Range, median, mean and mode

Data on its own is just a list of numbers. To make sense of it, we summarise it: we find a single value that represents the typical case, and a measure of how spread out the values are. This year you learn the four core summaries, the range, median, mean and mode, and, just as importantly, how to choose the one that tells the truest story about a data set.

These measures are everywhere, from a class average to a typical house price to the most popular shoe size. Knowing how to calculate each is the first step; knowing which one to trust in a given situation is the skill that separates careful reasoning from misleading statistics.

Centre and spread

Two questions describe almost any data set: where is its centre, and how spread out is it? The mean answers the first by sharing the total equally. Add every value and divide by how many there are, and you get the balancing point of the data. The range answers the second by subtracting the smallest value from the largest, giving a simple measure of how far the data stretches.

The mean is powerful because it uses every value, but that is also its weakness. One unusually large or small value drags it in that direction. The range is quick to find but blunt, since it depends only on the two extreme values and ignores everything in between. Each summary captures something real and misses something else, which is why we keep several.

The median and the mode

The median is the middle value once the data is in order, so half the values sit below it and half above. The mode is simply the value that occurs most often. Both are easy to find by hand, and both resist the pull of extreme values far better than the mean, which makes them valuable when a data set has unusual outliers.

Choosing the right measure is the heart of this topic. For a set of house prices with one enormous mansion, the mean is pulled upward and overstates the typical price, while the median stays close to what most houses actually cost. For shoe sizes a shop should stock, the mode matters most, since it names the size that sells the most. The maths of each measure is straightforward; deciding which one represents the data honestly is the real work, and it is a judgement you will make again and again.

Teaching tip: the clearest way to show why the choice of measure matters is with a small, deliberately skewed data set. List the salaries of a few staff plus one chief executive on a huge wage, then calculate the mean and the median together. Seeing the mean land above almost every individual salary, while the median sits right among them, makes the idea of a misleading average concrete and memorable.

A frequent slip is forgetting to order the data before finding the median. Build the habit of rewriting the values in order first, every time, since the middle of an unordered list is meaningless and this is where most errors creep in.

Why the mean balances

There is a deeper way to picture the mean. Imagine the values as weights on a beam: the mean is the point where the beam balances. Every value below the mean pulls one way and every value above pulls the other, and these pulls always cancel exactly, because the distances from the mean add up to the same amount on each side. This is why a single far-off value tips the balance and drags the mean toward it.

The median with even and odd counts

Finding the median depends on how many values there are. With an odd number of values, one value sits exactly in the middle once the data is ordered, and that is the median. With an even number there is no single middle, so the median is the average of the two central values, which can be a half-value that does not appear in the data at all. Either way, ordering the data first is essential.

Choosing and justifying a measure

The real skill is deciding which summary to trust. When data is fairly symmetric, the mean is reliable and uses every value. When one extreme outlier is present, the mean is pulled away and the median better represents the typical case. When the question is about what is most common, such as the shoe size a shop should stock, the mode is the measure that matters. Justifying the choice is as important as the calculation.