This Year 9 statistics lesson is written for pupils aged 13-14 and follows the National Curriculum. In short, it shows you how to calculate averages from frequency tables and grouped data and compare distributions. Before you start, it helps to revise similarity and enlargement and probability: sample spaces and Venn diagrams, because this lesson builds on them.
Year 9 statistics lesson: what you will learn
This lesson covers averages and range, choosing an average, frequency tables, working backwards with the mean, grouped data and comparing distributions.
Starter activity
Everyone in the class writes down the number of letters in their first name. Make a frequency table. Find the most common number, the middle number and the average number. Do the three values agree? What would happen to each if someone with a 20-letter name joined the class?
Mean, median, mode and range
Definition: averages and range
Averages (measures of central tendency) give a single number that represents a set of data.
- The mean is the total of all the values divided by how many values there are.
- The median is the middle value when the data is in order. With an even number of values, it is halfway between the two middle ones.
- The mode is the most common value. A set can have no mode, one mode or more than one.
- The range is the largest value minus the smallest. It is a measure of spread.
Worked example 1: five values
Find the mean, median and range of 4, 7, 9, 12, 8.
Mean \(= \dfrac{40}{5} = 8\). In order: 4, 7, 8, 9, 12, so the median is 8. Range \(= 12 - 4 = 8\).
Worked example 2: an even number of values
Find the median of 3, 9, 4, 12, 7, 5.
In order: 3, 4, 5, 7, 9, 12. The median is halfway between 5 and 7, which is 6.
Which average should I use?
Key fact: choosing an average
The mean uses every value, but it is pulled towards outliers (extreme values). The median is not affected by outliers. The mode is the only average that works for non-numerical data. For 2, 3, 3, 4, 48 the mean is \(\dfrac{60}{5} = 12\), which is far bigger than most of the values. The median, 3, is a better description of the typical value.
Frequency tables
Method: mean from a frequency table
- Add a column for \(x \times f\) (the value times its frequency).
- Add up the frequencies and the \(x \times f\) column.
- Divide \(\sum xf\) by \(\sum f\).
Worked example 3: goals scored
Find the mean, mode and median of the goals data.
| Goals \(x\) | Frequency \(f\) | \(x \times f\) |
|---|---|---|
| 0 | 2 | 0 |
| 1 | 5 | 5 |
| 2 | 8 | 16 |
| 3 | 3 | 9 |
| 4 | 2 | 8 |
| Total | 20 | 38 |
Mean \(= \dfrac{38}{20} = 1.9\). The mode is the value with the highest frequency, 2. With 20 values we need the 10th and 11th for the median. The cumulative frequencies are 2, 7, 15, so both are 2 and the median is 2.
Watch out: frequency tables
Do not add up the frequencies and divide by the number of rows. The total frequency is the number of data values.
Working backwards with the mean
Rule: total from the mean
Total \(=\) mean \(\times\) number of values.
Worked example 4: a missing value
The mean of five numbers is 8 and four of them are 6, 7, 9 and 10. Find the fifth. The mean of 6 numbers is 7; a seventh number makes the mean 8. Find it.
The total of the five is 40 and the four known add to 32, so the missing number is \(40 - 32 = 8\). For the second part, the total was 42 and becomes \(7 \times 8 = 56\), so the seventh number is \(56 - 42 = 14\).
Grouped data
Definition: class intervals, midpoint and modal class
Continuous data (heights or times) is often put into class intervals, for example \(140 \le h < 150\). We cannot know the exact values inside each group, so we estimate using the midpoint of each group. The modal class is the class with the highest frequency.
Worked example 5: estimating from grouped data
Estimate the mean height, and find the modal class and the class containing the median.
| Height \(h\) (cm) | Frequency \(f\) | Midpoint | \(f \times\) midpoint |
|---|---|---|---|
| \(140 \le h < 150\) | 4 | 145 | 580 |
| \(150 \le h < 160\) | 10 | 155 | 1550 |
| \(160 \le h < 170\) | 12 | 165 | 1980 |
| \(170 \le h < 180\) | 4 | 175 | 700 |
| Total | 30 | 4810 |
Estimated mean \(= \dfrac{4810}{30} = 160.3\) cm (1 d.p.); it is an estimate because we used midpoints. The modal class is \(160 \le h < 170\). With 30 values we need the 15th and 16th; the cumulative totals are 4, 14, 26, so the median lies in \(160 \le h < 170\). The range can only be estimated: at most \(180 - 140 = 40\) cm.
Comparing distributions
Worked example 6: comparing two teams
Team A has mean 12 and range 4; team B has mean 14 and range 10. Compare them.
Always compare one average and one measure of spread and refer to the context. Team B scores more on average, but team A is more consistent because its range is smaller.
Watch out: common mistakes
- Not putting the data in order before finding the median.
- Dividing by the number of rows in a frequency table rather than by the total frequency.
- Using the class boundaries rather than the midpoints in a grouped mean.
- Saying the mean of grouped data is exact (it is an estimate).
- Comparing only averages and ignoring the spread.
Worked example 7: finding the median from a frequency table
The table shows pets owned by 25 pupils: 0 pets: 6, 1 pet: 9, 2 pets: 7, 3 pets: 3. Find the median.
The median is the 13th value (since \((25+1)\div2=13\)). Cumulative frequencies: 6, 15, 22, 25. The 13th value is in the "1 pet" group, so the median is 1 pet.
Worked example 8: how an extra value changes the mean
The mean of 9 numbers is 12. A tenth number, 22, is added. Find the new mean.
Old total \(=9\times12=108\). New total \(=108+22=130\). New mean \(=130\div10=13\). The mean has increased by 1, because 22 is more than the old mean.
Worked example 9: estimated mean from a grouped table
Estimate the mean from this table: \(0\lt x\le20\): 10 values, \(20\lt x\le40\): 15 values, \(40\lt x\le60\): 5 values.
Midpoints 10, 30 and 50. Total \(\approx10\times10+15\times30+5\times50=100+450+250=800\) and there are 30 values, so the mean \(\approx800\div30=26.7\). It is an estimate because we do not know where in each group the values lie.
Worked example 10: comparing using an average and the range
Mia's five scores: 6, 7, 7, 8, 7. Noah's: 3, 10, 5, 9, 8. Both have a mean of 7. Compare them.
Mia's range is \(8-6=2\) and Noah's is \(10-3=7\). They have the same mean, but Mia is much more consistent. Always give one comment about the average and one about the spread.
Note: which average?
The mean uses every value and is best for symmetrical data. The median is not affected by extreme values and is best for skewed data, such as salaries or house prices. The mode is the only average for non-numerical data, such as favourite colours. The range tells you the spread, not the typical value.
Watch out: frequency tables
- In a frequency table the mean is (sum of value \(\times\) frequency) divided by the total frequency, not divided by the number of rows.
- The mode is the value with the highest frequency, not the frequency itself.
- In grouped data you can only estimate the mean, and you can only name the modal class, not the mode.
Did you know?
The word statistics comes from the Latin for state, because early statistics were facts collected by governments about their people, such as the Domesday Book of 1086 in England. Florence Nightingale, famous as a nurse, was also a pioneer of statistical charts and used them to persuade the government to improve hospitals.
Using statistics to answer questions
Method: comparing two distributions
- Calculate an average for each set of data, usually the mean or the median, and say which set has the higher average.
- Calculate a measure of spread, such as the range, and say which set is more consistent: the smaller the spread, the more consistent the data.
- Write two separate sentences, one about the average and one about the spread, and always refer to the context, for example "On average Team A scored more goals."
- Mention anything unusual, such as an extreme value, that might make one of the statistics misleading.
Worked example 11: the effect of an outlier
Five friends earn £8, £9, £9, £10 and £64 for a weekend job. Which average best represents a typical friend?
The mean is \(100\div5=£20\), but four of the five friends earn much less than this, so the mean does not describe them well. The value £64 is an outlier: it is far away from the rest. The median is the middle value, £9, and the mode is also £9. Both of these are close to what most of the friends earn, so the median is the best average here. The range, \(64-8=£56\), is also made very large by the single outlier.
Note: data in the news
Newspapers often report averages, such as the average house price or the average salary. These are usually medians, because a few very expensive houses or very high salaries would push the mean up and give a misleading picture.
Otto’s tip
For frequency tables, write the column \(x \times f\) first, add it up, then divide by the total frequency. It never fails. Hoot hoot!
How to use this Year 9 statistics lesson
First, read the key points so that you know where the Year 9 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: averages and grouped data 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: averages and grouped data 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 9 statistics lesson gives you everything you need for the chapter.
To sum up, regular practice works better than one long session. For instance, ten minutes a day is often enough. Besides, mistakes are useful, because they show what to revise next. Then, once a method feels easy, try a harder question. As a result, you build confidence step by step.
Similar pages
WorksheetStatistics: averages and grouped data – Year 9 maths worksheet with answers to download as PDF
AnswersStatistics: averages and grouped data – Year 9 maths worksheet answers to download as PDF
Multiple choice questionsStatistics: averages and grouped data – Year 9 maths multiple choice questions to download as PDF
QuizStatistics: averages and grouped data – Year 9 maths quiz to download as PDF
TestStatistics: averages and grouped data – Year 9 maths test to download as PDF
Test answersStatistics: averages and grouped data – Year 9 maths test answers to download as PDF
StartersStatistics: averages and grouped data – Year 9 maths starters to download as PDFOther chapters in this level
LessonNumber: approximation, error intervals and calculators – Year 9 maths lesson to download as PDF
LessonPercentage change and simple interest – Year 9 maths lesson to download as PDF
LessonExpanding binomials and taking out common factors – Year 9 maths lesson to download as PDF
LessonSolving linear equations and rearranging formulae – Year 9 maths lesson to download as PDF
LessonReal-life graphs – Year 9 maths lesson to download as PDF
LessonSequences and nth term – Year 9 maths lesson to download as PDF
LessonGraphs: linear and quadratic – Year 9 maths lesson to download as PDF
LessonDirect and inverse proportion – Year 9 maths lesson to download as PDF