GCSE · Maths · AQA · Spec 8300

Use statistics to describe a population

Two groups of friends both take 15 minutes, on average, to walk to school. So are the two groups the same? Put them on one line and find out.

Place it, then check

Same average. Same group?

Two made-up groups of five friends timed their walk to school, in minutes. Group A: 12, 14, 15, 16, 18. Group B: 5, 9, 15, 21, 25. For each group, slide its shortest time, its average time and its longest time onto the line. (Handy: in both groups the middle value and the mean come out the same.)

A min — Group A: shortest timeA avg — Group A: average timeA max — Group A: longest timeB min — Group B: shortest timeB avg — Group B: average timeB max — Group B: longest time

What does an average really tell you?

The class slept 7.5 hours. Or did it?

In a made-up class of 30 students, everyone recorded how many hours they slept last night. The mean is 7.5 hours.

Which of these is closest to what you think right now?

Most students in the class slept about 7.5 hours.Pick any student in the class and they slept 7.5 hours.The mean of 7.5 hours fully describes the class's sleep.7.5 hours is typical for the class, but I can't tell how much students differ.

How sure are you?

I'm sureI think soI'm guessing

Most students in the class slept about 7.5 hours.

This is a common misconception

Not necessarily. A mean of 7.5 hours fits a class where everyone slept 7 or 8 hours. It fits just as well a class where half slept 5 hours and half slept 10. The mean alone can't tell you which class you've got. That's exactly what a measure of spread is for.

Pick any student in the class and they slept 7.5 hours.

This is a common misconception

The mean belongs to the whole class, not to any one student. It's quite possible that nobody slept exactly 7.5 hours. Watch out for phrases like 'the average student': there may be no such person.

The mean of 7.5 hours fully describes the class's sleep.

This is a common misconception

It's only half a description. It gives the class's typical level, but not whether the students slept similar amounts or wildly different ones. You saw that on the number line: two groups, one average, two very different stories.

7.5 hours is typical for the class, but I can't tell how much students differ.

This matches the evidence

Exactly right. The mean summarises the class as a group. To finish the description you'd add how spread out the sleep times were, for example their range.

An average describes the group, not any one member. And on its own it's only half the story.

Spot a complete description

Which of these actually describes the runners?

Sort each description. Is it complete, or what is it missing?

Still to sort

Complete description (0)

A centre and a spread, each saying what was measured, with units.

Where the line is: The numbers alone aren't enough: a complete description reads as a sentence about the runners' times.

Needs a spread (0)

Says where the times are centred, but not how much they vary.

Where the line is: It may be perfectly worded, but with only an average you can't tell whether the runners finished close together or far apart.

Needs a centre (0)

Says how much the times vary, but not where they typically sit.

Where the line is: Knowing the times spread over 18 minutes doesn't tell you whether the runners were typically fast or slow.

Needs context and units (0)

Has both statistics, but doesn't say what was measured or in what units.

Where the line is: Both statistics are there, but a reader can't tell they are about finishing times, measured in minutes.

5 of 5 still to sort.

All 20 runners in a made-up fun run had their finishing times recorded. The times have a mean of 32 minutes and a range of 18 minutes. Here are five ways people described them.

Your turn to write it

Describe the eggs in one box

The masses of all 8 eggs in a box (made-up data) are, in grams: 58, 60, 61, 61, 62, 63, 64, 67. Describe the masses of the eggs in this box. Use statistics in your answer. [2 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

A correct average, stated as the mass of the eggs, with units: mean 62 g (or median 61.5 g, or mode 61 g). (1 mark)
A correct measure of spread, stated as a mass with units: the masses have a range of 9 g (from 58 g to 67 g). (1 mark)

Model answer

The eggs in this box have a mean mass of 62 g, and their masses have a range of 9 g, from the lightest at 58 g to the heaviest at 67 g.

'62' on its own describes nothing. 'The eggs have a mean mass of 62 g' describes the eggs. Put every statistic in the words and units of the context.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

One average can't describe a group on its own. Say where the group's values are centred, how much they vary, and say it in the words of the context.

What you need to know

A population is the whole group you are describing: every member of it, like all the friends in a group or all the matches in a season.
An average (such as the mean) gives the group's typical value: where its values are centred.
A measure of spread (such as the range) says how much the values vary, how different the members are from each other.
An average summarises the group as a whole. It doesn't tell you the value for any one member.
State each statistic in context, with units: 'group A's mean walking time is 15 minutes', not just '15'.

The big picture

A population is the whole group you're describing, and statistics let you describe it in a sentence. Say where its values typically sit (an average) and how much they vary (a measure of spread, such as the range), and put both in the context and units of the data. The average describes the group, not any one member, and on its own it tells only half the story.

Key points

1Two groups can share the same average and still be very different: both walking groups averaged 15 minutes, but one spread over 6 minutes and the other over 20.

2A small spread means the members are alike; a large spread means they differ a lot from each other.

3'The average person' is a summary, not a real member. There may be nobody whose value equals the average.

4A good description is one sentence about the group: what is typical, how much it varies, in the context's words and units.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

In its 6 matches this season, a made-up netball team scored 14, 18, 9, 22, 17 and 16 goals. Use statistics to describe the team's scoring this season.

⚠ Watch out

Giving bare numbers. 'Mean = 4.2, range = 3' could be about anything. 'The members of the chess club played a mean of 4.2 games last month, with a range of 3 games' actually describes the club.

🧠

Memory hook

Centre, spread, context. The average says where the group sits, the spread says how far it stretches, and the context says what you're talking about.

✓

Check yourself

All 7 houses on a made-up street hold 1, 2, 2, 3, 4, 4 and 5 people. Describe the households on this street using an average and a measure of spread, in context.

Flashcards

(6)

What is a population in statistics?

The whole group you want to describe: every member of it.

What does an average tell you about a population?

Where its values are centred: the group's typical level.

What does a measure of spread, such as the range, tell you?

How much the members' values differ from each other.

Two groups have the same mean. Must they be similar?

No. Their values can be spread very differently, like 12 to 18 minutes compared with 5 to 25 minutes.

The mean number of people per household in a town is 2.4. Does a typical household have 2.4 people?

No household can. The mean describes the town's households as a group, not any single household.

What turns a statistic into a description?

Stating it in context, with units: what was measured, and for which group.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Same average. Same group?

The class slept 7.5 hours. Or did it?

Which of these actually describes the runners?

Describe the eggs in one box

More AQA GCSE Maths topics