What is the difference between mean, median, and mode?

Mean (average): sum all values, divide by count. Sensitive to outliers. Median: middle value when sorted. Robust to outliers. Mode: most frequent value — can have multiple modes or none. Example set {1, 2, 2, 3, 100}: Mean = 21.6 (skewed by 100). Median = 2 (middle value, unaffected by 100). Mode = 2 (appears twice). When data has extreme outliers, median is a better 'typical' value than mean.

When should I use mean vs median vs mode?

Mean: use for symmetric, normally distributed data without outliers (e.g., heights, exam scores in a large class). Median: use for skewed distributions or data with outliers (e.g., household income — a few billionaires skew the mean far above the typical experience; US median household income is ~$80K while the mean is ~$105K). Mode: use for categorical data (most popular color, most common job title) or when identifying the most frequent value matters.

What is a weighted average (weighted mean)?

A weighted mean assigns different importance (weights) to each value. Formula: Σ(value × weight) ÷ Σ(weights). Example: grade calculation with homework (20%), midterm (30%), final (50%): homework 85, midterm 78, final 92 → (85×0.20 + 78×0.30 + 92×0.50) = 17 + 23.4 + 46 = 86.4. GPA calculations use credit hours as weights.

What is the range and why does it matter?

Range = Maximum − Minimum. It measures the total spread of data but is highly sensitive to outliers — a single extreme value changes the range completely. Example: {10, 12, 11, 13, 95} has a range of 85, which misrepresents the typical spread (most values are 10–13). For a better spread measure, use interquartile range (IQR = Q3 − Q1) or standard deviation.

Mean, Median & Mode Calculator

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Mean, Median & Mode — Definitions

Mean (Arithmetic Average)

x̄ = Σxᵢ / n — add all values and divide by count. Sensitive to outliers.

Median

Sort values. If n is odd: middle value. If n is even: average of the two middle values.
Robust to outliers; preferred for skewed distributions.

Mode

The value(s) that appear most often. Can be: no mode, unimodal, bimodal, or multimodal.

Frequently Asked Questions

What is the mean of {4, 8, 15, 16, 23, 42}?

Sum = 4+8+15+16+23+42 = 108. Count = 6. Mean = 108 ÷ 6 = 18.

What is the median of {3, 7, 2, 9, 5}?

Sorted: {2, 3, 5, 7, 9}. Middle value (5 numbers, position 3) = 5. For even count, average the two middle values.

Why is the US median income lower than the mean income?

Because income is right-skewed — a small number of very high earners pull the mean upward, while the median reflects the midpoint of all earners. Median income better represents the "typical" American household's financial reality.

Range

Range = Max − Min. A simple spread measure.

Quick Example: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23

Measure	Value	How
Mean	19.6	(3+7+5+…+23) / 10
Median	21.5	Sorted → (20+23)/2
Mode	23	Appears 3 times
Range	37	40 − 3

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References

Triola, M.F. (2018). Elementary Statistics (13th ed.). Pearson.
Moore, D.S., McCabe, G.P. & Craig, B.A. (2017). Introduction to the Practice of Statistics (9th ed.). Freeman.

Choosing the Right Measure of Center

Mean, median, and mode each describe the center of a dataset differently. The mean (average) is sensitive to outliers — a dataset {1, 2, 3, 4, 100} has a mean of 22, which doesn't represent the typical value well. The median (middle value when sorted) is robust to outliers — the median here is 3, a much better summary. The mode (most frequent value) is only useful when values repeat; for continuous data it may not exist or may be unhelpful.

Choose mean for normally distributed, symmetric data without outliers (test scores in a large class). Choose median for skewed data or data with outliers (household income, house prices). Choose mode for categorical data (most popular color, most common shoe size). A dataset can have multiple modes (bimodal, multimodal). The relationship between them indicates skewness: if mean > median > mode, the distribution is right-skewed.

When to Use Each Measure

Measure	Best For	Sensitive to Outliers?
Mean	Symmetric distributions	Yes — avoid if outliers present
Median	Skewed data, income, prices	No — preferred for skewed data
Mode	Categorical or nominal data	No — but may not be unique