Weighted Mean and std

numpy syntax for weighted mean and std:

import numpy as np
x = [x1,x2,x3,...]
w = [w1,w2,w3,...]  

np.average(x,weights=w)

numpy sample standard deviation:

import numpy as np
(np.cov(x,aweights=w))**(1/2)

Example: At the end of a term a student received a B in Biology (3 credits), A in History (3 credits), B in English Literature (4 credits), and C in Mathematics (5 credits). Find the students grade point average (GPA). Note: A = 4, B = 3, C = 2, D = 1 F = 0 grade points). GPAs are weighted mean. Find the weighted standard-deviation.

import numpy as np
x = [3,4,3,2]
w = [3,3,4,5]

np.average(x,weights=w)

Show expected output

2.8666666666666667

(np.cov(x,aweights=w))**.5

Show expected output

0.8359396992145376

Interpretation: The grade point average (GPA) is 2.87 with standard deviation 0.84

Mean and Standard deviation of Grouped Frequency Tables

Sometimes instead of the raw data, we are given a summary frequency table.

Example: Below is a frequency table of quiz scores from 26 students. Find the mean and standard deviation. In this case, we take the midpoint of each class to represent the class score.

class	midpoint	freq
0-2	1	1
3-5	4	6
6-8	7	10
9-11	10	7
12-14	13	0
15-17	16	2

numpy syntax for mean and std with frequency as weight:

import numpy as np
np.average(x,weights=freq)

numpy sample standard deviation:

import numpy as np
(np.cov(x,fweights=freq),ddof=1)**(1/2)

Example: Below is a frequency table of quiz scores from 26 students. Find the mean and standard deviation. In this case, we take the midpoint of each class to represent the class score.

class	midpoint	freq
0-2	1	1
3-5	4	6
6-8	7	10
9-11	10	7
12-14	13	0
15-17	16	2

Click + Code
Type import numpy as np
Type mid_points = [1,4,7,10,13,16]
Type freq = [1,6,10,7,0,2]
Type np.average(mid_points,weights=freq)
Click play button

import numpy as np
mid_points = [1,4,7,10,13,16]
freq = [1,6,10,7,0,2]
np.average(mid_points,weights=freq)

Show expected output

7.576923076923077

Interpretation: the mean is 7.576923076923077

import numpy as np
mid_points = [1,4,7,10,13,16]
freq = [1,6,10,7,0,2]
(np.cov(mid_points,fweights=freq,ddof=1))**.5

Show expected output

3.5005494074282333

Interpretation: the sample standard deviation is 3.5005494074282333

import numpy as np

mid_points = [1,4,7,10,13,16]
freq = [1,6,10,7,0,2]
np.sqrt(np.cov(mid_points,fweights=freq,ddof=0))

Show expected output

3.432571103225427

Interpretation: The population standard deviation is 3.432571103225427

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.