This Year 8 statistics lesson is written for pupils aged 12-13 and follows the National Curriculum. In short, it shows you how to draw pie charts, scatter graphs and compare averages. Before you start, it helps to revise constructions and congruence and transformations, because this lesson builds on them.
Year 8 statistics lesson: what you will learn
This lesson covers averages and range, frequency tables, grouped data, pie charts, scatter graphs and comparing distributions.
Starter activity
Ask five friends how many hours of sleep they had last night and how many glasses of water they drank today. Write your results as a list. Find the mean and the median of each list. Do the two averages tell the same story? Which would you use if one friend had been awake all night?
Averages and spread
Definition: mean, median and mode
An average is a single value that represents a whole set of data.
- The mean is the total divided by the number of values. It uses every value but is pulled by very large or small ones.
- The median is the middle value when the data are in order. It is not affected by extreme values.
- The mode is the most common value. It is the only average for non-numerical data such as favourite colours.
Definition: range
The range is the largest value minus the smallest. It measures spread: a small range means the data are consistent.
Worked example 1: finding averages
Find the mean, median, mode and range of 4, 7, 9, 10, 10.
Mean \(=\dfrac{40}{5}=8\). Median (the middle of five ordered values) \(=9\). Mode \(=10\). Range \(=10-4=6\). For an even number of values the median is halfway between the two middle ones: the median of 2, 5, 8, 11 is \(\dfrac{5+8}{2}=6.5\).
Frequency tables
Method: mean from a frequency table
- Multiply each value by its frequency.
- Add these to get the total of all the values.
- Divide by the total frequency.
Worked example 2: frequency table
Scores of 1, 2, 3 and 4 occur 3, 5, 4 and 3 times. Find the mean, mode and median.
Total frequency \(=15\). Total of scores \(=3+10+12+12=37\). Mean \(=\dfrac{37}{15}=2.47\) (2 d.p.). The mode is 2 (highest frequency). The cumulative frequencies are 3, 8, 12, 15, so the 8th value, the median, is 2.
Grouped data
Definition: midpoint and modal class
When data are in class intervals you cannot find the exact mean. Use the midpoint of each class as an estimate. The modal class is the class with the highest frequency.
Worked example 3: grouped data
Classes 0-10, 10-20 and 20-30 have frequencies 4, 6 and 11. Estimate the mean and find the modal class.
Midpoints are 5, 15 and 25. Estimated total \(=20+90+275=385\), total frequency 21, so the estimated mean is \(\dfrac{385}{21}\approx18.3\). The modal class is 20-30. The 11th of 21 values also falls in the class 20-30, so the median lies there.
Pie charts
Rule: pie chart angles
The whole circle is \(360^\circ\). Angle \(=\dfrac{\text{frequency}}{\text{total}}\times360^\circ\).
Worked example 4: drawing and reading a pie chart
In a survey of 40 pupils, 18 chose football, 10 rugby, 8 tennis and 4 other. Find the angles. Then, for a chart of 72 pupils, how many are in a \(55^\circ\) sector?
The angles are \(162^\circ\), \(90^\circ\), \(72^\circ\) and \(36^\circ\), which add to \(360^\circ\). For 72 pupils, one pupil is \(5^\circ\), so a \(55^\circ\) sector is 11 pupils.
Scatter graphs
Definition: scatter graph and correlation
A scatter graph plots pairs of values to see whether there is a relationship, called correlation.
- Positive correlation: as one variable increases, the other increases (height and shoe size).
- Negative correlation: as one variable increases, the other decreases (hours of television and test score).
- No correlation: no pattern (shoe size and maths score).
Key fact: line of best fit
A line of best fit is a straight line drawn by eye through the middle of the points, with roughly equal numbers of points on each side. Use it to estimate values within the range of the data; predicting far outside is unreliable. Correlation does not prove that one thing causes the other.
Worked example 5: using a line of best fit
A line of best fit goes through \((2,10)\) and \((6,30)\). Estimate \(y\) when \(x=4\).
The gradient is \(\dfrac{20}{4}=5\) and the equation is \(y=5x\), so at \(x=4\) the estimate is 20.
Outliers and comparing data
Definition: outlier
An outlier is a value that is far from the rest. Use the median for skewed data (such as salaries), the mean when values are fairly symmetrical, and the mode for categories.
Worked example 6: effect of an outlier
In 12, 14, 15, 13, 48, which value is an outlier? Compare the mean and median.
The value 48 is an outlier. The mean is \(\dfrac{102}{5}=20.4\), but without 48 it is \(13.5\). The median is 14 either way, so it is a better summary here.
Note: comparing distributions
To compare two sets of data, write one comment about an average and one about the spread. Class A has mean 62 and range 20; class B has mean 58 and range 8. Class A did better on average, but class B was more consistent.
Watch out: common mistakes
- Finding the median before putting the data in order.
- Dividing the total by the number of classes instead of the total frequency.
- Giving the frequency instead of the value as the mode.
- Extending a line of best fit far beyond the data.
- Saying that correlation proves cause.
Worked example 7: finding a missing value
The mean of five numbers is 9. Four of the numbers are 4, 7, 11 and 12. Find the fifth number.
The total of five numbers with mean 9 is \(5\times9=45\). The four known numbers add up to \(4+7+11+12=34\). The fifth number is \(45-34=11\). Check: \((34+11)\div5=9\).
Worked example 8: comparing distributions
The boys' median time for a 100 m race is 14.2 s with range 3.1 s. The girls' median is 14.8 s with range 1.9 s. Compare the groups.
The boys' median is lower, so on average the boys were faster. The girls' range is smaller, so the girls' times were more consistent. Always mention both an average and the spread, and write your comparison in the context of the question.
Worked example 9: a pie chart angle
In a survey of 80 people, 18 chose tea. What angle is needed?
Each person is \(360\div80=4.5^\circ\). Tea: \(18\times4.5=81^\circ\). Or as a fraction: \(\dfrac{18}{80}\times360=81^\circ\).
Method: estimating the mean from grouped data
- Find the midpoint of each class.
- Multiply each midpoint by its frequency.
- Add up the products and divide by the total frequency.
- State that it is an estimate, because the exact values are not known.
Note: choosing an average
The mean uses all the data but is pulled by outliers. The median is the middle value and is not affected by extreme values. The mode is the most common value and is the only average that works for non-numerical data such as favourite colours. Say why you chose your average, using the shape of the data.
Watch out: common mistakes
- Forgetting to order the data before finding the median.
- Finding the mean of a frequency table by adding the frequencies instead of multiplying value × frequency.
- Joining the points on a scatter graph: draw a straight line of best fit instead.
- Using a line of best fit far outside the data (extrapolation).
Did you know?
Florence Nightingale, the famous nurse, was also a pioneer of statistics. In 1858 she used pie-chart-like diagrams, which she called coxcombs, to show that more soldiers in the Crimean War were dying from disease than from wounds. Her charts convinced the government to improve hospital hygiene, and she became the first woman elected to the Royal Statistical Society.
Writing a good comparison
Many statistics questions end with “compare the two sets of data”. A full answer always contains two comparisons, each linked to the context. First compare an average, for example “the median time for Class A is 2 minutes shorter, so Class A were usually faster”. Then compare a measure of spread, for example “Class B has a smaller range, so their times were more consistent”. Saying only that one number is bigger than another does not earn the marks; you must say what it means for the pupils, the plants or the matches in the question.
Did you know?
Florence Nightingale, famous as a nurse, was also a pioneering statistician. In 1858 she drew a new kind of circular chart to show that more soldiers in the Crimean War died from disease than from wounds, and her diagrams persuaded the government to improve hospital hygiene and saved many lives.
Otto's tip
Mean, median, mode, range. Always order the data first and write the average you are finding on each line of your working.
How to use this Year 8 statistics lesson
First, read the key points so that you know where the Year 8 statistics lesson is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the statistics: scatter graphs and averages worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the statistics: scatter graphs and averages test and mark it with the test answers.
Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 8 statistics lesson gives you everything you need for the chapter.
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AnswersStatistics: scatter graphs and averages – Year 8 maths worksheet answers to download as PDF
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TestStatistics: scatter graphs and averages – Year 8 maths test to download as PDF
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