-mean;

square=diff*diff;

System.out.println("Day "+i + " " + data + " " + mean + " " + diff + " " + square);

temp += square;

}

System.out.println("----------------------------------------------------------------------------------- " );

if (str.equals("P"))

variance= temp/args.length;

else

variance= temp/(args.length-1);

System.out.println("\nVariance = " + variance+"\n");

return variance;

}

//Calculate the Average

double getMean( double...args)

{

double sum = 0.0;

for(double data : args)

sum += data;

return sum/args.length;

}

public static void main(String args[])

{

Stdev std=new Stdev();

System.out.println("standard deviation of a population = " + std.getStdDev("P", 99, 100, 96, 100, 106, 102 , 98 ) +"\n");

System.out.println("standard deviation of a sample = " + std.getStdDev("S", 99, 100, 96, 100, 106, 102 , 98 ) +"\n");

}

} ```

Output :