Mean, median, and mode are the three primary measures of central tendency. They each describe the “typical” or “average” value in a dataset, but in different ways. Understanding when to use each is critical for proper data analysis.

Mean (Average)

The mean is the sum of all values divided by the number of values. It’s the most commonly used measure of central tendency but can be affected by extreme values.

Mean for Ungrouped Data

Formula:

Mean = (x₁ + x₂ + ... + xₙ) / n
Mean = Σxᵢ / n

Example: Dataset: 10, 15, 20, 25, 30 Mean = (10 + 15 + 20 + 25 + 30) / 5 = 100 / 5 = 20

Mean for Grouped Data (Frequency Distribution)

When data is organized into groups with frequencies:

Formula:

Mean = Σ(fᵢ × xᵢ) / Σfᵢ

where:
fᵢ = frequency of class i
xᵢ = midpoint of class i (for continuous data)
Σfᵢ = total number of observations (N)

Example:

Class Midpoint (x) Frequency (f) f × x
10-20 15 5 75
20-30 25 8 200
30-40 35 7 245
Total 20 520

Mean = 520 / 20 = 26

Key point for continuous data: Use class midpoint as the representative value for each group.

Median (Middle Value)

The median is the middle value when data is arranged in order. It’s robust to outliers and useful for skewed distributions.

Median for Ungrouped Data

Process:

  1. Arrange data in ascending order
  2. If odd number of values: median = middle value
  3. If even number of values: median = average of two middle values

Example (odd n): Dataset: 10, 15, 20, 25, 30 Median = 20 (middle value, 3rd position)

Example (even n): Dataset: 10, 15, 20, 25 Median = (15 + 20) / 2 = 17.5 (average of 2nd and 3rd values)

Median for Grouped Data

Formula:

Median = L + ((N/2 - CF) / f) × h

where:
L = lower boundary of median class
N = total frequency
CF = cumulative frequency before median class
f = frequency of median class
h = class width

Steps:

  1. Find cumulative frequencies
  2. Locate median class (where cumulative frequency ≥ N/2)
  3. Apply formula

Example:

Class Frequency Cumulative
10-20 5 5
20-30 8 13
30-40 7 20
Total 20

Median class location: N/2 = 20/2 = 10 Median class is 20-30 (cumulative frequency 13 ≥ 10)

Median = 20 + ((10 - 5) / 8) × 10 Median = 20 + (5/8) × 10 Median = 20 + 6.25 = 26.25

Mode (Most Frequent Value)

The mode is the value that appears most frequently. It’s the only measure of central tendency for purely categorical data.

Mode for Ungrouped Data

Process: Simply identify the value that appears most often.

Example: Dataset: 10, 15, 15, 20, 20, 20, 25 Mode = 20 (appears 3 times)

Types:

  • Unimodal: One mode
  • Bimodal: Two modes
  • Multimodal: Multiple modes
  • No mode: All values appear with equal frequency

Mode for Grouped Data

For grouped data, we identify the modal class (class with highest frequency).

Formula (Approximate):

Mode = L + ((f₁ - f₀) / (2f₁ - f₀ - f₂)) × h

where:
L = lower boundary of modal class
f₁ = frequency of modal class
f₀ = frequency of class before modal class
f₂ = frequency of class after modal class
h = class width

Example (using data above): Modal class: 30-40 (frequency 7 is highest)

Mode = 30 + ((7 - 8) / (2×7 - 8 - 0)) × 10 Mode = 30 + (-1 / 6) × 10 Mode ≈ 28.33

Note: For grouped data, if there’s minimal information, use the class midpoint as the mode.

Comparing Mean, Median, Mode

Characteristic Mean Median Mode
Definition Average Middle value Most frequent
Ungrouped data Simple to calculate Position-based Count-based
Grouped data Uses frequencies Cumulative formula Highest frequency class
Affected by outliers Yes (sensitive) No (robust) No
Good for Normal distributions Skewed distributions Categorical data
Best use General purposes Skewed/real data Categories/modes

When to Use Each

Use Mean when:

  • Data is approximately normally distributed
  • No extreme outliers
  • You need mathematical properties (algebra, further analysis)

Use Median when:

  • Data is skewed or has outliers
  • You want the “typical” value for highly variable data
  • Income, home prices, or other right-skewed data

Use Mode when:

  • Data is categorical (colors, brands, preferences)
  • Identifying most popular/common value
  • Data distribution is highly multimodal

Relationship: Mean vs Median vs Mode

Symmetrical distribution: Mean = Median = Mode

Right-skewed (positive skew): Mean > Median > Mode (Tail pulls mean to the right)

Left-skewed (negative skew): Mean < Median < Mode (Tail pulls mean to the left)

Quick Reference

Ungrouped Data Formulas

Mean = Σx / n
Median = middle value(s) when sorted
Mode = most frequent value

Grouped Data Formulas

Mean = Σ(f × x) / Σf
Median = L + ((N/2 - CF) / f) × h
Mode = L + ((f₁ - f₀) / (2f₁ - f₀ - f₂)) × h

Alternative Measures of Central Tendency:

Related Concepts:

Applications: