Year 8StatisticsDeveloping knowledge from dataYear 8 Maths

Mean, median, mode and range

What this lesson covers

  • A variable is an attribute or measurement of the people or objects being studied. A categorical variable classifies objects or individuals into groups. Discrete numerical variables are counted. Continuous numerical variables are measured.
  • The response to a statistical question can be summarised by a measure of central tendency. The mean is the average of numerical data. The median is the middle value for sorted numerical data. The mode is the data value with the highest frequency for categorical data or discrete numerical data.
  • The response to a statistical question can be summarised by the range as a measure of spread. The range for numerical data is the highest value minus the lowest value.
Key idea

Are our feet the same length?

  • That is a real question, and you cannot answer it by thinking.
  • You have to go and measure some feet.
  • Then you have to say something honest about the numbers.
Step
1. Ask a question
2. Collect data that could answer it
3. Summarise it
4. Answer the question
Key idea

What you measure is the variable

  • A variable is the thing you record about each person.
  • Foot length. Eye colour. Number of siblings.
  • They are not all the same kind of thing.
VariableBecauseKind
Eye coloursorts into groupscategorical
Number of siblingscounteddiscrete numerical
Foot lengthmeasuredcontinuous numerical
Favourite sportsorts into groupscategorical
Key idea

Categorical sorts into groups

  • Eye colour puts each person into a group.
  • Blue, brown, green. There is no arithmetic to do.
  • You cannot find the average eye colour of a class.
VariableBecauseKind
Eye coloursorts into groupscategorical
Number of siblingscounteddiscrete numerical
Foot lengthmeasuredcontinuous numerical
Favourite sportsorts into groupscategorical
Key idea

And numerical splits two ways

  • Counted things are discrete. Measured things are continuous.
  • The test: could it sensibly be a half?
  • Not half a sibling. But 24.5 centimetres is fine.
QuestionAnswerMeans
Can it be a half?Nocounted, so discrete
Yesmeasured, so continuous
Siblingsno half siblings herediscrete
Foot length24.5 cm is finecontinuous
Worked example

So: six people, feet measured

  • 22, 24, 24, 25, 26 and 29 centimetres.
  • Continuous numerical data, because we measured it.
  • Already sorted, which saves a step later.
So: six people, feet measured
Worked example

The mean: share it out evenly

  • Add every value: 150.
  • Divide by how many there are: 150 ÷ 6.
  • 25 centimetres. That is the mean.
StepWorkingGives
Add them all22+24+24+25+26+29150
Divide by how many150 ÷ 625
Mean25 cm
Key idea

It is the balancing point

  • Put the mean on the plot and look at it.
  • The values below it balance the values above it.
  • Which is why one very large value drags it upwards.
It is the balancing point
Worked example

The median: the middle one

  • Sort the data first. Always. Otherwise this is meaningless.
  • Six values, so there is no single one in the middle.
  • Take the middle two, 24 and 25, and go halfway: 24.5.
StepWorkingGives
Sort them first22, 24, 24, 25, 26, 29
Six valuesno single middle one
Take the middle two24 and 25
Halfway between(24 + 25) ÷ 224.5 cm
Key idea

Odd is easier than even

  • Five values: the third one is the middle. Done.
  • Four values: the middle two, averaged.
  • 3, 5, 8, 10 gives (5 + 8) ÷ 2, which is 6.5.
How manyMiddleMedian
Odd, say 5the 3rd valuethat value
Even, say 6the 3rd and 4thhalfway between
3, 5, 8, 105 and 86.5
Worked example

The mode: the most common

  • Which value turns up most often?
  • 24 appears twice, everything else once.
  • So the mode is 24 centimetres.
StepWorkingGives
Which value repeats?24 appears twice
Everything elseappears once
Mode24 cm
Worked example

And the range is not an average at all

  • Highest minus lowest. 29 − 22 = 7 centimetres.
  • It does not tell you where the data sits.
  • It tells you how spread out it is. Different question.
StepWorkingGives
Highest29 cm
Lowest22 cm
Range29 − 227 cm
Key idea

Three centres, three answers

  • Mean 25. Median 24.5. Mode 24.
  • All three are correct. They disagree.
  • So "the average foot is 25 cm" needs saying which one you used.
MeasureValueWhat it is
Mean25 cmthe balancing point
Median24.5 cmthe middle one
Mode24 cmthe most common
Range7 cmhow spread out
Key idea

Back to the question we asked

  • Are our feet the same length? No.
  • They differ by up to 7 centimetres, which is the range.
  • The centre tells you the typical foot. The range answers this.
Back to the question we asked
Key idea

One honest warning about the mode

  • Our 24 repeated because we rounded to whole centimetres.
  • Measure to the millimetre and probably nothing repeats at all.
  • On measured data the mode is often useless.
DataMode is
Eye colourthe only optionessential
Number of siblingssensibleuseful
Foot length to the mmprobably nothing repeatsuseless
Key idea

So which one should you use?

  • Categorical data leaves you only the mode.
  • If a few values are extreme, the median resists them better.
  • And if you want spread rather than centre, use the range.
SituationBecauseUse
Categorical datamean and median impossiblemode
Measured datamode often meaninglessmean or median
A few extreme valuesmean gets draggedmedian
Spreadnone of the threerange
Your turn

Your turn: three questions

  • The data is 5, 8, 8, 11 and 13. Pause here.
  • Five values, so the median is straightforward.
  • Label your answers. Two of them come out the same number.
Question
The data: 5, 8, 8, 11, 13
1Find the mean.
2Find the median.
3Find the mode and the range.
Show answer

1. 9. They add to 45, and 45 ÷ 5 = 9. 2. 8. Five values, so the middle one is the third, and they are already sorted. 3. Mode 8, the only repeated value. Range 13 − 5 = 8. Note the mode and the range are both 8 here and mean completely different things, which is a good reason to label your answers.

Worked example

The answers

  • They add to 45, so the mean is 9.
  • The median is the third value, which is 8.
  • The mode is 8 too. And the range is 13 − 5, which is also 8.
Answer
Mean9
Median8
Mode8
Range8
Summary

What to take away

  • Start with a question. The arithmetic is there to answer it.
  • Mean, median and mode are three answers and they disagree.
  • The range is spread, not centre. Say which one you used.
Remember
Start with a questionnot with the arithmetic
Counted or measuredtells discrete from continuous
Meanadd them all, divide by how many
Mediansort first, then take the middle
Rangehighest minus lowest, a spread not a centre

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