Mean, Median, Mode, and Range
Calculate mean, median, mode, and range for data sets. Understand the effect of outliers on each measure and choose the most appropriate measure of center for a given context.
A lesson of Pre-Algebra, in Statistics & Probability, about 10 minutes.
Measures of Center and Spread
By the end of this lesson, you will be able to calculate and compare mean, median, and mode, and use range and IQR to describe spread.
Measures of Center
Mean (average): $\bar{x} = \dfrac{\text{sum of values}}{n}$
Median: the middle value when data is ordered. If even $n$: average the two middle values.
Mode: the most frequent value. There can be no mode, one mode, or multiple modes.
When to use which:
- Symmetric data, no outliers → mean
- Skewed data or outliers → median (resistant to extremes)
Quick Example
Data: 5, 8, 3, 8, 11, 7, 3, 8
Ordered: 3, 3, 5, 7, 8, 8, 8, 11
Mean = $(5+8+3+8+11+7+3+8)/8 = 53/8 = 6.625$
Median = $(7+8)/2 = 7.5$ (two middle values)
Mode = 8 (appears 3 times)
Measures of Spread
Range: Max − Min. Simple but affected by outliers.
$11 - 3 = 8$
IQR (Interquartile Range): $Q_3 - Q_1$ — range of middle 50%.
From the ordered data: Q₁ = 4, Q₃ = 8 → IQR = 4
Outlier rule: below $Q_1 - 1.5(\text{IQR})$ or above $Q_3 + 1.5(\text{IQR})$.
Effect of Outliers
Data: 10, 11, 12, 13, 50
Mean = 19.2 (pulled up by 50). Median = 12. Mode = none.
The median (12) better represents the typical value than the mean (19.2).
Summary
Mean: average; use when data is symmetric. Median: middle; use when outliers are present. Mode: most frequent.
Range: Max − Min. IQR: Q₃ − Q₁ (resistant to outliers).