Mean, Median, and Mode

Salary surveys regularly report an "average" salary that almost nobody earns. This is not an accident — it is a deliberate (or careless) choice of the wrong summary statistic. Understanding when the mean lies to you is one of the first skills that separates a data scientist from someone who just runs .mean() on everything.

Learning Objectives

• Compute mean, median, and mode in Python using NumPy and pandas
• Explain why the mean is misleading for skewed data
• Choose the correct measure of central tendency for a given dataset
• Understand weighted mean and when to use it
• Recognize how outliers shift the mean while leaving the median stable

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What Is Central Tendency?

Central tendency answers the question: "What is the typical value in this dataset?" There are three common answers, and they give different results depending on the shape of your data.

Measure	Definition	Best when
Mean	Sum divided by count	Data is symmetric, no extreme outliers
Median	Middle value when sorted	Data is skewed or has outliers
Mode	Most frequent value	Categorical data, or finding the peak of a distribution

The right choice is not a matter of preference — it follows from the shape of your data.

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Mean — The Arithmetic Average

The mean adds everything up and divides by the number of values.

Formula

mean = (x₁ + x₂ + ... + xₙ) / n In NumPy: np.mean(arr) | In pandas: series.mean()

import numpy as np
import pandas as pd

salaries = pd.Series([45000, 48000, 52000, 55000, 60000, 58000, 950000])

print("Mean:  ", salaries.mean())
print("Median:", salaries.median())
# Output:
# Mean:   181142.857...
# Median: 55000.0

One CEO salary of 950,000 pulls the mean to 181,000 — a number that describes nobody in this dataset. The median of 55,000 is what a typical employee actually earns.

The mean lies on skewed data

Income, house prices, customer spend, website response times — these are all right-skewed. Reporting the mean for these variables without also checking the median is a common mistake. In any EDA, always compute both.

Why the Mean Still Matters

The mean is not wrong — it is the right tool for symmetric data. It is also the foundation of many statistical methods (regression, t-tests) and is what your ML model minimizes when you use mean squared error. Know when to trust it.

# Symmetric data — mean and median are close
exam_scores = pd.Series([68, 72, 74, 75, 76, 77, 79, 81, 83, 85])

print("Mean:  ", exam_scores.mean())    # Output: 77.0
print("Median:", exam_scores.median())  # Output: 76.5
# Close to each other — either is fine here

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Median — The Middle Value

Sort your data. The median is the value that sits at the exact center. Half the values are below it, half above.

# Odd number of values — middle element
odd_series = pd.Series([10, 20, 30, 40, 50])
print(odd_series.median())  # Output: 30.0

# Even number of values — average of two middle elements
even_series = pd.Series([10, 20, 30, 40])
print(even_series.median())  # Output: 25.0

The classic skew test

Compute mean - median. If the result is large and positive, your data is right-skewed. Large and negative means left-skewed. Near zero means roughly symmetric. This one comparison tells you which summary statistic to report.

housing_prices = pd.Series([250000, 275000, 290000, 310000, 320000, 340000, 2500000])

gap = housing_prices.mean() - housing_prices.median()
print(f"Mean:   {housing_prices.mean():,.0f}")    # Output: 612,142
print(f"Median: {housing_prices.median():,.0f}")  # Output: 310,000
print(f"Gap:    {gap:,.0f}")                      # Output: 302,142 — heavily right-skewed

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Mode — The Most Common Value

The mode is the value that appears most often. It is the only measure of central tendency that works for categorical data.

customer_cities = pd.Series(["Mumbai", "Delhi", "Mumbai", "Bangalore", "Mumbai", "Delhi"])

print(customer_cities.mode())
# Output:
# 0    Mumbai
# dtype: object

For numeric data, mode finds the peak of the distribution — useful when you want to know the most popular price point or the most common transaction size.

purchase_amounts = pd.Series([99, 149, 99, 199, 99, 249, 149, 99])
print(purchase_amounts.mode())
# Output:
# 0    99
# dtype: int64

Multiple modes

series.mode() returns all modes if there are ties. Always check len(series.mode()) before assuming a single peak. A dataset with two modes (bimodal) often signals two distinct subgroups — worth investigating.

bimodal = pd.Series([1, 1, 2, 3, 4, 5, 5])
print(bimodal.mode())
# Output:
# 0    1
# 1    5
# dtype: int64
# Two modes — suggests two clusters in the data

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Weighted Mean

The standard mean treats every value equally. The weighted mean lets you give some values more importance — used in GPA calculations, index construction, and recommendation systems.

Weighted Mean Formula

weighted_mean = sum(value × weight) / sum(weights)

# A student's grades across courses with different credit hours
grades  = np.array([85, 90, 78, 92])
credits = np.array([4,  3,  4,  2])

weighted_mean = np.average(grades, weights=credits)
simple_mean   = np.mean(grades)

print(f"Simple mean:   {simple_mean:.2f}")    # Output: 86.25
print(f"Weighted mean: {weighted_mean:.2f}")  # Output: 84.69
# The 4-credit course with 78 drags the weighted mean below the simple mean

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Real-World Example: Salary Data

This is the canonical example. Run this and internalize the numbers.

import pandas as pd
import numpy as np

np.random.seed(42)

# Simulate a small company: 20 employees + 1 founder
employee_salaries = np.random.normal(loc=60000, scale=8000, size=20)
founder_salary    = np.array([2_500_000])

all_salaries = pd.Series(np.concatenate([employee_salaries, founder_salary]))

print(f"Mean salary:   ${all_salaries.mean():>12,.0f}")
print(f"Median salary: ${all_salaries.median():>12,.0f}")
print(f"Mode salary:   ${all_salaries.mode()[0]:>12,.0f}")  # Will be approximate
print(f"Min:           ${all_salaries.min():>12,.0f}")
print(f"Max:           ${all_salaries.max():>12,.0f}")

# Output (approximate):
# Mean salary:   $    179,412
# Median salary: $     59,711
# Mode salary:   ~  varies
# Min:           $     45,083
# Max:           $  2,500,000

The mean of $179K describes none of the 20 employees and misrepresents the company's compensation. The median of $59K is what most people actually earn. This gap is why public salary discussions almost always use median — and why some companies prefer to report mean.

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Computing All Three in a DataFrame

df = pd.DataFrame({
    "employee_id":  range(1, 8),
    "department":   ["Engineering", "Engineering", "Sales", "Sales", "HR", "Engineering", "Sales"],
    "annual_salary": [70000, 85000, 55000, 60000, 48000, 95000, 52000]
})

# Per-column summaries
print(df["annual_salary"].mean())    # Output: 66428.57
print(df["annual_salary"].median())  # Output: 60000.0
print(df["department"].mode()[0])    # Output: Engineering

# Group-level summaries — more useful in practice
print(df.groupby("department")["annual_salary"].agg(["mean", "median"]))
# Output:
#               mean  median
# department
# Engineering  83333   85000
# HR           48000   48000
# Sales        55667   55000

Always group before summarizing

A company-wide average salary hides the fact that engineering earns 70% more than HR. Always break your summary statistics down by relevant groups — you will find the real story there.

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Missing Values

Pandas skips NaN by default when computing these statistics. This is usually what you want, but be aware.

data_with_gaps = pd.Series([100, 200, np.nan, 400, 500])

print(data_with_gaps.mean())    # Output: 300.0 (ignores NaN, computes over 4 values)
print(data_with_gaps.count())   # Output: 4 (not 5)

NaN changes your denominator silently

If you have 30% missing values and compute the mean, you are computing the mean of the 70% that remain. Whether that is a valid estimate depends on WHY the values are missing. Never report a mean without checking .isnull().sum() first.

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Key Takeaways

• Mean is sensitive to outliers. Use it on symmetric data.
• Median is robust. Use it when data is skewed or has outliers.
• Mode works on categories and tells you the most popular value.
• The gap between mean and median is your first diagnostic for skew.
• Weighted mean matters when values have different levels of importance.
• Always check for missing values before computing any summary statistic.

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Back to Day 4 Agenda | Next: Variance and Standard Deviation