Unimodal — Definition, Formula & Examples
Unimodal describes a distribution or data set that has exactly one peak — one value or region that occurs most frequently. When you graph a unimodal distribution, the frequencies rise to a single highest point and then fall.
A distribution is unimodal if its probability density function or frequency histogram possesses a single local maximum (mode). The function increases monotonically up to the mode and decreases monotonically after it.
How It Works
To determine whether a distribution is unimodal, look at its histogram or frequency plot. Count the number of distinct peaks. If there is only one peak, the distribution is unimodal. A normal (bell-shaped) distribution is the most common example, but unimodal distributions do not have to be symmetric — they can be skewed left or skewed right and still have just one peak.
Worked Example
Problem: A teacher records test scores for 20 students: 55, 60, 62, 65, 68, 70, 72, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 88, 90, 95. Determine whether the distribution is unimodal.
Group into intervals: Create bins of width 10: 50–59 (1), 60–69 (4), 70–79 (8), 80–89 (5), 90–99 (2).
Identify peaks: The frequencies rise from 1 to 4 to 8, then fall from 8 to 5 to 2. There is exactly one highest bin (70–79 with frequency 8).
Answer: The distribution is unimodal with its single peak in the 70–79 range.
Visualization
Why It Matters
Recognizing that a distribution is unimodal tells you important things about the data: it likely represents one underlying group rather than a mixture of distinct populations. In AP Statistics and introductory college courses, describing shape — unimodal, bimodal, symmetric, or skewed — is the first step in any distribution analysis.
Common Mistakes
Mistake: Assuming unimodal means the same thing as symmetric.
Correction: A distribution can be unimodal and heavily skewed. Unimodal only requires one peak — it says nothing about whether the left and right sides are mirror images.