6.3 The Standard Deviation

The standard deviation is the square root of the variance.

The sample variance of the data set $x sub 1, x sub 2, x sub 3,..., x sub n$ is given by $s = \sqrt{\frac{Σ (x_{i} - μ)^{2}}{n - 1}}$

numpy for sample standard deviation:

import numpy as np
x = [x1,x2,x3,...]
np.std(x,ddof=1)

The population variance of the data set $x sub 1, x sub 2, x sub 3,..., x sub N$ is given by $σ = \sqrt{\frac{Σ (x_{i} - μ)^{2}}{N}}$

numpy for population standard deviation:

import numpy as np
x = [x1,x2,x3,...]
np.std(x)

Example: find the sample standard deviation (std) of the data set 1,2,3,4,5

import numpy as np
x = [1,2,3,4,5]
np.std(x,ddof=1)

Show expected output

1.5811388300841898

Interpretation: The sample standard deviation is 1.5811388300841898

Example: find the population standard deviation of the data set 1,2,3,4,5

import numpy as np
x = [1,2,3,4,5]
np.std(x)

Show expected output

1.4142135623730951

Interpretation: The population standard deviation is 1.4142135623730951

Example: The 2010 life expectancy of the 77 Chicago community areas are listed below. Write a numpy program to compute the population standard devaition.

77.3, 80.3, 76, 80.5, 81.5, 81.9, 81.3, 85.2, 81, 80.9, 79.7, 83.4, 84.4, 80.6, 80.3, 79.6, 79.8, 79.6, 79.5, 80.5, 79.8, 80.3, 74.5, 79.5, 71.9, 68.8, 71.7, 78.9, 72.1, 82.2, 81.4, 85, 80.6, 81.9, 74.1, 73.6, 69.8, 74.1, 80.5, 68.9, 82.3, 74.5, 72.5, 74, 74.7, 72.7, 71.9, 77.1, 72.9, 76.4, 76.2, 78.4, 71.6, 76.4, 77.1, 79.8, 79.5, 80.8, 80.4, 80.2, 74.4, 81, 79.2, 77.5, 80.5, 75.2, 70.1, 70.7, 71.1, 78.2, 72.6, 80.5, 74.9, 79.6, 75.3, 82.2, 79.8, 77.8

import numpy as np
x = [77.3, 80.3, 76, 80.5, 81.5, 81.9, 81.3, 85.2, 81, 80.9, 79.7, 83.4, 84.4, 
     80.6, 80.3, 79.6, 79.8, 79.6, 79.5, 80.5, 79.8, 80.3, 74.5, 79.5, 71.9, 
     68.8, 71.7, 78.9, 72.1, 82.2, 81.4, 85, 80.6, 81.9, 74.1, 73.6, 69.8, 74.1, 
     80.5, 68.9, 82.3, 74.5, 72.5, 74, 74.7, 72.7, 71.9, 77.1, 72.9, 76.4, 76.2, 
     78.4, 71.6, 76.4, 77.1, 79.8, 79.5, 80.8, 80.4, 80.2, 74.4, 81, 79.2, 77.5, 
     80.5, 75.2, 70.1, 70.7, 71.1, 78.2, 72.6, 80.5, 74.9, 79.6, 75.3, 82.2, 
     79.8, 77.8]

np.std(x,ddof=0)

Show expected output

4.036198185451496

Example: A random sample of 35 annual salaries of Animal Control employees of the City of Chicago is listed below. Write a numpy program to compute the sample variance and standard deviation.

45288, 96096, 69468, 79872, 92520, 49968, 76248, 110076, 84324, 101496, 100668, 66864, 49968, 101496, 49968, 69468, 69468, 96096, 100716, 73380, 52044, 76248, 66864, 66864, 76848, 66864, 139392, 56280, 69468, 66864,73380, 87564, 49968, 49968, 76848

import numpy as np

x = [45288, 96096, 69468, 79872, 92520, 49968, 76248, 110076, 84324, 101496, 
     100668, 66864, 49968, 101496, 49968, 69468, 69468, 96096, 100716, 73380, 
     52044, 76248, 66864, 66864, 76848, 66864, 139392, 56280, 69468, 66864,
     73380, 87564, 49968, 49968, 76848]

print('The sample variance and standard deviation are')
print(np.var(x,ddof=1))
print(np.std(x,ddof=1))

Show expected output

The sample variance and standard deviation are
438099876.7865547
20930.835549173727

Interpretation: The sample variance and standard deviation are 438099876.7865547 and 20930.835549173727 respectively.

Interpretation: The population standard deviation is 4.036198185451496

Adapted from Python for Introductory Statistics, by Simon Aman (Truman College, City Colleges of Chicago), licensed under CC BY 4.0. Changes were made: reformatted as an accessible XYZ web edition with live in-browser code cells. License: CC-BY-4.0.