KS3 · Maths

Range and spread of data

Two data sets can have the same mean and still look completely different on a line. One quick number tells them apart: the range.

Maths · Statistics

Put the range on the line

Here is a data set: −3, 0, 1, 2, 4, 6. Slide the two ends of the bar onto the smallest and the largest value. Tick both ends, because they are real data values (filled circles). Then check. After that, press Try again and drag the upper end out to 38. That is what one far-away extra value does.

Where do the extremes live?

Find the line where the working goes wrong

Fifty people were timed at a ticket desk. Wait in minutes: 3, 4, 5, 6, 7, 8. Frequency: 5, 12, 20, 9, 3, 1. Find the range of the waiting times.

A student's answer — which line goes wrong?

Working backwards

You know the range and one end. Find the other.

The lowest temperature one week was −6 °C and the range was 15 °C. What was the highest temperature?

  1. The range is highest − lowest. We know the lowest (−6 °C) and the range (15 °C), so the highest is the one we are hunting for.
  2. missing step
Which line is step 2?

Average or range?

An average (mean, median, mode)vsThe range

Set P is 8, 9, 10, 11, 12. Set Q is 2, 3, 10, 17, 18. Both have a mean of 10.

Focus

What kind of measure is it?

An average (mean, median, mode)

A measure of central tendency: it describes the centre of the data.

The range

A measure of spread: it describes how far apart the data are.

The insight

They do different jobs, which is why one of them is a useful extra to have.

Sets P and Q

An average (mean, median, mode)

The mean of P is 10. The mean of Q is 10.

The range

The range of P is 12 − 8 = 4. The range of Q is 18 − 2 = 16.

Which set is more consistent?

An average (mean, median, mode)

The mean can't separate them, because both are 10.

The range

P has the smaller range, so it is less varied and more consistent. This is a guide, not a guarantee.

A range of zero

An average (mean, median, mode)

Five values that are all 7 have a mean of 7.

The range

Their range is 7 − 7 = 0. A range of zero means all the values are the same.

What does it use?

An average (mean, median, mode)

Each of these summarises the centre of the data.

The range

It uses only the two extreme values, so one outlier can make it much larger than the spread of most of the data.

Predict, then check

Data set: 32, 32, 44, 51, 60, 97. The range is 97 − 32 = 65. You are about to remove one value.

Which removal changes the range?

Outliers: remove, keep or investigate?

What would you do with each outlier?

Each of these is an outlier. Decide whether it can be removed before you recalculate the range, whether it stays, or whether you need to dig further.

Still to sort

Recorded in error: can be removed (0)

It is a mistake, so removing it and recalculating the range is fine.

Correctly recorded: keep it (0)

It is real data, and it shows the true variety of the data.

Could be either: investigate (0)

You can't tell yet, so find out before you decide.

Where the line is: If you can't tell whether it is a mistake or a genuine value, investigate further before removing anything.

4 of 4 still to sort.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two numbers decide it. Everything in between gets no vote.

What you need to know

  • Spread is how far apart the values are. Less spread-out data is more consistent.
  • The range measures spread: range = largest value − smallest value, the two extreme values.
  • Write the range as one number, like 7, not '3 to 10' or '10 − 3'.
  • Negatives and decimals change nothing: the range is still the difference. From −9 to 6, it is 15.
  • Have a goZara says the range of −4, 1, 3, 8 is 4, because '8 − 4'. She sounds very sure. Is she right?

    No. The range is 12.

    The smallest value is −4, so the gap is 8 − (−4) = 12. Zara dropped the minus sign and measured from 4 instead of from −4.

  • In a frequency table, the extremes are the data values, never the frequencies.
  • A range of zero means every value is the same. Sets with the same mean can still have very different ranges.
  • Have a goDev says a set of five numbers has a range of 0, and that the numbers are 'all different, just very close together'. Can he be right?

    No. All five values must be the same.

    A range of 0 means largest − smallest = 0, so there is no gap at all. 'Very close together' still leaves a gap bigger than zero.

  • You can't find a range for things like pets or colours, because they have no order to give extremes.
  • Have a goA class lists its favourite pets, and Mia counts 4 different pets and says 'the range is 4'. What is wrong with looking for a range here?

    Pets have no order, so there are no extreme values. There is no range.

    Counting kinds of pet is not subtracting extremes, and without an order there is no largest or smallest value to subtract.

  • An outlier is extremely large or small compared with the rest, so it can stretch the range a lot.
  • Have a goQuiz scores are 12, 14, 15, 15, 17. A score of 60 is added. Does the range grow a little, or a lot?

    A lot: from 5 to 48.

    The range uses only the two extremes. One far-away value becomes the new largest, so it drags the range up even though the other scores have not moved.

  • Removing a value changes the range only if it is the only largest or only smallest value.
  • Remove an outlier only if it was recorded in error. Keep correct ones and investigate doubtful ones.

The big picture

The range is the largest value minus the smallest value, written as one number. It measures spread, so alongside an average it helps you judge how varied or consistent data are. One outlier can stretch it, removing or adding a value changes it only if that value is (or becomes) the only largest or smallest, and an outlier can be removed only if it was recorded in error.

Key points

1Range = largest value − smallest value, written as one number.
2It is a measure of spread. Means, medians and modes are measures of central tendency.
3The range uses only the two extreme values, so an outlier can make it much larger than the spread of most of the data.
4Removing a value changes the range only if it is the only largest or only smallest. An added value changes it only if it becomes a new largest or smallest.
5Only outliers recorded in error can be removed. Correct ones are kept, and doubtful ones are investigated.

Worked example

Problem

Overnight temperatures (°C) over a week were: 2.5, −1.5, 0, 3, −0.5, 1.5, 4.5. Find the range.

⚠ Watch out

Writing the range as '3 to 10' or '10 − 3'. The range is a single value, the answer to the subtraction: 7.

🧠

Memory hook

Think of a washing line. The range is how long the line must be to reach from the first peg to the last. Only the two end pegs decide it. Pegs in between can come and go, but one peg dragged far away stretches the whole line.

✓

Check yourself

Try this without looking back: for 14, 18, 18, 21, 95, find the range, spot the outlier, and say what you must check before removing it.

Flashcards

(14)
Range of −8, −3, 2? Write it the right way.
10. One number: 2 − (−8) = 10, not '−8 to 2'.
Is the range a measure of spread or of central tendency?
Spread: how far apart the values are. Mean, mode and median are measures of central tendency.
In a frequency table, where are the extreme values?
In the data values (the category labels), not in the frequencies.
Where do you find the extremes on a line graph? On a bar chart of single data points?
Line graph: the highest and lowest points. Bar chart of single data points: the tallest and shortest bars.
What does it mean if data are more consistent?
They are less spread out, so the values are less varied.
What does a range of zero tell you?
Every value in the data set is the same.
Two data sets have the same mean. Must their ranges match?
No. Same mean can come with very different ranges, so the range is a useful extra measure of how varied the data are.
Why can't you find the range of favourite colours?
Colours have no fixed order, so you can't identify extreme values to subtract.
What is an outlier?
A data point extremely large or extremely small compared with the rest. On a dot plot it lies far from where most points are clustered.
Why can one outlier make the range misleading?
The range uses only the two extreme values, so one outlier can make it much larger than the spread of most of the data.
When does removing a value change the range?
Only if it is the only largest or only smallest value. Removing a middle value, or one of two equal lowest values, leaves it unchanged.
When does adding a value change the range?
Only if the new value becomes a new largest or smallest value.
When can an outlier be removed?
Only if it was recorded in error, for example a misplaced decimal point or a value that makes no sense. Keep correct ones and investigate doubtful ones.
A correct outlier makes a graph hard to read. What can you do?
It may be left off the graph, but not removed from the data. Its existence should still be shown or stated separately.

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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