Mean, median and mode: drag the points, watch the markers move

How to use this simulation

1. In raw-data mode, drag any dot on the number line: the mean, median and mode markers update instantly.
2. Tick "Add an outlier value": the mean marker jumps a long way, while the median marker barely shifts.
3. Watch the shaded band: it shows the mean ± 1 standard deviation, and it widens as the data spreads out more.
4. Switch to grouped-data mode and slide each class interval's frequency: watch the median class and modal class change.
5. Set one class frequency to zero: its histogram bar disappears, and the median and mode both shift toward the surviving classes.

A class test and a very large salary

Seven friends score 2, 4, 5, 5, 7, 9, 10 out of 10 on a class test. If someone asks "how did your group do", and you can only answer with one number, what do you say? Finding that one representative number, in three different ways, is what this page is about.

Now imagine a small office where nine employees earn roughly the same modest salary, and a tenth person — the owner — earns many times more than everyone else. Report the “average salary” and you get a number nobody in the office is actually close to. Report the “median salary” instead, and that one very large number barely moves it. That gap is one of the most useful lessons in statistics.

The simulation above shows exactly this: dots sit on a number line, and ticking the outlier switch adds one far-away value, just like that office owner's salary.

Starting from zero: central tendency and spread

Summarizing a set of numbers with one typical value is called a measure of central tendency. There are three common ones: the mean, the median and the mode.

The mean is every value added up and divided by how many there are: x̄ = (Σx) ÷ n. The median is the middle value once the data is sorted from smallest to largest (the average of the two middle values if there is an even number of them). The mode is whichever value occurs most often.

Two groups can share the same mean and still be spread out very differently. That is measured separately, by a measure of dispersion: the simplest is the range (highest minus lowest), and a more careful one is the standard deviation, which averages how far every value sits from the mean.

Key terms

Get the vocabulary straight before the formulas; definition questions come straight from this table.

The mean, median and mode formulas

Each formula follows directly from its definition.

Why the median resists an outlier

Calculating the mean needs the exact magnitude of every value, so one unusually large or small value pulls the mean toward itself. Calculating the median only needs the order of the values — where each one sits once they are sorted — not how large it is. An outlier sits at one end of the sorted list, but the position in the middle barely moves.

In the example (7 values): mean 6, median 5. Adding one outlier of 35 (now 8 values): the mean jumps to 9.625 — a huge change. But the median becomes just 6 — barely different, because it is still the average of the two middle values, and those two values barely changed.

That is exactly why real-world reports often use "median" instead of "mean" wherever extreme values are common, such as income or house prices: a country's "median income" is reported precisely because a handful of extremely wealthy people would drag the mean far above what a typical person earns.

Grouped data: summarizing a long list

With hundreds of individual values, writing each one down is impractical. Instead, the values are split into equal-width class intervals (say, 0–5, 5–10, 10–15, …) and only the count in each class — the frequency — is recorded. That summary is called grouped data.

Grouped data no longer records each exact value, only which class it fell into, so the formulas make one assumption: every value in a class is treated as sitting at that class's midpoint. That gives the mean formula its shape, while the median and mode use an interpolation formula instead, which assumes the values are spread evenly across the class.

Which measure to use, and when

All three measures are correct, but they are not equally useful in every situation.

Try these in the simulation

Predict the outcome before you run the simulation, then check.

• In raw-data mode, drag two dots onto the exact same value: the mode readout locks onto that value, and the dots visibly stack on the number line.
• Spread every dot to a different value, with no repeats at all: the mode readout reports that no value repeats.
• Turn the outlier on and watch the shaded band (mean ± 1 SD): it widens a lot once the outlier is added, because the spread has grown.
• In grouped mode, push one middle class's frequency to the maximum: that class becomes the modal class, and the median class often lands there too.
• In grouped mode, set the first and last class frequencies to zero: the remaining three classes still give a valid median and mode, using the exact same formulas.

Solved problems

Every solution sorts the data first, then states the formula, then substitutes — write it the same way in an exam.

Quick reference: the standard deviation, step by step

For problem 1's data, here is every value's deviation from the mean and its square. Averaging the squares and taking the square root gives σ.

Mistakes almost everyone makes

Avoiding these keeps exam marks from slipping away on mean, median and mode questions.

• Forgetting to sort the data before finding the median — the definition of median depends entirely on the sorted order.
• Taking only one middle value when n is even, instead of averaging the two middle values.
• Thinking the mode is the largest value — the mode is the most frequent value, not the biggest one.
• Using n instead of n/2 in the grouped median formula — the formula needs half the total frequency.
• Averaging the deviations directly in a standard deviation calculation, without squaring first — that always gives zero, because positive and negative deviations cancel.
• Calling the mean "wrong" just because one outlier changed it a lot — the mean is not wrong, only sensitive to outliers, which is exactly why the median is more trustworthy for that kind of data.

Mean, median and mode in real life

These measures show up in newspapers and reports well outside the exam hall.

• A country reports its "median income", not its "average (mean) income", because a small number of extremely wealthy people would drag the mean far above what a typical person earns.
• A shoe shop tracks its best-selling size (the mode) so it can keep more stock of that size on the shelf.
• Exam results are analysed using both the class mean and the class median; a big gap between them signals that some students scored unusually high or low.
• A weather forecast reporting a month's average temperature also needs its spread, since two months can share the same mean temperature while one stays steady and the other swings wildly.
• Factories aim for a low standard deviation in manufactured part sizes, since a small standard deviation means most parts come out close to the target size.

Exam tips

Mean, median and mode questions appear on almost every statistics syllabus. Two habits earn marks reliably: always sort the data before finding the median, and always check whether n is odd or even before picking the median rule.

Revision: one-screen summary

The night before an exam, this list and the tables above are enough to refresh everything.

• Mean x̄ = (Σx)/n — uses every value, sensitive to outliers.
• Median = the middle value of sorted data — depends on position, resistant to outliers.
• Mode = the most frequent value — none if nothing repeats, more than one if there is a tie.
• Range = highest − lowest; standard deviation σ = √(Σ(x−x̄)²/n).
• Grouped data finds the mean from midpoints, and the median and mode from an interpolation formula.
• When outliers are present, the median is the more trustworthy representative value.

Frequently asked questions