Weighted Mean — Formula, Applications, and Python

Weighted Mean

The weighted mean assigns different weights to different observations, allowing some values to have more influence on the average than others.

$\bar{x}_w = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}$

When to Use Weighted Mean

Python Implementation

import numpy as np
import pandas as pd

# ========================================
# Example 1: GPA Calculation
# ========================================
courses = pd.DataFrame({
    'Course': ['Statistics', 'Linear Algebra', 'Machine Learning', 'Databases', 'Ethics'],
    'Grade_Points': [4.0, 3.7, 3.3, 4.0, 3.0],
    'Credits': [4, 3, 4, 3, 1]
})

weighted_gpa = np.average(courses['Grade_Points'], weights=courses['Credits'])
simple_gpa = courses['Grade_Points'].mean()

print("Courses:")
print(courses.to_string(index=False))
print(f"\nWeighted GPA: {weighted_gpa:.4f}")
print(f"Unweighted GPA: {simple_gpa:.4f}")
print(f"Difference: {weighted_gpa - simple_gpa:+.4f} (credits matter!)")

# ========================================
# Example 2: Portfolio Return
# ========================================
portfolio = pd.DataFrame({
    'Asset': ['Stock A', 'Stock B', 'Bonds', 'Cash'],
    'Weight_pct': [40, 35, 20, 5],
    'Return_pct': [12.5, -3.2, 4.1, 0.5]
})

portfolio_return = np.average(portfolio['Return_pct'], 
                               weights=portfolio['Weight_pct'])
simple_avg_return = portfolio['Return_pct'].mean()

print("\nPortfolio:")
print(portfolio.to_string(index=False))
print(f"\nWeighted portfolio return: {portfolio_return:.2f}%")
print(f"Simple average return:     {simple_avg_return:.2f}%")

# ========================================
# Example 3: Survey Weighting (post-stratification)
# ========================================
survey = pd.DataFrame({
    'Group': ['18-34', '35-54', '55+'],
    'Survey_pct': [15, 50, 35],    # % in survey sample
    'Pop_pct': [30, 40, 30],       # % in true population
    'Support_pct': [75, 55, 40]    # % supporting the policy
})

# Unweighted (biased — overrepresents 35-54 group)
unweighted = np.average(survey['Support_pct'], weights=survey['Survey_pct'])
# Weighted to population (correct)
weighted = np.average(survey['Support_pct'], weights=survey['Pop_pct'])

print("\nSurvey with nonrepresentative sample:")
print(survey.to_string(index=False))
print(f"\nUnweighted mean: {unweighted:.1f}% support (biased!)")
print(f"Population-weighted mean: {weighted:.1f}% support (correct)")

Properties of the Weighted Mean

# Property 1: Reduces to arithmetic mean when all weights equal
equal_weights = [1, 1, 1, 1, 1]
data = [10, 20, 30, 40, 50]
print(f"Equal weights → weighted mean = {np.average(data, weights=equal_weights):.1f}")
print(f"Simple mean = {np.mean(data):.1f}")

# Property 2: Extreme weights force convergence toward that value
extreme_weights = [1, 1, 1, 1, 100]
print(f"Extreme weight on last → weighted mean = {np.average(data, weights=extreme_weights):.1f}")
print(f"Last value: {data[-1]}")  # Should be close to 50

# Property 3: Normalized weights must sum to 1 for percentages to work
weights = [4, 3, 4, 3, 1]
norm_weights = [w/sum(weights) for w in weights]
print(f"Normalized weights: {[f'{w:.3f}' for w in norm_weights]}")
print(f"Sum: {sum(norm_weights):.4f}")

Key Takeaways

1. Weighted mean is the correct tool when observations differ in importance
2. np.average(data, weights=w) in Python computes it exactly
3. GPA, portfolio returns, price indices — all weighted means in disguise
4. Survey post-stratification weighting corrects for nonprobability samples
5. Equal weights → arithmetic mean (special case of weighted mean)
6. A weight of zero excludes an observation; very large weights dominate the result