__Finding the variance and the standard deviation__.

Variance and Standard Deviation are related measures of dispersion that tell us more about a list of numbers,like are they all similar, clustered near the mean or are they all different. To calculate either of these, we first need to find the difference of each of the numbers from the mean. The variance is the average of the square of those differences. The variance is calculated using the formula:

**Variance = (****Σ****(x _{i }- x_{mean}) ^{^ 2)} / n**.

Where, x_{i }= individual numbers

x_{mean = }mean of these numbers

and n = number of values in the list.

So,now, in order to calculate the variance, for each value in the list, we take the difference between that number and the mean and square it. Then we add all those squared differences together and, finally, divide the whole sum by n to find the variance.

And to calculate the standard deviation as well, all we have to do is to calculate the square root of the variance.

Now, we will write a Python program to find out the variance and the standard deviation of a list of numbers.

__PYTHON PROGRAM.__

import os

import sys

def calculate_mean(numbers):

s = sum(numbers)

N = len(numbers)

mean = s/N

return mean

def find_differences(numbers):

mean = calculate_mean(numbers)

diff = []

for num in numbers:

diff.append(num-mean)

return diff

def calculate_variance(numbers):

diff = find_differences(numbers)

squared_diff = []

for d in diff:

squared_diff.append(d**2)

# Find the variance

sum_squared_diff = sum(squared_diff)

variance = sum_squared_diff/len(numbers)

return variance

donations = [100,60,70,900,100,200,500,500,503,600,1000,1200]

variance = calculate_variance(donations)

std = variance ** 0.5

print("The standard deviations of the list of numbers is {0}".format(std))

__Program anaylsis__

The function **calculate_variance()** calculates the variance of the list of numbers passed to it. Firstly, it calls the **find_differences()** function to calculate the difference of each of the numbers from the mean and appends each difference so found to a list. We notice that the mean is calculated by another function, calculate_mean(),called from the **find_differences()** function. Then, using the sum function, it sums up the differences so found. Then, the variance is calculated using the formula, dividing the sum of the differences by the total numbers in the list of numbers.

**OUTPUT OF THE PROGRAM**.

>>>

The standard deviations of the list of numbers is 375.5627166887931

>>>