# Variance And Standard Deviation

## Preliminary

import math

## Create Data

data = [3,2,3,4,2,3,5,2,2,33,3,5,2,2,5,6,62,2,2,3,6,6,2,23,3,2,3]

## Calculate Population Variance

Variance is a measurement of the spread of a data's distribution. The higher the variance, the more "spread out" the data points are. Variance, commonly denoted as \(S^{2}\), is calculated like this:

$$ \text{Population Variance} = S_n^{2} = \frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^{2}$$

$$ \text{Sample Variance} = S_{n-1}^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^{2}$$

Where \(n\) is the number of observations, \(\bar{x}\) is the mean of the observations, and \(x_i-\bar{x}\) is an individual observation's from the mean of the data. Note that if we were estimating the variance of a population based on a sample from that population, we should use the second equation, replacing \(n\) with \(n-1\).

# Calculate n n = len(data) # Calculate the mean mean = sum(data)/len(data) # Create a list of all deviations from the mean all_deviations_from_mean_squared = [] # For each observation in the data for observation in data: # Calculate the deviation from the mean deviation_from_mean = (observation - mean) # Square it deviation_from_mean_squared = deviation_from_mean**2 # Add the result to our list all_deviations_from_mean_squared.append(deviation_from_mean_squared) # Sum all the squared deviations in our list sum_of_deviations_from_mean_squared = sum(all_deviations_from_mean_squared) # Divide by n population_variance = sum_of_deviations_from_mean_squared/n # Show variance population_variance

160.78463648834017

## Calculate Population Standard Deviation

Standard deviation is just the square root of the variance.

# Find the square root of the population variance population_standard_deviation = math.sqrt(population_variance) # Print the populaton standard deviation population_standard_deviation

12.68008818929664