How to calculate the variance and the standard deviation of data using Python?

by Amrita Mitra | Feb 9, 2023 | Statistics For Machine Learning

Calculate Variance And Standard Deviation In Python

Let’s say we are given n numbers [x₁, x₂, x₃, … x_n]. The variance of the given n numbers is given by the formula:

$\sigma^{2}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n}\ where\ \bar{x}=the\ mean\ of\ the\ given\ n\ numbers=\frac{\sum_{i=1}^{n}x_{i}}{n}$

And the standard deviation of the given n numbers is given by the formula:

$\sigma=\sqrt{variance}=\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n}}$

Please note that the variance we calculate using the mentioned formula is the true variance. If we want to calculate the sample variance of a sample and use the sample variance to infer the variance of the population, then we need to use (n-1) in the denominator instead of n. So, the sample variance of the given n numbers is given by the formula:

$\sigma^{2}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n-1}\ where\ \bar{x}=the\ mean\ of\ the\ given\ n\ numbers=\frac{\sum_{i=1}^{n}x_{i}}{n}$

And the sample standard deviation is given by the formula:

$\sigma=\sqrt{variance}=\sqrt{\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n-1}}$

In Python, we can easily calculate the true variance, sample variance, true standard deviation and sample standard deviation of a set of n numbers using the numpy module.

import numpy
 
data = [1, 2, 3, 3, 4, 5, 5, 5, 8, 10]
 
true_variance = numpy.var(data)
sample_variance = numpy.var(data, ddof=1)
true_std = numpy.std(data)
sample_std = numpy.std(data, ddof=1)
 
print("True Variance: ", true_variance)
print("Sample Variance: ", sample_variance)
print("True Standard Deviation: ", true_std)
print("Sample Standard Deviation: ", sample_std)

The output of the above program will be:

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