GCSE · Maths · AQA · Spec 8300 · Foundation

Compare distributions — graphical and central tendency/spread

Two buses go to your school. One is usually quicker, but it once took 35 minutes. Which one would you trust?

Higher

Maths · Comparing distributions

Route A or Route B? Twenty journeys to school on each bus

The five-number summaries
	Min	LQ	Median	UQ	Max	IQR
Route A	14	17	19	22	26	5
Route B	12	15	16	18	35	3

The thick line inside each box is the median. Which route is typically quicker?

Route ARoute B

The whiskers stretch from the quickest journey to the slowest. Which route has the bigger range?

Route ARoute B

The box covers the middle half of the journeys. Comparing the boxes, which route is more consistent?

Route ARoute B

Two questions, every time. Which is typically quicker? Compare an average — here, the medians. Which is more consistent? Compare a spread. The range uses only the two most extreme journeys, so one very slow day can stretch it. The IQR uses the middle half, so how slow the slowest day was makes no difference to it. That's why the range and the IQR can disagree.

Maths · Reading data

Zoom in: one week on Route B

Each bar is one day's journey. Tap a bar to read it, then switch the lens — median, range, mean — and watch what Wednesday does to each one.

On Wednesday the bus broke down. Tap a bar to see its value.

Mean

19.8

Median

Mode

Range

Watch out: The mean is 19.8 minutes — slower than four of the five journeys, so it doesn't describe a typical day at all. Cover up Wednesday and see: the mean drops to 16 and the range shrinks from 20 to 2, but the median and the mode are still 16. One extreme value pulls the mean and stretches the range; the median and mode don't budge.

Maths · Types of data

Two labels for every data set

Sort each data set by who collected it — then switch the rule and sort the same six by whether they're counted or measured.

Did you collect it yourself, or is someone else's data being re-used?

Still to sort

Primary data (0)

Collected first-hand by you (or your team) for your own investigation.

Where the line is: It's about WHO collected it, not how accurate it is. Your own rough stopwatch times are still primary.

Secondary data (0)

Collected by someone else — you're re-using it from records, a website or a table.

Where the line is: Official or very accurate data is still secondary if someone else collected it.

6 of 6 still to sort.

One question is about who collected the data. The other is about what kind of values it can take. They're independent — any combination is possible.

Maths · Choosing your measures

Who's the better player?

Jess and Sam each play eight games of the same video game. Jess scores 42, 45, 44, 47, 43, 46, 45, 44. Sam scores 30, 38, 52, 35, 92, 33, 36, 40.

Which is closest to what you think?

Sam. He scored 92 — the best score of anyone.It's a draw. Add up and divide: they both average 44.5.Sam. His range is 62 and Jess's is only 5, so his scores go much higher.Jess. Her median is higher, her scores are steadier, and Sam's 92 is an outlier.

How sure are you?

I'm sureI think soI'm guessing

Sam. He scored 92 — the best score of anyone.

This is a common misconception

One brilliant game doesn't tell you about a typical game. Six of Sam's eight scores are lower than Jess's worst score of 42. To compare, use a typical value (an average), never the single best result.

It's a draw. Add up and divide: they both average 44.5.

This is a common misconception

The means really are equal: 356 ÷ 8 = 44.5 for both. But Sam's mean is propped up by that 92 — without it, his mean would be about 37.7. When one value sits far from the rest, the median is the fairer average: Jess 44.5, Sam 37.

Sam. His range is 62 and Jess's is only 5, so his scores go much higher.

This is a common misconception

A bigger range means the scores are more spread out, not higher. How high and how spread out are two separate questions. Sam's range of 62 says his scores are all over the place — which is a point against him, not for him.

Jess. Her median is higher, her scores are steadier, and Sam's 92 is an outlier.

This matches the evidence

Yes. Median for 'typical': Jess 44.5, Sam 37. Spread for 'consistent': Jess's scores all sit within 5 points of each other; even ignoring the 92, Sam's still cover 22 points (30 to 52). And you've named the outlier and what it does — it drags Sam's mean up to match Jess's.

Don't let one extreme value decide a comparison. Name the outlier, then compare an average it can't drag (often the median) and a spread.

Maths · Putting it into words

Write your comparison, then check it

Group X: median 48 seconds, range 30 seconds. Group Y: median 62 seconds, range 12 seconds.

Two groups of students timed how long they took to finish the same puzzle. Compare the times of the two groups. [2 marks]

Your answer

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

The mark scheme

Compares the medians AND says what that means for the puzzle: Group X were typically quicker (48 s against 62 s). (1 mark)
Compares the ranges AND says what that means for the puzzle: Group Y's times were more consistent (12 s against 30 s). (1 mark)

Model answer

On average, Group X were quicker at the puzzle, because their median time was 48 seconds compared with 62 seconds for Group Y. Group Y's times were more consistent, because their range was 12 seconds compared with 30 seconds for Group X.

One sentence for typical, one for consistent — and in each, say what the numbers mean for the people in the question. Careful with times: lower means quicker.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Every comparison comes down to two questions: which is typically higher, and which is more consistent?

What you need to know

An average (a measure of central tendency) tells you what's typical. Mean: add up the values and divide by how many there are. Median: the middle value once the data is in order. Mode: the most common value. Modal class: in grouped data, the group with the highest frequency.
A measure of spread tells you how consistent the data is. Range = largest value − smallest value. Higher: interquartile range (IQR) = upper quartile − lower quartile, the spread of the middle half of the data.
An outlier is a value far away from the rest of the data. It pulls the mean towards it and stretches the range; the median and the IQR hardly move.
Higher: a box plot shows the minimum, lower quartile, median, upper quartile and maximum on a scale, so two box plots on one axis can be compared at a glance.
Primary data is collected first-hand by you; secondary data was collected by someone else. Discrete data can only take separate values (usually counted); continuous data can take any value in a range (measured).
A full comparison makes two statements — one about an average and one about a spread — and says what each means in context.

The big picture

To compare two distributions, answer two questions. Which is typically higher? Compare an average: the median, mean, mode or, for grouped data, the modal class. Which is more consistent? Compare a spread: the range or, at Higher, the interquartile range. Look out for outliers, which pull the mean and stretch the range, and always say what each comparison means in the context of the data.

Key points

1Ask two questions every time: which is typically higher, and which is more consistent?

2A smaller spread means more consistent. A bigger spread says nothing about which set is higher.

3Read the context before you say 'better': for journey or race times, a lower median means quicker.

4How extreme the largest or smallest value is changes the range but not the IQR (Higher) — so the two can disagree about which set is more consistent.

5Grouped data hides the exact values: you can compare modal classes, but you can only estimate the range.

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Worked example

Problem

Two classes recorded how many books each student read over the summer. Class P — 0–2 books: 4 students; 3–5 books: 11; 6–8 books: 7; 9–11 books: 3. Class Q — 0–2 books: 9 students; 3–5 books: 8; 6–8 books: 5; 9–11 books: 3. Compare the two classes using the modal class. Then explain why you can't find the exact range for either class.

⚠ Watch out

Reading numbers back instead of comparing them. 'A's median is 19, B's is 16' just repeats the values. Say what they mean: 'Route B is typically quicker — its median journey is 3 minutes shorter.' With times, lower means quicker.

🧠

Memory hook

Typical and consistent: an average for 'typical', a spread for 'consistent', and a sentence for each that says what it means.

✓

Check yourself

Sunlight plants: median height 24 cm, range 6 cm. Shade plants: median 17 cm, range 14 cm. Compare the average and the spread in context. (Sunlight plants: typically taller, and more consistent in height.)

Flashcards

(13)

Median

The middle value when the data is in order. With an even number of values, it's halfway between the middle two.

Mean

Add up all the values and divide by how many there are. An extreme value pulls it towards itself.

Mode and modal class

Mode: the most common value. Modal class: in grouped data, the group with the highest frequency.

Range

Largest value − smallest value. It uses only the two extremes, so one outlier can stretch it.

Interquartile range (IQR) — Higher

Upper quartile − lower quartile: the spread of the middle half of the data. How extreme the largest value is makes no difference to it.

Outlier

A value far away from the rest of the data. If it's genuine, mention it rather than deleting it.

Reading a box plot — Higher

Whisker ends: smallest and largest values. Box: lower quartile to upper quartile. Line in the box: the median.

What does a smaller spread tell you?

The data is more consistent: the values are closer together. It says nothing about which set is higher.

What two statements make a full comparison?

One about an average and one about a spread, each saying what it means in context.

Primary data

Data you collect first-hand yourself, for your own investigation.

Secondary data

Data someone else collected that you re-use — from records, a website or a published table.

Discrete data

Data that can only take separate values, like counts or shoe sizes.

Continuous data

Data that can take any value in a range — measurements like time, height and mass.

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.

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Route A or Route B? Twenty journeys to school on each bus

Zoom in: one week on Route B

Two labels for every data set

Who's the better player?

Write your comparison, then check it

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