Averages and Range

1 · Data, and why we summarise it

Data is information that has been collected, often as a list of numbers. Each number in the list is called a value, and the whole list is a data set. For example, a teacher asks nine pupils how many minutes it takes them to get to school and writes down their answers: 12, 5, 20, 8, 15, 8, 25, 10, 8. That's a data set with nine values.

A jumbled list is hard to make sense of. Quick: what's a typical journey? What's the longest? Your eyes have to jump around. The first trick statisticians use is simple: put the values in order, from smallest to largest. Try it.

Even in order, nine or twelve numbers are a lot to remember. So we summarise them: we describe the whole data set with just a couple of numbers. There are two big questions to ask about any data set.

• What is a typical value? A number that represents the whole set is called an average. There are three different averages, each with its own idea of "typical".
• How spread out are the values? Are they all bunched close together, or scattered far apart? This is called the spread, and at KS3 we measure it with the range.

The colours above stay the same all the way down the page: plum for the mode, orange for the median, teal for the mean and dark ink for the range. Grey is just raw data.

2 · The mode: the most common value

The mode is the value that appears most often. It's the "most popular" answer. If you ask your class what pet they have and more people say "dog" than anything else, the mode is dog. Another name for it is the modal value ("the modal pet is a dog").

To find it, count how many times each value appears. That count is called the frequency. The value with the highest frequency is the mode.

Below is a class survey of favourite pets. Each dot is one pupil's vote. Tap a column to add a vote (or tick the box and tap to take one away), and watch the plum mode change.

Three things this shows:

• The mode works for words, not just numbers. Pets, colours, shoe brands: you can't add up "dog + cat", but you can still say which is most common. The mode is the only average that works for this kind of non-numerical data.
• There can be more than one mode. If two values tie for most common, both are modes. (Data with two modes is called bimodal.)
• There can be no mode. If every value appears only once, nothing is more common than anything else, so there is no mode.

Try it: find the mode

3 · The median: the middle value

Imagine your class lines up in order of height, shortest to tallest. The person standing exactly in the middle of the line has the median height: half the class is shorter, half is taller. The median is the middle value once the data is in order.

The method: (1) put the values in order; (2) cross off one from each end, again and again, until you reach the middle. Step through it below.

With an odd number of values, one card is left in the middle: that's the median. With an even number, you are left with two middle cards. The median is halfway between them: add them and divide by 2. For 12 and 15 that's (12 + 15) ÷ 2 = 27 ÷ 2 = 13.5. The median doesn't have to be one of the values.

Shortcut for big data sets: with n values (n just means "how many"), the median is the (n + 1) ÷ 2th value in order. For 7 values: (7 + 1) ÷ 2 = 4, the 4th value. For 25 values: the 13th. For 8 values: 4.5, meaning halfway between the 4th and 5th values. Careful: this tells you where the median is, not what it is.

Try it: find the median

4 · The mean: sharing out equally

Five friends have some sweets: 3, 7, 2, 6 and 7. That's not fair! So they put all the sweets in a pile and share them out equally. The pile has 3 + 7 + 2 + 6 + 7 = 25 sweets, and 25 shared between 5 people is 5 each. That fair share, 5, is the mean.

Think of each value as a tower of blocks. The mean is the height you get if you level the towers: move blocks from tall towers to short towers until they're all the same. Drag a tower up or down to change the data, then press Level.

Levelling always gives the same answer as this quick method:

mean = total ÷ how many values

Add up all the values to get the total (all the blocks in one pile), then divide by how many values there are (the number of towers). Notice:

• The mean doesn't have to be one of the values, or even a whole number. 26 blocks in 5 towers gives 26 ÷ 5 = 5.2. That's why "2.4 children" is a perfectly sensible mean, even though no family has 2.4 children!
• If the answer doesn't come out exactly, round it sensibly, e.g. 17 ÷ 3 = 5.666… ≈ 5.7 (to 1 decimal place).
• A value of 0 still counts! It adds nothing to the total but it does count as one of the values.
• When data is given in a frequency table, there's a faster way to find the total: see Frequency tables.

Try it: find the mean

5 · The range: how spread out?

An average only tells half the story. Amir and Bea both play seven basketball games and both have a mean of 10 points a game. Are they the same kind of player? Move the slider and look.

Amir scores between 8 and 11 every game: he's consistent (reliable, always about the same). Bea might score 4 one game and 16 the next. Their averages are identical, but the spread is very different. The simplest measure of spread is the range:

range = largest value − smallest value

A small range means the values are close together (consistent). A large range means they are spread out (varied, less predictable). If a coach needs a player who reliably scores about 10, they'd pick Amir; if they need someone who might win the game with a huge score, maybe Bea.

Try it: find the range

6 · Outliers, and choosing the best average

An outlier is a value that is much bigger or much smaller than the rest: it "lies outside" the group. Here is the weekly pocket money (in £) of nine friends, shown as a dot plot (each dot is one person, stacked above their amount; see Statistical charts for more on these). The cards above it show the same values as a list.

Drag any dot along the line to change the data. Then tick the box to add Zak, who gets £30 a week, and watch what each measure does.

Zak's £30 is an outlier. Look at what happens:

• The mean jumps up, because the mean uses every value, and Zak's big number adds a lot to the total. Now only Zak gets more than the mean, so it's not very "typical".
• The median hardly moves (here it doesn't move at all). It only cares about the middle, and Zak just joins the end of the line.
• The mode doesn't change: Zak's £30 appears only once.
• The range shoots up, because Zak is the new largest value.

Sometimes an outlier is a mistake (someone typed 300 instead of 3.00) and should be checked or removed. Sometimes it's real and important. Either way, it's a reason to think about which average to use.

Try it: which average would you choose?

7 · Working backwards from the mean

The mean is total ÷ how many. Turn that round and you get a very useful fact:

total = mean × how many values

If 4 friends have a mean of 5 sweets each, there must be 4 × 5 = 20 sweets altogether, whatever the individual amounts. This lets you find a missing value: work out what the total should be, then take away the values you know. In the picture, the teal line is the mean you need. Type a guess to see where the mean would end up, then check.

8 · Comparing two data sets

Which team is better? Which class did better in the test? To compare two data sets properly you need two comparisons:

1. Compare an average (usually the mean or median): which set is higher on average?
2. Compare the range: which set is more consistent (smaller range)?

And each comparison must say what it means in the real situation, not just quote numbers. "The Lions' mean is 2.6 and the Tigers' is 1.9" is only half an answer. "The Lions scored more goals on average, because their mean is higher" is a full one. Below are the goals scored by two teams in 10 matches. Build the two sentences.

The best team on average isn't always the most consistent one. Real comparisons often say "Team A scored more on average, but Team B was more consistent." Looking at the whole shape of the data is the next step: see Distributions and correlation.

Practice: mixed questions

New questions every time. If you go wrong, the feedback tries to spot which mistake you made.

Worked examples

Quick check

What you've learned

• Data is a set of values. We summarise it with an average (what's typical) and a measure of spread (how spread out).
• The mode is the most common value. It works for words as well as numbers; there can be several modes or none. Don't give its frequency by mistake.
• The median is the middle value once the data is in order. With an even number of values it's halfway between the two middle ones. It's at position (n + 1) ÷ 2.
• The mean is the fair share: total ÷ how many, like levelling towers of blocks. It needn't be a whole number or one of the values.
• The range = largest − smallest: one number that measures spread. A small range means consistent.
• An outlier drags the mean and stretches the range, but barely moves the median and doesn't affect the mode. With outliers, prefer the median; for categories, the mode; otherwise the mean uses all the data.
• Work backwards with total = mean × how many to find a missing value.
• Compare two data sets using an average and the range, with a sentence that says what each means in context.