Mean vs. Median vs. Mode

Q: What is the mode?

The mode is the value that occurs most frequently in a data set. Unlike mean and median, the mode can be used with categorical (non-numeric) data — for example, the most popular car color.

The mean is the arithmetic average (sum ÷ count). The median is the middle value when data is sorted. The mode is the value that appears most often. For symmetric data they are roughly equal; for skewed data they can differ significantly — which is why choosing the right measure matters.

Mean vs. Median

	Mean	Median
Definition	Sum of all values divided by count	Middle value of sorted data
Formula	$\bar{x} = \frac{\sum x_i}{n}$	Middle position: $\frac{n+1}{2}$ th value
Uses all data?	Yes — every value affects the mean	No — only the middle value(s)
Affected by outliers?	Yes — sensitive to extreme values	No — resistant to outliers
Best for	Symmetric distributions without outliers	Skewed distributions or data with outliers
Example: {1, 2, 3, 4, 100}	Mean = 22	Median = 3
Data type	Quantitative only	Quantitative (ordinal for median)

When to Use Each

Use Mean when...

Data is roughly symmetric with no major outliers
You need a measure that accounts for every data point
Further statistical calculations (standard deviation, z-scores)
Financial averages like average price or average return

Use Median when...

Data is skewed (incomes, house prices, wait times)
There are significant outliers
You want a 'typical' value that isn't pulled by extremes
Ordinal data where arithmetic mean doesn't apply

Examples

Symmetric data

Test scores: {72, 78, 82, 85, 88}. Mean = 81, Median = 82. They are close because the data is roughly symmetric.

Skewed data

Salaries: {$30k, $35k, $40k, $42k, $250k}. Mean = $79.4k, Median = $40k. The mean is pulled up by the $250k outlier, making the median a better measure of 'typical' salary.

Mode example

Shoe sizes sold: {8, 9, 9, 10, 9, 11, 10, 9}. Mode = 9 (appears 4 times). The mode tells the store which size to stock most.

Common Confusion Points

Students often default to the mean for every situation. But for skewed data like income distributions, the median is almost always more informative. 'The average American salary' is misleading — the median salary tells a more accurate story.

A data set can have no mode (all values unique), one mode (unimodal), or multiple modes (bimodal, multimodal). The mean and median always have exactly one value.

Frequently Asked Questions

Which is better: mean or median?

Neither is universally better. Use the mean for symmetric data when you want a measure influenced by every value. Use the median for skewed data or when outliers would distort the average. In practice, report both to give a complete picture.

Can mean, median, and mode all be different?

Yes. In a right-skewed distribution (like income), the mode < median < mean. In a left-skewed distribution, mean < median < mode. They are equal only for perfectly symmetric distributions.

What is the mode?