KS3 · Maths

Comparing data using averages and range

Two footballers both score a mean of three goals a match. Are they the same player? Not even close, and you're about to see why.

Comparing data

Same mean, different players

Two footballers both have a mean of 3 goals per match. Sort each feature of their goal records: tick Footballer A, Footballer B, both, or neither. Predict first, then check.

A Footballer A
B Footballer B

A mean of 3 goals per match
A range of 7 goals
A range of 3 goals
A modal score of 0
Two modal scores, 3 and 4 (bimodal)
Sometimes scores a lot of goals, other times not many at all
Scores sit close to the mean
Scored more consistently around three goals
A mean of 4 goals per match

Where do the averages sit?

Twelve journeys to school

Twelve pupils timed their journey to school in minutes: 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 21. The times add up to 90. Drag each tag to where it belongs, then check. The last tag is a slip: on a dot plot of these times there are 4 columns (5, 6, 7 and 21 minutes), and the slip divides 90 by 4 instead of by 12.

Min — Smallest valueMax — Largest valuePlace MedianPlace ModePlace MeanSlip — 'Mean' = 90 ÷ 4 columns

What does the range really tell you?

Same range, so what?

Two classes sat the same maths test. Both sets of scores have a range of 20.

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

Same range, so the two classes vary by exactly the same amount.Same range, so the two classes must have about the same mean.The class with the smaller range is always the more consistent one.Same range just means the smallest and largest scores are the same distance apart. It doesn't promise equal variation, and it says nothing about the means.

How sure are you?

I'm sureI think soI'm guessing

Same range, so the two classes vary by exactly the same amount.

This is a common misconception

Easy to believe, but no. An equal range only means the smallest and largest scores are the same distance apart. The scores in between could be bunched up in one class and scattered in the other, so an equal range does not guarantee equal variation.

Same range, so the two classes must have about the same mean.

This is a common misconception

The range says how far apart the smallest and largest scores are. It doesn't tell you the value of the mean, or any other average. One class could sit much higher than the other and still have the same range.

The class with the smaller range is always the more consistent one.

This is a common misconception

Usually, yes. 'Always' is the trap. The range only measures the distance between the smallest and largest values, so one or a few outliers can make a range large even when most scores are close together. Look at the values behind the range too.

Same range just means the smallest and largest scores are the same distance apart. It doesn't promise equal variation, and it says nothing about the means.

This matches the evidence

Yes. That's exactly what an equal range gives you. To compare the classes properly you'd read the range together with an average, in context.

The range measures only the distance between the smallest and largest values. Usually a bigger range means a more varied dataset and a smaller range a more consistent one, but it can't promise equal variation and it never tells you the mean.

Outliers

Remove it or keep it?

Each of these is an outlier. Pick one, then decide: remove it or keep it? Commit before you read the reason.

Still to sort

Remove: recorded in error (0)

The value can't be right.

Where the line is: Extreme isn't enough. A value only goes if it was recorded in error.

Keep: a real data point (0)

Correctly recorded, so it is valuable.

Where the line is: A correct outlier stays, even though it is extreme.

6 of 6 still to sort.

Your turn to compare

Two businesses, one comparison

Business A: mean 33.9 hours, range 17 hours, modal 37 hours. Business B: mean 32.3 hours, range 6 hours, modal 34 hours.

Write a comparison of the weekly hours worked at Business A and Business B. Use an average and the range, and say what they mean for the two businesses. [4 marks]

Your answer

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

The mark scheme

Uses at least one average to compare. For example, A's mean of 33.9 is higher than B's 32.3, or A's mode of 37 is higher than B's 34. (1 mark)
Uses the range to compare spread: B's range of 6 is smaller than A's range of 17. (1 mark)
Interprets at least one statistic in context. For example, A's typical week is longer, or B's weekly hours are more consistent, with less spread between the smallest and largest. (1 mark)
Recognises the comparison is open to interpretation, especially because the means are close. (1 mark)

Model answer

Business A has the higher mean (33.9 hours against 32.3), and its higher modal value (37 against 34) supports that, so A's typical week is longer. Business B has the smaller range (6 hours against 17), so its weekly hours are more consistent, with less spread between its smallest and largest. The means are quite close, so the comparison is open to interpretation.

Pick an average and the range, say what each means in the situation, and add a further statistic if it supports you.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

One number can make two datasets look identical. Here's how to tell them apart.

What you need to know

One statistic isn't a comparison: two datasets can share a mean and still be very different.
Statistical summaries sum up a dataset using statistics such as the mean, median, mode and range.
The range is the largest value take away the smallest, and it measures spread.
Have a goYour friend swears a dataset with a range of 0 is hiding some wild values. Is your friend onto something?
No. Every value in it must be the same.
The range is the largest value take away the smallest, so a range of 0 means the largest and smallest are equal, which leaves no room for anything different.
Usually, a bigger range means a more varied dataset and a smaller range a more consistent one.
Equal ranges don't guarantee equal variation, and the range doesn't tell you the value of the mean.
Every average sits between the smallest and largest values, so a mean outside that interval is guaranteed wrong.
Outliers are extremely large or small values; remove one only if it was recorded in error.
Have a goSam spots the biggest number in a list and yells, 'Outlier! Delete it!' What should Sam find out before touching the delete key?
Whether it was recorded in error.
Being extreme isn't the same as being wrong. A value recorded correctly is a real, valuable data point, so it stays.
The range changes only when a value added becomes a new extreme, or a value removed was the only largest or smallest.
Have a goA dataset has largest value 97 and smallest value 32, so its range is 65. You remove the value 44. What is the new range?
Still 65.
44 is neither the largest nor the smallest value, so the two ends haven't moved and the range can't change.
Compare with at least one average and the range, read in context; close results are open to interpretation.
Bar charts compare frequencies quickly; pie charts show one group against the whole. Add the bars to get a bar chart's total.
Have a goA bar chart of 40 pets has a dog bar that is 10 tall. What fraction of the pets are dogs?
10 out of 40, which is a quarter.
A proportion compares one subgroup with the whole, and the whole is the bars added together, not the number of bars.

The big picture

One statistic is never a full comparison. Read at least one average and the range together, in context, and keep an eye on outliers and on what changing a data point does.

Key points

1Never compare on one statistic alone. Pair an average with the range, and say what they mean in the situation.

2Range = largest − smallest. It usually tracks variation, but it only looks at the two extremes.

3Averages can't leave the smallest-to-largest interval, and the mean is the one an outlier drags most.

4Keep outliers that were recorded correctly; remove only the ones recorded in error.

5The mode is the only average that works for unordered categories, such as favourite colours.

6Bar chart: compare frequencies. Pie chart: compare one group with the whole.

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Worked example

Problem

The mean of 10 numbers is 50. One of the numbers, 45, is changed to 55. What is the new mean? And can you tell whether the range changes?

⚠ Watch out

Treating one statistic as the whole comparison, usually 'same mean, so same data'. Equal means or equal ranges don't make datasets alike. Read an average and the range together, in context.

🧠

Memory hook

An average says what's typical. The range says how far apart the extremes are. A comparison needs both.

✓

Check yourself

A friend says, 'Class 8A has the higher mean, so 8A is better.' Name two more things you'd check. (The range, and the median or mode, then interpret in context.)

Flashcards

(12)

What is a statistical summary?

A set of statistics, such as the mean, median, mode and range, that sum up the properties and features of a dataset. It's most insightful when comparing two datasets.

What does the mean tell you?

A typical value: the total of all the data points shared out equally among them.

How do you find the median when there are two middle values?

Put the data in order, add the two middle values and divide by two (12 and 14 give 26 ÷ 2 = 13).

What is the mode, and what does bimodal mean?

The mode is the most frequently occurring value. A dataset with two modes is bimodal.

How do you find the range, and what does it measure?

Largest value take away smallest value (97 − 32 = 65). It measures spread: the distance between the extremes.

What does a bigger range usually show, and what's the catch?

Usually a more varied dataset. The catch: the range only measures the distance between the smallest and largest values, so an outlier can make it large even when most values are close together.

Two datasets have the same range. What can't you conclude?

That they vary equally, or anything about the value of the mean. An equal range only means the same distance between the smallest and largest values.

Between which values must every average sit?

Between the smallest and largest data points. A mean outside that interval is guaranteed wrong, for example from dividing by the number of dot plot columns instead of the number of data points.

Which average works for unordered categories like favourite colours?

The mode. It's the only one of the three averages that can be used for non-numerical data.

What is an outlier, and when do you remove it?

A data point extremely large or small compared with the rest (it can be negative or a decimal). Remove it only if it was recorded in error; keep correct ones.

Add a value above the mean. What happens to the mean?

It increases. A value below the mean makes it decrease, and a value equal to the mean leaves it the same.

Bar chart or pie chart: which is quick for what?

Bar chart: reading frequencies and comparing subgroups. Pie chart: comparing one subgroup against the whole (a 90° sector is a quarter).

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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Comparing data using averages and range

Same mean, different players

Twelve journeys to school

Same range, so what?

What does each change do?

Remove it or keep it?

Two businesses, one comparison

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