Mean, Median, and Mode

Overview

The mean, median, and mode are three measures of central tendency—ways to describe the "center" of a dataset.

Definitions

Measure	Definition	When to Use
Mean	Arithmetic average	Symmetric data, no outliers
Median	Middle value	Skewed data, ordinal data
Mode	Most frequent value	Categorical data, bimodal

Formulas

Arithmetic Mean

For a dataset with $n$ values:

$$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum x_i}{n}$$

Population Mean

$$\mu = \frac{\sum x_i}{N}$$

Weighted Mean

$$\bar{x}_w = \frac{\sum (w_i \times x_i)}{\sum w_i}$$

Median Calculation

1. Order the data from smallest to largest
2. If $n$ is odd: median = middle value
3. If $n$ is even: median = average of two middle values

$$\text{Median position} = \frac{n + 1}{2}$$

Examples

Odd number of values (5 values): 2, 5, 7, 9, 12

• Position: $(5+1)/2 = 3$
• Median = 7 (the 3rd value)

Even number of values (6 values): 2, 5, 7, 9, 12, 15

• Positions: 3rd and 4th values
• Median = $(7 + 9)/2 = 8$

Mode

The value(s) that appear most frequently.

Type	Description
Unimodal	One mode
Bimodal	Two modes
Multimodal	Multiple modes
No mode	All values appear equally

Properties

Mean Properties

• Uses all data values
• Affected by outliers
• Sum of deviations from mean = 0: $\sum (x_i - \bar{x}) = 0$
• Can be calculated for interval/ratio data

Median Properties

• Not affected by outliers
• 50% of data above, 50% below
• Can be calculated for ordinal, interval, or ratio data

Mode Properties

• Can be used for categorical data
• Not affected by outliers
• May not exist or may not be unique

Relationship in Distributions

Distribution	Relationship
Symmetric	Mean ≈ Median ≈ Mode
Right-skewed	Mode < Median < Mean
Left-skewed	Mean < Median < Mode

Choosing the Right Measure

Situation	Best Measure
Normal distribution	Mean
Skewed distribution	Median
Outliers present	Median
Categorical data	Mode
Ordinal data	Median or Mode

Examples

Example 1: Test Scores

Data: 85, 90, 78, 92, 88, 85, 95, 85

$$\text{Mean} = \frac{85 + 90 + 78 + 92 + 88 + 85 + 95 + 85}{8} = \frac{698}{8} = 87.25$$

Ordered: 78, 85, 85, 85, 88, 90, 92, 95

$$\text{Median} = \frac{85 + 88}{2} = 86.5$$ $$\text{Mode} = 85 \text{ (appears 3 times)}$$

Example 2: With Outlier

Salaries: $40k, $42k, $45k, $48k, $200k

$$\text{Mean} = \frac{40 + 42 + 45 + 48 + 200}{5} = \$75\text{k}$$ $$\text{Median} = \$45\text{k} \text{ (3rd value)}$$

The median ($45k) better represents typical salary.

Example 3: Weighted Mean

Grades: Homework (20%): 90, Midterm (30%): 80, Final (50%): 85

$$\bar{x}_w = \frac{(0.20 \times 90) + (0.30 \times 80) + (0.50 \times 85)}{0.20 + 0.30 + 0.50}$$ $$= \frac{18 + 24 + 42.5}{1} = 84.5$$