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Mean, Median, Mode

Overview

Central tendency measures identify the center of a data distribution, summarizing a dataset with a single representative value. The three most commonly used measures are the mean, median, and mode. Each has distinct properties, strengths, and weaknesses that make it appropriate for different situations.

1. Mean

The mean is the arithmetic average of a set of numbers, calculated by summing all values and dividing by the count.

Formulas

\[ \begin{array}{lllll} \text{Population Mean} && \mu &=& \displaystyle\frac{\sum_{i=1}^N x_i}{N} \\[10pt] \text{Sample Mean} && \bar{x} &=& \displaystyle\frac{\sum_{i=1}^n x_i}{n} \\[10pt] \text{Expected Value} && \mathbb{E}[X] &=& \displaystyle\sum_i x_i \, \mathbb{P}(X = x_i) \quad \text{(discrete)} \\[6pt] &&& =& \displaystyle\int_{-\infty}^{\infty} x \, f_X(x) \, dx \quad \text{(continuous)} \end{array} \]

Example

For the dataset 70, 85, 90, 95, 100:

\[ \bar{x} = \frac{70 + 85 + 90 + 95 + 100}{5} = \frac{440}{5} = 88 \]

Mean as Balancing Point

The mean is the value where the sum of deviations equals zero:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} \quad \Rightarrow \quad \sum_{i=1}^N (x_i - \mu) = 0 \]

This means the mean is the "center of gravity" of the data.

Mean: Income Example

import pandas as pd
import matplotlib.pyplot as plt

url = 'https://raw.githubusercontent.com/gedeck/practical-statistics-for-data-scientists/master/data/loans_income.csv'
loans_data = pd.read_csv(url)

mean_income = loans_data['x'].mean()

fig, ax = plt.subplots(figsize=(12, 3))
ax.hist(loans_data['x'], bins=20, density=True, alpha=0.3,
        color='blue', edgecolor='black')
ax.plot([mean_income, mean_income], [0, 1.6e-5], '--c',
        alpha=0.7, label="Mean")
ax.legend()
ax.set_title("Histogram of Income Data with Mean Indicator")
ax.set_xlabel("Income")
ax.set_ylabel("Density")
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
plt.show()

2. Median

The median is the middle value when data is arranged in order. It divides the dataset into two equal halves.

How to Calculate

  1. Sort the data in ascending order.
  2. If \(n\) is odd, the median is the middle value at position \((n+1)/2\).
  3. If \(n\) is even, the median is the average of the two middle values.

Example

For the dataset 70, 85, 90, 95, 100 (odd count): Median = 90.

For the dataset 70, 85, 90, 95 (even count): Median = \((85 + 90)/2 = 87.5\).

Median vs. Mean: Income Data

import pandas as pd
import matplotlib.pyplot as plt

url = 'https://raw.githubusercontent.com/gedeck/practical-statistics-for-data-scientists/master/data/loans_income.csv'
loans_data = pd.read_csv(url)

mean_income = loans_data['x'].mean()
median_income = loans_data['x'].median()

fig, ax = plt.subplots(figsize=(12, 3))
ax.hist(loans_data['x'], bins=20, density=True, alpha=0.3,
        color='blue', edgecolor='black')
ax.plot([mean_income, mean_income], [0, 1.6e-5], '--c',
        alpha=0.7, label="Mean")
ax.plot([median_income, median_income], [0, 1.6e-5], '--r',
        alpha=0.7, label="Median")
ax.legend()
ax.set_title("Histogram of Income Data with Mean and Median")
ax.set_xlabel("Income")
ax.set_ylabel("Density")
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
plt.show()

Median Is Robust Against Outliers

The median is far less affected by extreme values than the mean. This is demonstrated by adding outliers to income data:

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

url = 'https://raw.githubusercontent.com/gedeck/practical-statistics-for-data-scientists/master/data/loans_income.csv'
loans_data = pd.read_csv(url)
income_data = loans_data['x'].values

mean_income = income_data.mean()
median_income = np.median(income_data)

fig, (hist_ax, box_ax) = plt.subplots(1, 2, figsize=(12, 3))
plt.suptitle("Original Income Data", fontsize=20)

n, bin_edges, _ = hist_ax.hist(income_data, bins=20, density=True, alpha=0.5,
                                color='skyblue')
hist_ax.plot([mean_income, mean_income], [0, n.max()], '--b', label='Mean')
hist_ax.plot([median_income, median_income], [0, n.max()], '--r', label='Median')
hist_ax.legend()
hist_ax.set_title("Histogram")

box_ax.boxplot(income_data, vert=False, patch_artist=True)
box_ax.set_title("Boxplot")

for ax in (hist_ax, box_ax):
    ax.spines['top'].set_visible(False)
    ax.spines['right'].set_visible(False)

plt.show()

# Now add outliers
outliers = np.array([20_000_000] * 20)
data_with_outliers = np.concatenate((income_data, outliers))

mean_outliers = data_with_outliers.mean()
median_outliers = np.median(data_with_outliers)

fig, (hist_ax, box_ax) = plt.subplots(1, 2, figsize=(12, 3))
plt.suptitle("Income Data with Outliers", fontsize=20)

n, bin_edges, _ = hist_ax.hist(data_with_outliers, bins=bin_edges,
                                density=True, alpha=0.5, color='skyblue')
hist_ax.plot([mean_outliers, mean_outliers], [0, n.max()], '--b', label='Mean')
hist_ax.plot([median_outliers, median_outliers], [0, n.max()], '--r', label='Median')
hist_ax.legend()
hist_ax.set_title("Histogram")

box_ax.boxplot(data_with_outliers, vert=False, patch_artist=True)
box_ax.set_title("Boxplot")

for ax in (hist_ax, box_ax):
    ax.spines['top'].set_visible(False)
    ax.spines['right'].set_visible(False)

plt.show()

The mean shifts dramatically when outliers are introduced, while the median barely changes.

Real-Life Examples Where Median Is Preferred

Income Distribution: A few extremely high earners skew the mean upward. Median household income provides a more accurate picture of what the "typical" person earns.

Real Estate: Median home prices better represent the housing market than mean prices, which can be inflated by a few luxury sales.

Michael Jordan Case (NBA Salary): Jordan's salary was so much higher than typical NBA players that it significantly inflated the mean. The median salary better reflects what most players earned.

Public Policy: Governments report median household income to assess financial well-being because it is not distorted by extreme wealth.

3. Trimmed Mean

The trimmed mean (or truncated mean) is the arithmetic mean calculated after removing a specified percentage of observations from both tails of the sorted distribution. This hybrid approach offers robustness against outliers while still utilizing most of the data.

Definition

For a dataset with \(n\) observations sorted as \(x_{(1)} \le x_{(2)} \le \cdots \le x_{(n)}\), the \(p\)-trimmed mean removes \(\lceil p \cdot n / 2 \rceil\) observations from each tail and averages the remaining values:

\[ \bar{x}_{p\%} = \frac{1}{n - 2\lceil p \cdot n / 2 \rceil} \sum_{i=\lceil p \cdot n / 2 \rceil + 1}^{n - \lceil p \cdot n / 2 \rceil} x_{(i)} \]

Example: Population Data

Using U.S. state population data, compare the mean, 10% trimmed mean, and median:

import pandas as pd
from scipy.stats import trim_mean

# Load state data
state = pd.read_csv('state.csv')

# Regular mean (sensitive to outliers like California)
mean_pop = state['Population'].mean()
print(f"Mean Population: {mean_pop:,.0f}")

# 10% trimmed mean (removes 5% from each tail)
trimmed_mean_pop = trim_mean(state['Population'], 0.1)
print(f"10% Trimmed Mean: {trimmed_mean_pop:,.0f}")

# Median (completely robust)
median_pop = state['Population'].median()
print(f"Median Population: {median_pop:,.0f}")

Output: 

Mean Population: 6,162,876
10% Trimmed Mean: 4,783,697
Median Population: 4,436,370

The trimmed mean occupies a middle ground: it's less influenced by extreme values (California's 37M population) than the mean, yet uses more data than the median alone. This makes it valuable when a moderate level of robustness is desired without completely ignoring the tails.

When to Use Trimmed Mean

  • Moderate robustness: You want outlier resistance but don't want to discard data entirely.
  • Academic traditions: Some fields prefer trimmed means for hypothesis testing (e.g., psychology, education).
  • Olympic scoring: Judges' scores are often averaged after trimming the highest and lowest.

4. Weighted Mean and Weighted Median

When observations have differing importance or frequency, the weighted mean and weighted median assign each value a weight reflecting its significance.

Weighted Mean

The weighted mean is the sum of weighted values divided by the sum of weights:

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

where \(w_i\) are the weights.

Example: Murder Rate by State Population

When computing the national murder rate, states with larger populations should influence the average more heavily. Use the state's population as a weight:

import pandas as pd
import numpy as np

state = pd.read_csv('state.csv')

# Unweighted mean murder rate
unweighted_mean = state['Murder.Rate'].mean()
print(f"Unweighted Mean Murder Rate: {unweighted_mean:.3f}")

# Weighted mean (weighted by population)
weighted_mean = np.average(state['Murder.Rate'], weights=state['Population'])
print(f"Weighted Mean Murder Rate: {weighted_mean:.3f}")

Output: 

Unweighted Mean Murder Rate: 4.066
Weighted Mean Murder Rate: 4.446

The weighted mean is higher because highly populated states (CA, TX, FL, NY) tend to have higher murder rates than small states. The unweighted mean treats Montana (population 990K) and California (population 37M) as equal—a distortion that the weighted mean corrects.

Weighted Median

The weighted median is the value where the cumulative weight reaches 50% of the total weight. Unlike the weighted mean, it requires a specialized function:

import pandas as pd
import wquantiles

state = pd.read_csv('state.csv')

# Unweighted median
unweighted_median = state['Murder.Rate'].median()
print(f"Unweighted Median: {unweighted_median:.1f}")

# Weighted median (weighted by population)
weighted_median = wquantiles.median(state['Murder.Rate'],
                                    weights=state['Population'])
print(f"Weighted Median: {weighted_median:.1f}")

Output: 

Unweighted Median: 4.0
Weighted Median: 4.4

When to Use Weighted Statistics

Financial Data: Portfolio returns are weighted by asset values.

Survey Data: Responses are weighted to match population demographics.

Aggregated Data: When data represents groups (e.g., state-level statistics), weight by group size.

Importance Weighting: Some observations are more reliable or relevant than others.

5. Mode

The mode is the value that occurs most frequently in a dataset. A dataset may be unimodal (one mode), bimodal (two modes), multimodal (more than two modes), or have no mode if all values appear equally often.

Computing Mode in Python

import statistics

data = [4, 1, 2, 2, 3, 5]
mode = statistics.mode(data)
print(f"{mode = }")  # mode = 2

For datasets with multiple modes:

import statistics

data = [4, 1, 2, 2, 3, 3, 5]
mode = statistics.mode(data)
print(f"{mode = }")  # Returns the first mode encountered

modes = statistics.multimode(data)
print(f"{modes = }")  # Returns all modes: [2, 3]

6. Comparing Mean, Median, Mode, and Other Measures

Mean is best for continuous, symmetrically distributed data without outliers. It uses all data points but is sensitive to extremes.

Median is preferred for skewed distributions or data with outliers. It represents the middle value and is robust to extreme observations.

Mode is most useful for categorical data or for identifying the most common value. It can be used with nominal data (e.g., most popular color).

Relationship to Distribution Shape

  • Symmetric distribution: Mean ≈ Median ≈ Mode
  • Right-skewed distribution: Mode < Median < Mean
  • Left-skewed distribution: Mean < Median < Mode

Summary

Each measure of central tendency serves a distinct purpose. The mean provides a mathematical average but is vulnerable to outliers, the median offers a robust center that resists extreme values, and the mode identifies the most frequent observation. Choosing the appropriate measure depends on the data's distribution shape and the analytical question at hand.