Weighted standard deviation in NumPy

There doesn’t appear to be such a function in numpy/scipy yet, but there is a ticket proposing this added functionality. Included there you will find Statistics.py which implements weighted standard deviations.

There is a class in statsmodels that makes it easy to calculate weighted statistics: statsmodels.stats.weightstats.DescrStatsW.

Assuming this dataset and weights:

import numpy as np
from statsmodels.stats.weightstats import DescrStatsW

array = np.array([1,2,1,2,1,2,1,3])
weights = np.ones_like(array)
weights[3] = 100

You initialize the class (note that you have to pass in the correction factor, the delta degrees of freedom at this point):

weighted_stats = DescrStatsW(array, weights=weights, ddof=0)

Then you can calculate:

• .mean the weighted mean:

  >>> weighted_stats.mean      
  1.97196261682243
  

• .std the weighted standard deviation:

  >>> weighted_stats.std       
  0.21434289609681711
  

• .var the weighted variance:

  >>> weighted_stats.var       
  0.045942877107170932
  

• .std_mean the standard error of weighted mean:

  >>> weighted_stats.std_mean  
  0.020818822467555047
  

  Just in case you’re interested in the relation between the standard error and the standard deviation: The standard error is (for ddof == 0) calculated as the weighted standard deviation divided by the square root of the sum of the weights minus 1 (corresponding source for statsmodels version 0.9 on GitHub):

  standard_error = standard_deviation / sqrt(sum(weights) - 1)
  

There is a very good example proposed by gaborous:

import pandas as pd
import numpy as np
# X is the dataset, as a Pandas' DataFrame
mean = mean = np.ma.average(X, axis=0, weights=weights) # Computing the 
weighted sample mean (fast, efficient and precise)

# Convert to a Pandas' Series (it's just aesthetic and more 
# ergonomic; no difference in computed values)
mean = pd.Series(mean, index=list(X.keys())) 
xm = X-mean # xm = X diff to mean
xm = xm.fillna(0) # fill NaN with 0 (because anyway a variance of 0 is 
just void, but at least it keeps the other covariance's values computed 
correctly))
sigma2 = 1./(w.sum()-1) * xm.mul(w, axis=0).T.dot(xm); # Compute the 
unbiased weighted sample covariance

Correct equation for weighted unbiased sample covariance, URL (version: 2016-06-28)

Here’s one more option:

np.sqrt(np.cov(values, aweights=weights))

A follow-up to "sample" or "unbiased" standard deviation in the "frequency weights" sense since "weighted sample standard deviation python" Google search leads to this post:

def frequency_sample_std_dev(X, n):
    """
    Sample standard deviation for X and n,
    where X[i] is the quantity each person in group i has,
    and n[i] is the number of people in group i.
    See Equation 6.4 of:
    Montgomery, Douglas, C. and George C. Runger. Applied Statistics 
     and Probability for Engineers, Enhanced eText. Available from: 
      WileyPLUS, (7th Edition). Wiley Global Education US, 2018.
    """
    n_groups = len(n)
    n_people = sum(n)
    lhs_numerator = sum([ni*Xi**2 for Xi, ni in zip(X, n)])
    rhs_numerator = sum([Xi*ni for Xi, ni in zip(X,n)])**2/n_people
    denominator = n_people-1
    var = (lhs_numerator - rhs_numerator) / denominator
    std = sqrt(var)
    return std

Or modifying the answer by @Eric as follows:

def weighted_sample_avg_std(values, weights):
    """
    Return the weighted average and weighted sample standard deviation.

    values, weights -- Numpy ndarrays with the same shape.
    
    Assumes that weights contains only integers (e.g. how many samples in each group).
    
    See also https://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Frequency_weights
    """
    average = np.average(values, weights=weights)
    variance = np.average((values-average)**2, weights=weights)
    variance = variance*sum(weights)/(sum(weights)-1)
    return (average, sqrt(variance))

print(weighted_sample_avg_std(X, n))

I was just searching for an API equivalent of the numpy np.std function that also allows the axis parameter to be set:

(I just tested it with two dimensions, so feel free for improvements if something is incorrect.)

def std(values, weights=None, axis=None):
    """
    Return the weighted standard deviation.
    axis -- the axis for std calculation
    values, weights -- Numpy ndarrays with the same shape on the according axis.
    """
    average = np.expand_dims(np.average(values, weights=weights, axis=axis), axis=axis)
    # Fast and numerically precise:
    variance = np.average((values-average)**2, weights=weights, axis=axis)
    return np.sqrt(variance)

Thanks to Eric O Lebigot for the original answer.