Interactive guide
Three ways to answer "what's typical here?"
Mean, median, and mode all try to summarize a pile of numbers with a single one, but they disagree about what "typical" means, and that disagreement is the entire point of having three of them. Drag some points around and watch.
The mean chases every point; the median mostly ignores them
The mean is the arithmetic average: add every value, divide by how many there are. Because every single point contributes to that sum, moving any point by some amount moves the mean too, proportionally.
The median is just the middle value once everything is sorted; it only cares about order, not size. Drag the largest point from 9 to 900 and the median doesn't move at all; it's still the same point sitting in the middle.
The mode is simply whichever value shows up most often: the tallest stack below. With continuous, all-different data there may be no mode at all; it only means something once values repeat.
11 points below, snapped to whole numbers so repeats (and a mode) are actually possible. Drag any of them.
The three definitions
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1
Mean = sum ÷ count
x̄ = (x₁+x₂+...+xₙ) / n. Sensitive to every value, including extreme ones.
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2
Median = the middle, once sorted
Odd n: the single middle value. Even n: the average of the two middle values. Unaffected by how extreme the extremes are.
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3
Mode = the most frequent value
Whatever repeats most. Can be one value, several (tied), or none at all if nothing repeats.
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4
None of them is "the" right answer
Household income is usually reported as a median for exactly this reason: a handful of billionaires would drag the mean far from what's actually typical.
Drag any point along the line; it snaps to the nearest whole number.
a point stacked dots at the same value show up as a tower: the tallest tower is the mode.
Drag a point to see how each measure reacts.
- Mean
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- Median
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- Mode
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Numbers = [ ]
Skewness: the three measures come apart in a fixed order
When a distribution is symmetric, mean, median, and mode all sit in the same place. Stretch a long tail out to one side, though, and they split apart, always in the same order relative to that tail, because the mean is pulled toward it while the mode (the peak of the bars) barely moves.
Try each preset below and watch where the three markers land.
The pattern to remember
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1
Symmetric
Mean = median = mode. No tail pulling in either direction.
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2
Right-skewed tail points right
mean > median > mode. The rare high values pull the mean up more than the median.
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3
Left-skewed tail points left
mean < median < mode. Mirror image: the tail now drags the mean down instead.
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4
Bimodal
Two peaks tie for mode. Mean and median both land in the valley between them, describing neither peak.
Pick a shape and see where mean, median, and mode land.
mean median mode
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- Mean
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- Median
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- Mode
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The jargons
Mean, median, and mode all claim to describe a dataset’s “typical” value — they only agree when the data is symmetric and unimodal, and the rest of this list is about what happens once it isn’t.
The arithmetic average. Uses every value's exact size, which is exactly why it's sensitive to extreme ones.
The middle value in sorted order. Depends only on rank, not magnitude: an extreme value only matters that it's the biggest, not how big.
The most frequent value. The only one of the three that also makes sense for non-numeric (categorical) data, "most common favorite color," for instance.
A measure of a distribution's asymmetry: which direction its tail stretches, and how far. Zero for a symmetric distribution.
A value far from the rest of the data. Disproportionately affects the mean while leaving the median almost untouched: the core reason robust statistics prefer the median for skewed data.
A distribution with two distinct peaks, often a sign that the data is secretly a mix of two different groups, each with its own typical value.