Statistics › 📊 Describing Data

Mean, Median & Mode

Three ways to name the "center" of a data set — and how to tell which one is telling you the truth.

Intro StatisticsAP Statistics level
💡
The big idea: Given a pile of numbers, three classic summaries claim to capture the "typical" value: the mean (the balance point), the median (the middle of the ordered data), and the mode (the most frequent value). They often disagree — and the disagreement is informative. The mean gets tugged around by extreme values; the median shrugs them off. Knowing which measure answers your question, and why, is the core skill of describing data.
🎯 By the end, you'll be able to
  • Define the mean, median, and mode and compute each from a small data set
  • Explain why the mean is the balance point and gets pulled toward outliers
  • Explain why the median is robust to outliers and skew
  • Choose the appropriate measure of center for symmetric, skewed, or categorical data
📎 You should already know
  • Reading a number line and ordering values from low to high
  • Basic arithmetic: sums and division

What do we mean by 'the center'?

Suppose you have a pile of numbers — test scores, house prices, wait times — and someone asks for a single number that captures the typical value. That one summary is a measure of center.

There are three classic ones, and they can disagree with each other. Knowing which one to trust for a given data set is the whole skill of this lesson.

🔑 Three measures of center
  • Mean — the everyday average: add up all the values and divide by how many there are.
  • Median — the middle value once the data is put in order.
  • Mode — the value that appears most often.

Mean: the balance point

The mean is what most people simply call 'the average'. You add every value and divide by the count \(N\):

\[ \bar{x} = \frac{1}{N}\sum_{i=1}^{N} x_i \]
The mean: the sum of all the values divided by how many values there are.

Physically, the mean is the balance point of the data. Picture each value as an equal weight sitting on a number line: the mean is the spot where the line would balance on a fingertip. Because every single value pulls on that balance point, one far-off value can tug it a long way — a fact we will see live in a moment.

Median: the middle of the ordered data

To find the median, sort the values from smallest to largest and take the one in the middle.

  • With an odd count there is a single middle value — for \(3, 5, 8\) the median is \(5\).
  • With an even count, average the two middle values — for \(3, 5, 8, 10\) the median is \((5+8)/2 = 6.5\).

The median cares only about position in the ordered list, not about how far away the extremes are. That is exactly what makes it robust to outliers.

Mode: the most common value

The mode is simply the value that shows up most often. A data set can have one mode, several, or none at all if every value is unique.

The mode is also the only measure of center that works for categorical data — things you can count but not average. 'The most common blood type is O' or 'the most-ordered flavor is vanilla' are statements about the mode; there is no meaningful 'mean flavor'.

🎮 Interactive: watch the mean chase an outlier LIVE
The dots are your data on a 0–100 number line. The amber fulcrum below the line marks the mean; the green pin above marks the median. Drag a dot out toward the edge, or press 'Add outlier' — the mean slides after it while the median holds its ground.
✨ The mean chases; the median resists

Drag one dot far out to the right in the tool above. The mean — the balance point — slides toward it, because that extreme value pulls hard on the average. The median barely moves: shifting one point further out does not change which value sits in the middle.

This is the single most useful fact about center: the mean is sensitive to outliers, the median is not.

So which one should you report?

  • Roughly symmetric data with no wild values — the mean is efficient and familiar.
  • Skewed data, or data with outliers (incomes, house prices, response times) — the median gives a more honest 'typical' value.
  • Categorical data (colors, brands, yes/no) — only the mode makes sense.

A quick tell: if the mean and median are far apart, the data is skewed, and you should think hard about which number actually answers the question you were asked.

📝 Worked example: Seven employees receive year-end bonuses (in thousands of dollars): 3, 4, 4, 5, 6, 7, 40. Find the mean, median, and mode, and decide which best describes a typical bonus.
  1. Order the data (already sorted): 3, 4, 4, 5, 6, 7, 40.
  2. Mean = sum / N = (3 + 4 + 4 + 5 + 6 + 7 + 40) / 7 = 69 / 7 ≈ 9.9 thousand.
  3. Median = the middle (4th) of the 7 ordered values = 5 thousand.
  4. Mode = the most frequent value = 4 (it appears twice).
✓ The single $40k bonus drags the mean up to about $9.9k — well above what almost everyone actually received. The median of $5k describes a typical bonus far better, because it ignores how extreme that one outlier is.

Check your understanding

1. Which measure of center uses every value and acts as the data's balance point?
The mean is the sum divided by the count, so every value pulls on it — it is literally the balance point of the data.
2. A country's household incomes are strongly right-skewed, with a few very high earners. Which single number best describes a 'typical' income?
The high earners inflate the mean. The median sits at the middle of the ordered incomes and resists those outliers, so it better reflects a typical household.
3. For the data 2, 3, 3, 6, 11, the mean is 5 and the median is 3. Adding one more value, 40, to the set will…
New set 2, 3, 3, 6, 11, 40: the mean rises from 5 to 65/6 ≈ 10.8 (up ~5.8), while the median only moves from 3 to (3+6)/2 = 4.5 (up 1.5). The outlier tugs the mean far more.
4. Students name their favorite ice-cream flavor (vanilla, chocolate, mint…). Which measure of center is meaningful here?
Flavors are categories, not numbers, so you cannot add or order them to get a mean or median. Only the mode — the most-chosen flavor — makes sense.
✅ Key takeaways
  • The mean adds all values and divides by the count; it is the balance point and uses every value.
  • The median is the middle of the ordered data; it is robust to outliers and skew.
  • The mode is the most frequent value, and it is the only measure of center that works for categorical data.
  • Outliers and skew pull the mean but barely move the median — a large gap between them signals skew.
  • Report the median for skewed data, the mean for roughly symmetric data, and the mode for categories.