Mean-Median-Mode

Overview — What Are Measures of Central Tendency?

In statistics, a measure of central tendency is a single value that summarises a dataset by identifying where the centre of the data lies. The three most common measures are:

The Mean (Arithmetic Average)

The mean is calculated by adding all values and dividing by the number of values. It takes into account every single value in the dataset — which makes it powerful but also vulnerable to extreme values (outliers).

Formula

Real Example — Employee Salaries

A small company has 10 employees with the following monthly salaries (₹):

The Outlier Problem — When Mean Fails

Now suppose the CEO joins with a salary of ₹5,00,000/month:

Python Code

import numpy as np

salaries = [32000, 35000, 37000, 38000, 40000,
            42000, 45000, 47000, 50000, 54000]

print("Mean (no CEO)  :", np.mean(salaries))        # 42000.0

salaries_with_ceo = salaries + [500000]
print("Mean (with CEO):", np.mean(salaries_with_ceo))  # 83636.36
print("Median (robust):", np.median(salaries_with_ceo)) # 42000.0

Industry Applications of Mean

When to Use the Mean

• Data is normally distributed (symmetric, bell-shaped)
• No significant outliers present
• Variables are continuous (height, weight, temperature, exam scores)
• You need the value in further calculations (standard deviation, regression)

The Median (Middle Value)

The median is the middle value in a sorted dataset. Exactly half the values fall above it and half fall below. Because the median only depends on position — not magnitude — it is completely unaffected by extreme outliers.

Formula

Real Example — House Prices in a City

Seven properties recently sold in a neighbourhood (₹ Lakhs):

Python Code

import numpy as np

house_prices = [45, 52, 47, 55, 50, 48, 420]

print("Mean  :", np.mean(house_prices))    # 109.57 — misleading!
print("Median:", np.median(house_prices))  # 50.0  — accurate ✓

Industry Applications of Median

When to Use the Median

• Data is skewed (not symmetric — income, property prices, wealth)
• Outliers are present and significant
• Ordinal data — values have order but distances are not equal (e.g., survey ratings 1–5)
• Reporting typical experience rather than a mathematical average

The Mode (Most Frequent Value)

The mode is the value that appears most frequently in a dataset. Unlike the mean and median, the mode can be used with categorical (non-numeric) data. A dataset may have one mode, two modes (bimodal), multiple modes, or no mode at all.

Real Example — Retail Shoe Store

A shoe store records sizes purchased by 238 customers to decide which size to order the most:

Real Example — Favourite Colour Survey (Categorical Data)

A school surveys 200 students: "What is your favourite colour?"

Unimodal, Bimodal & Multimodal Distributions

Python Code

from scipy import stats
import numpy as np

shoe_sizes = [6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5, 10, 10.5, 11]
counts      = [5, 8,  15, 22, 38, 45, 40, 30,  10,  10,  5]

data = np.repeat(shoe_sizes, counts)
print("Mode:", stats.mode(data).mode[0])   # 8.5

# Categorical example
colours = ["Blue"]*72 + ["Red"]*58 + ["Green"]*35
print("Favourite:", stats.mode(colours).mode[0])  # Blue

Industry Applications of Mode

When to Use the Mode

• Data is categorical (colours, brands, yes/no, city names)
• You need to identify the most popular item or preference
• Detecting clusters or groups in data (bimodal = two distinct groups)
• Quality control — finding the most common defect type
• Imputing missing values in categorical ML columns

Skewness — How Distribution Shape Changes Everything

The relationship between mean, median, and mode reveals the shape of a distribution. Understanding skewness is essential for choosing the right measure and avoiding misleading analysis.

Complete Real Example — Student Exam Scores

A class of 20 students takes a maths exam (out of 100). Here are their scores:

Interpretation for the Teacher

Decision Guide — Which Measure to Choose?

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