Year 8 · Chapter 13

Averages from grouped data, scatter graphs and correlation

By the end of this chapter you will be able to

Find the mean, median, mode and range of a list of values, including from a set of data in order
Find the midpoint of each class, and use midpoints to estimate the mean of grouped data
Find the modal class and the class containing the median of grouped data
Compare two sets of data using an average and the range
Plot a scatter graph, describe the correlation, and use a line of best fit to make predictions

You should already know

Mean, median, mode and range of small lists (Year 7)
Frequency tables, and tallies (Year 7)
Plotting coordinates and reading graphs (Year 8, Chapters 7 and 8)

Where this leads

Cumulative frequency, histograms and box plots (GCSE)
Correlation coefficient and regression lines (GCSE and A-level)
Sampling and the reliability of statistical conclusions (GCSE)

National Curriculum for England, Key Stage 3 mathematics, Year 8. Content: Find the modal class, the class containing the median, and an estimate of the mean from a grouped frequency table, using midpoints, Compare distributions using averages and the range, Plot a scatter graph from a table, and describe correlation as positive, negative or none, Use a line of best fit by eye to interpolate and extrapolate, and discuss when extrapolation is not reliable.

Exercise P Do you remember?

Non-calculator

FWork out 3 × 4 + 5.

Done

FPut these in order from smallest to largest: 5, 2, 9, 4.

Done

FWork out 40 ÷ 8.

Done

FWhat is the midpoint of 4 and 10?

Done

FWhich number is the middle of the list 2, 6, 8, 11, 15?

Done

13.1

Mean, median, mode and range of a list

Year 8
Find the mean, median, mode and range of a list of values

Video lesson: Mean, median, mode and range of a listComing soon

Ali records the number of goals scored by his football team in five matches: 3, 7, 4, 9, 7. He wants one number to describe a typical match, and a number to show how much the scores vary.

An average is one number that represents a set of data. There are three averages:

mean = total ÷ number of values;
median = the middle value, when the values are put in order;
mode = the value that appears most often.

The range measures spread: range = biggest value − smallest value.

Key ideas

Mean = total ÷ number of values.
Median: put the values in order first, then find the middle (average the two middle values if there are an even number).
Mode: the most common value. There can be more than one mode.
Range = biggest − smallest.

Worked example 13.1.1

Find the mean of 3, 7, 4, 9, 7.

Method. The mean is the total divided by the number of values.

Add the values. Find the total of all the values.
3 + 7 + 4 + 9 + 7 = 30
Count the values. There are 5 values.
n = 5
Divide. Mean = total ÷ number of values.
30 ÷ 5 = 6
Write the answer.
6

Check. 6 × 5 = 30, the total.

Worked example 13.1.2

Find the median of 4, 9, 1, 7, 3.

Method. Put the values in order, then find the middle.

Put the values in order. Write them from smallest to largest.
1, 3, 4, 7, 9
Find the middle. The middle value is the 3th value.
n = 5
Write the answer.
4

Check. There are 2 values below and 2 above the median.

Worked example 13.1.3

Find the mode and the range of 2, 3, 3, 5, 7.

Method. Count the values for the mode, and subtract the smallest from the biggest for the range.

Find the mode. The value 3 appears twice; the others appear once.
mode = 3
Find the range. The biggest value is 7 and the smallest is 2.
7 − 2 = 5
Write the answer.
Mode 3, range 5

Check. The mode is the most common, and the range is the spread.

Watch out

Watch out: the median needs the values in order first. Finding the middle of an unordered list is a common mistake.

Exam tip

Exam tip. Write the values in order before you find the median. For an even number of values, average the two middle ones.

13.2

Grouped frequency tables

Year 8
Find the modal class and the class containing the median, Estimate the mean using midpoints

Video lesson: Grouped frequency tablesComing soon

A library records how many minutes each of 20 children spent reading. The data is in classes, such as 0 to 10 minutes, so the head of the library can see the pattern quickly.

When data is grouped, we do not know the exact values. We use the midpoint of each class to estimate the mean.

To estimate the mean: multiply each frequency by its midpoint, add these totals, then divide by the total frequency.

The modal class is the class with the highest frequency. The median class is the class that contains the middle position, (n + 1) ÷ 2.

Key ideas

The midpoint of a class is halfway between its two ends.
Estimated mean = (sum of f × midpoint) ÷ (sum of f).
Modal class = the class with the highest frequency.
Median class = the class where the running total reaches (n + 1) ÷ 2.

Worked example 13.2.1

A table has classes 0 < t ≤ 10, 10 < t ≤ 20, 20 < t ≤ 30, 30 < t ≤ 40, with frequencies 4, 6, 8, 2. Find the estimated mean.

Method. Estimate the mean with the midpoints: multiply each frequency by the midpoint, add, and divide by the total frequency.

Find the midpoint of each class. The midpoint is the middle of the class, the mean of its two ends.
5, 15, 25, 35
Multiply each frequency by its midpoint. This gives an estimate of the total for each class.
4 × 5 = 20, 6 × 15 = 90, 8 × 25 = 200, 2 × 35 = 70
Add the totals. The total of all the frequencies is 20.
Σfx = 380
Divide. Estimated mean = Σfx ÷ Σf.
380 ÷ 20 = 19
Write the answer.
19

Check. The estimate is close to the mean of the midpoints weighted by the frequencies.

Worked example 13.2.2

Using the same table, find the modal class.

Method. The modal class is the class with the highest frequency.

Find the highest frequency. Look down the frequency column.
highest = 8
Find its class. Read across to the class.
20 < t ≤ 30
Write the answer.
20 < t ≤ 30

Check. The modal class is the class, not the frequency.

Worked example 13.2.3

Using the same table (n = 20), find the class that contains the median.

Method. The median is in the class that contains the middle position, (n + 1) ÷ 2.

Find the position of the median. n = 20, so the median is at position (20 + 1) ÷ 2 = 21/2.
position 21/2
Add the frequencies up to each class. Find the class where the running total reaches the position.
running totals
Write the answer.
20 < t ≤ 30

Check. Check the running total before the class is below the position.

Watch out

Watch out: the estimated mean is an estimate, because the exact values are not known. Use the midpoint, not the class label, when you multiply.

Exam tip

Exam tip. Write a column for the midpoint, a column for f × midpoint, and add a total row. This layout is what examiners look for.

Exercise 13B Grouped data: midpoints, modal and median classes

Non-calculator

FA grouped table has classes 0 < t ≤ 10, 10 < t ≤ 20, 20 < t ≤ 30, 30 < t ≤ 40 with frequencies 4, 6, 8, 2. Find the estimated mean.

Time (minutes)	Frequency
0 < t ≤ 10	4
10 < t ≤ 20	6
20 < t ≤ 30	8
30 < t ≤ 40	2

Done

FFind the modal class of the table above.

Done

FFind the class that contains the median of the table above.

Done

FA table has classes 0 to 4, 5 to 9, 10 to 14, 15 to 19 with frequencies 2, 2, 4, 2. Use midpoints 2, 7, 12 and 17 to estimate the mean.

Score	Frequency
0 – 4	2
5 – 9	2
10 – 14	4
15 – 19	2

Done

FFind the modal class of the table with classes 0 to 4, 5 to 9, 10 to 14, 15 to 19 and frequencies 2, 2, 4, 2.

Done

FFor the table with frequencies 2, 2, 4, 2 (n = 10), find the class that contains the median.

Done

RThe midpoint of the class 20 < t ≤ 30 is 25. Explain how you know, using the two ends of the class.

Done

RA table has classes 0 to 10 and 10 to 20 with frequencies 5 and 15. Ali says the estimated mean must be 10. Is Ali right? Explain.

Done

PSA table of heights has frequencies 3, 5, 7 and 5 for four classes with midpoints 150, 160, 170 and 180 cm. Estimate the mean height, rounding to the nearest cm.

Done

RA class of 20 pupils has times in a race. The modal class has frequency 7. Explain why the modal class does not have to be the middle class.

Done

RFor the first table (n = 20), find the estimated mean using midpoints 5, 15, 25, 35 with frequencies 4, 6, 8, 2. Explain why it is only an estimate.

Done

13.3

Comparing two sets of data

Year 8
Compare distributions using averages and the range

Video lesson: Comparing two sets of dataComing soon

Two classes take the same spelling test. Mrs Lee wants to know which class did better, and which class was more consistent. She uses an average to compare the typical score, and the range to compare the spread.

When you compare two sets of data, always say something about the average (which is typical) and the spread (the range), in the context of the question.

Key ideas

Compare averages to say which set is higher or lower overall.
Compare ranges to say which set is more consistent (smaller range) or more spread out (larger range).
Always write your conclusion in the context of the question.
Use the same kind of average for both sets.

Worked example 13.3.1

Class A scores are 6, 8, 9, 10, 12. Class B scores are 7, 9, 9, 9, 11. Compare the means.

Method. The mean is the total divided by the number of values.

Add the values. Find the total of all the values.
6 + 8 + 9 + 10 + 12 = 45
Count the values. There are 5 values.
n = 5
Divide. Mean = total ÷ number of values.
45 ÷ 5 = 9
Write the answer.
9

Check. 9 × 5 = 45, the total.

Worked example 13.3.2

Using the same scores, compare the ranges.

Method. The range is the biggest value minus the smallest value.

Find the biggest value. Look for the largest number.
11
Find the smallest value. Look for the smallest number.
7
Subtract. Range = biggest − smallest.
11 − 7 = 4
Write the answer.
4

Check. 7 + 4 = 11.

Worked example 13.3.3

Use your results to say which class is more consistent.

Method. A smaller range means the scores are closer together.

Compare the ranges. Class A has range 6 and class B has range 4.
6 and 4
Decide. Class B has the smaller range, so its scores are more consistent.
B is more consistent
Write the answer.
Class B is more consistent

Check. The means are both 9, so the averages are the same; the spread is the difference.

Watch out

Watch out: equal means do not mean the same spread. Always check the range too.

Exam tip

Exam tip. Write a comparison using both the average and the spread, and add the context. For example: “On average ... but ... varies more.”

13.4

Scatter graphs and correlation

Year 8
Plot a scatter graph from a table, Describe correlation as positive, negative or none

Video lesson: Scatter graphs and correlationComing soon

Sam records how many hours he studied for each of eight tests, and the mark he got. He plots the pairs on a scatter graph to see whether more study gives higher marks.

A scatter graph plots pairs of values as points. Look at the pattern to describe the correlation:

positive: as one value goes up, the other tends to go up;
negative: as one value goes up, the other tends to go down;
no correlation: there is no clear pattern.

A line of best fit is a straight line drawn by eye, close to most of the points, with about as many points above as below it.

Key ideas

Each point is one pair of values (x, y).
Positive correlation: the points go up from left to right.
Negative correlation: the points go down from left to right.
No correlation: the points have no clear pattern.

Worked example 13.4.1

Describe the correlation in the scatter graph of hours studied and marks, if the points rise from left to right.

Method. Look at the direction of the points.

Look at the pattern. The points go up as we move to the right.
up to the right
Name the correlation. When both values rise together, the correlation is positive.
positive
Write the answer.
Positive correlation

Check. More hours tend to go with higher marks.

Worked example 13.4.2

A scatter graph shows the age of a car and its value. As age goes up, the value goes down. Describe the correlation.

Method. Look at how the two values move together.

Look at the pattern. As one value goes up, the other goes down.
down to the right
Name the correlation. This is a negative correlation.
negative
Write the answer.
Negative correlation

Check. Older cars tend to be worth less.

Worked example 13.4.3

Describe the correlation of shoe size and a test score, if the points are scattered with no pattern.

Method. If there is no pattern, there is no correlation.

Look at the pattern. The points are scattered with no clear direction.
no pattern
Name the correlation. So there is no correlation.
none
Write the answer.
No correlation

Check. Shoe size does not predict the test score here.

Watch out

Watch out: correlation does not show that one thing causes the other. Hours studied and marks may be related, but other things also affect marks.

Exam tip

Exam tip. Describe the correlation in words, and say what it means in the context of the question. Do not write only “positive”.

Exercise 13D Scatter graphs and correlation

Non-calculator

FPlot the points (1, 40), (2, 45), (3, 50), (4, 58), (5, 62), (6, 70), (7, 75), (8, 80) on the grid. Describe the correlation.

Done

FA scatter graph shows temperature and ice cream sales. The points rise to the right. What type of correlation is this?

Done

FA scatter graph shows the age of cars and their values. The points fall to the right. What type of correlation is this?

Done

FA scatter graph has points scattered with no pattern. What type of correlation is this?

Done

FPlot the points (1, 9), (2, 7), (3, 6), (4, 4), (5, 2). What type of correlation is shown?

Done

RExplain why a scatter graph with points along a straight line sloping up has strong positive correlation.

Done

RAli says that a scatter graph of hours studied and marks shows that studying causes higher marks. Is Ali right? Explain.

Done

PSPlot the points (2, 3), (4, 5), (6, 6), (8, 9), (10, 10). Describe the correlation.

Done

RThe scatter graph of hours studied and marks has a positive correlation. Give one reason why the points might not lie exactly on a line.

Done

13.5

Line of best fit, interpolation and extrapolation

Year 8
Use a line of best fit by eye to interpolate and extrapolate, Discuss when extrapolation is unreliable

Video lesson: Line of best fit, interpolation and extrapolationComing soon

Sam drew a line of best fit on his graph of hours studied and marks. He uses the line to predict the mark for 4.5 hours of study, and for 12 hours. He wants to decide which prediction is more reliable.

Interpolation means using the line to estimate a value inside the range of the data. It is usually reliable.

Extrapolation means using the line to estimate a value outside the range of the data. It is less reliable, because the pattern may not continue.

Key ideas

Interpolation: inside the data range, usually reliable.
Extrapolation: outside the data range, less reliable.
Use the line of best fit, not a single point, for a prediction.
Check the units and the meaning of the values in the answer.

Worked example 13.5.1

The line of best fit is y = 5x + 40, where x is hours and y is marks. Predict the mark for 4.5 hours.

Method. Put x = 4.5 into the line, which is inside the data range (1 to 8).

Put x = 4.5 into the line. y = 5 × 4.5 + 40.
5 × 4.5 + 40
Work it out. 5 × 4.5 = 22.5, then 22.5 + 40 = 62.5.
62.5
Write the answer. 4.5 hours is inside the data range, so this is interpolation.
62.5 marks

Check. The prediction is between the marks for 4 and 5 hours.

Worked example 13.5.2

Using the same line, predict the mark for 12 hours. Is this reliable?

Method. Put x = 12 into the line, which is outside the data range.

Put x = 12 into the line. y = 5 × 12 + 40.
5 × 12 + 40
Work it out. 60 + 40 = 100.
100
Decide. 12 hours is outside the data, so this is extrapolation and is less reliable.
extrapolation
Write the answer.
100 marks, but not reliable

Check. A mark of 100 may not be possible, and the pattern may change.

Worked example 13.5.3

Using the same line, predict the mark for 6 hours. Is this reliable?

Method. Put x = 6 into the line. The value 6 is inside the data range (1 to 8), so this is interpolation.

Put x = 6 into the line. y = 5 × 6 + 40.
5 × 6 + 40
Work it out. 30 + 40 = 70.
70
Decide. 6 hours is inside the data, so this is interpolation and is usually reliable.
interpolation
Write the answer.
70 marks, reliable

Check. 70 is between the marks for 5 and 8 hours.

Watch out

Watch out: extrapolation can give silly answers. A mark above 100 is not possible, so the prediction must be checked against the context.

Exam tip

Exam tip. Say whether your prediction is interpolation (reliable) or extrapolation (less reliable), and give a reason.

Exercise 13E Line of best fit, interpolation and extrapolation

Non-calculator

FThe line of best fit is y = 5x + 40. Predict the mark for 4 hours of study.

Done

FUsing y = 5x + 40, predict the mark for 4.5 hours.

Done

PSUsing y = 5x + 40, predict the mark for 12 hours. Is this interpolation or extrapolation?

Done

PSA line of best fit is y = 3x + 10, where x is the number of ice creams and y is the total cost in pence. Find the cost of 6 ice creams.

Done

RAli says that the line y = 5x + 40 predicts that a pupil who studies for 0 hours scores 40. Is this interpolation? Explain.

Done

RExplain why extrapolation may not be reliable for predicting marks after 20 hours of study. Accept any answer that says the pattern may not continue, and marks have a maximum.

Done

PSA line of best fit gives y = 2x + 30. Use it to find the value of y when x = 5, and say whether this is reliable if the data is for x from 1 to 10.

Done

RTwo pupils draw different lines of best fit. Explain why their predictions might be different, even if both lines are reasonable.

Done

Chapter summary

Skill	Method
Mean	Total ÷ number of values
Median	Middle value after ordering (average of the two middle values if needed)
Mode and range	Most common value; biggest − smallest
Estimated mean from a grouped table	Sum of (f × midpoint) ÷ sum of f
Modal and median class	Highest frequency; class where the running total reaches (n + 1) ÷ 2
Correlation	Positive (rising), negative (falling), or none (no pattern)
Interpolation and extrapolation	Inside the data range: reliable. Outside: less reliable

Can you…?

I can find the mean, median, mode and range of a list
I can find midpoints and estimate a mean from a grouped table
I can find the modal class and the median class
I can compare two sets of data using an average and the range
I can plot a scatter graph and describe the correlation
I can use a line of best fit to interpolate and extrapolate, and say when a prediction is not reliable

Key terms

mean: the total divided by the number of values
median: the middle value when the values are in order
mode: the value that appears most often
range: the biggest value minus the smallest value
midpoint: the middle value of a class, halfway between its two ends
modal class: the class with the highest frequency
scatter graph: a graph that plots pairs of values as points
correlation: how the two values in a scatter graph are related
line of best fit: a straight line drawn by eye, close to most of the points

Exercise 13Z Check-up

Non-calculator

Find the mean of 4, 6, 8, 10, 12.

Done

Find the median of 9, 3, 7, 1, 5.

Done

Find the mode of 2, 4, 4, 6, 6, 6, 8.

Done

Find the range of 15, 3, 9, 12.

Done

Find the estimated mean from the table with midpoints 5, 15, 25, 35 and frequencies 4, 6, 8, 2.

Done

Find the modal class of the table with classes 0 < t ≤ 10, 10 < t ≤ 20, 20 < t ≤ 30, 30 < t ≤ 40 and frequencies 4, 6, 8, 2.

Done

Describe the correlation of a scatter graph with points that fall from top left to bottom right.

Done

The line of best fit is y = 4x + 20. Predict y when x = 3.

Done

Compare the means of two classes: 10, 12, 14 and 11, 12, 13. Which mean is bigger, or are they equal?

Done

Total 25 marks. Allow about 25 minutes.

Exercise 13X Exam-style questions

Non-calculator

FFind the mean and the median of 3, 8, 5, 10, 4.[2 marks]

Done

FEstimate the mean of the grouped data with midpoints 2, 7, 12 and 17 and frequencies 2, 2, 4, 2.[3 marks]

Done

FFind the modal class and the class containing the median of the table with frequencies 2, 2, 4, 2 (n = 10).[3 marks]

Done

FTwo classes have a mean of 50. Class X has a range of 30 and class Y has a range of 12. Which class has more consistent marks? Explain.[2 marks]

Done

FA scatter graph shows the number of hours of sunshine and the number of ice creams sold. The points rise from left to right. Describe the correlation.[2 marks]

Done

FThe line of best fit is y = 3x + 12. Predict the value of y when x = 10. Is this interpolation or extrapolation if the data is from x = 1 to 5?[3 marks]

Done

FTwo scatter graphs have points with positive correlation. Explain why it is not certain that a high value of x gives a high value of y for every point.[3 marks]

Done

FFind the estimated mean of the grouped data: classes 0 to 4, 5 to 9, 10 to 14, 15 to 19 with frequencies 1, 3, 4, 2 (midpoints 2, 7, 12, 17).[4 marks]

Done

FA set of six values has a mean of 7 and a range of 6. The smallest value is 4. Find the largest value.[3 marks]

Done

← Chapter 12: Probability of combined events

Averages from grouped data, scatter graphs and correlation

By the end of this chapter you will be able to

You should already know

Where this leads

Exercise P Do you remember?

Mean, median, mode and range of a list

Key ideas

Watch out

Exam tip

Exercise 13A Mean, median, mode and range

Grouped frequency tables

Key ideas

Watch out

Exam tip

Exercise 13B Grouped data: midpoints, modal and median classes

Comparing two sets of data

Key ideas

Watch out

Exam tip

Exercise 13C Comparing two sets of data

Scatter graphs and correlation

Key ideas

Watch out

Exam tip

Exercise 13D Scatter graphs and correlation

Line of best fit, interpolation and extrapolation

Key ideas

Watch out

Exam tip

Exercise 13E Line of best fit, interpolation and extrapolation

Chapter summary

Can you…?

Key terms

Exercise 13Z Check-up

Exercise 13X Exam-style questions