A list: 3, 7, 2, 7, 6
In order: 2, 3, 6, 7, 7
Mean = 25 ÷ 5 = 5; median = 6; mode = 7
Range = 7 − 2 = 5
Averages and range What does each average tell you about the data?
Foundation and Higher · AQA · 45 minutes
Students watch one value grow and see the mean slide while the median stays put. Then mean, median, mode and range from lists and frequency tables, a missing value from a mean, and comparing two sets. Slides, a vote that shows who's stuck, practice and homework.
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Five values sit as dots over a number line: 1, 4, 5, 6 and a. The little peak under the line is the pivot they balance on: the mean (the words under the graph say where it touches the line). The upright line at 5 is the median. Make a bigger. Which one moves, and which stays put?
Once: it touches at x = 5
The idea in one minute
An average is one number that stands for a whole set of data. The mean shares the total out equally: it is where the data balance on a number line. The median is the middle value once the data are in order. The mode is the value that comes up most often. The range, biggest take away smallest, says how spread out the data are.
The lesson goes picture first. Data are dots on a number line, so students see the mean as a balance point and see why one extreme value drags it while the median stays put. A frequency table is then shown as the same dots, grouped, so 'six matches had 1 goal' means six 1s, not one 6. Students find a missing value from a mean, choose the best average, and compare two sets with an average and the range, in context. The stretch is an estimate of the mean from grouped data.
By the end, students can
- find the mean, median, mode and range of a list
- find them from a frequency table, without mixing up values and frequencies
- find a missing value when the mean is given
- choose the best average for a set of data and say why
- compare two sets of data with an average and the range, in context
Key words
- mean
- median
- mode (modal value)
- range
- frequency
- outlier (an extreme value)
- estimate (from grouped data)
Before this lesson
Students should add, multiply and divide, with a calculator where it helps, order numbers, decimals included, and read a simple table.
The lesson, step by step
45 minutes in total. The stretch is optional.
5 min 1 · Hook · students think
Where does it balance?
Five values as dots; one grows. Students predict which average moves.
Say“Five test scores as dots on a number line: 1, 4, 5, 6 and 9. Where would a ruler balance on one finger? Now I make the 9 bigger. Which average changes?”
Ask- If the 9 became 20, would the middle score change?
- Would the total change? So would the mean?
Watch forExpecting both averages to move together: ask what the middle value is before and after.
For the teacher
Both averages start at 5: the total is 25, shared by 5 values, and 5 is the middle score.
As the last value grows, the total grows, so the mean slides right. The middle value is still 5. One extreme value drags the mean and leaves the median alone: that is why we choose between them.
Vote first (Ask the class): mean moves, median moves, both, neither. Then move a up from 9 on the slide (keep it at 6 or more, so the median stays 5).
Shorter: Show the slide once at a = 9 and once at a = 19; skip the vote (−2 min).
12 min 2 · Explain · you explain
Four numbers from a list, then a table
Mean, median, mode and range of a list; then of a frequency table, seen as the same dots.
Say“Order first, every time. Then the mean shares the total, the median is the middle, the mode is the most common and the range is the spread.”
Ask- Six values: which one is in the middle?
- In the table, how many 1s are there? Write them out.
- Is the mode 6, or 1? What does the 6 count?
Watch forTaking a frequency as a value: 'six matches had 1 goal' is six 1s.
An even number: 4, 8, 1, 9, 3, 5
In order: 1, 3, 4, 5, 8, 9
Two in the middle, 4 and 5: median = 4.5
Mean = 30 ÷ 6 = 5
A table: goals in 20 matches
Goals 0, 1, 2, 3, 4; matches 3, 6, 5, 4, 2
Total goals = 0×3 + 1×6 + 2×5 + 3×4 + 4×2 = 36
Mean = 36 ÷ 20 = 1.8 goals
Mode = 1 goal; median = 2 goals; range = 4
For the teacher
First slide: a list. Order it, then find all four. The mean line sits where the dots balance; the median is the third of five dots.
Second slide: six values. The median is halfway between the two middle values, and need not be one of the data.
Third slide: a frequency table is a dot plot written as numbers. Each row is a stack. The total of the data is each value times its frequency, added up; divide by the number of matches.
Fourth slide: the median of 20 values is halfway between the 10th and 11th. Add up the frequencies as you go (3, 9, 14…) to find them. The mode is the value with the biggest frequency, not the frequency.
Last slide: which average, and how to compare. The mean uses every value but an extreme one drags it; the median ignores extremes; the mode is the only one for words, such as colours. To compare two sets, say one thing about an average and one about the range, each in context.
On the table slide, point to each stack of dots: a row of the table is a stack. Divide by 20 matches, not by 5 rows.
Shorter: Skip the even list; one slide for the table (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one. Then you do its twin, which is almost the same.”
Ask- What do you divide by: the number of rows, or the number of people?
Watch forA missing value: dividing the mean by the count instead of finding the total first.
Median of an even list
4, 8, 1, 9, 3, 5
In order: 1, 3, 4, 5, 8, 9
Median = (4 + 5) ÷ 2 = 4.5
Then theirs
Find the median of 7, 2, 10, 4, 6, 3
Mean from a table
Goals 0 to 4; matches 3, 6, 5, 4, 2
Total goals = 36, matches = 20
Mean = 36 ÷ 20 = 1.8
Then theirs
Pets 0, 1, 2, 3; people 4, 7, 6, 3. Mean number of pets?
A missing value
The mean of 5, 8, x and 11 is 9
Total = 4 × 9 = 36
x = 36 − 24 = 12
Then theirs
The mean of 6, x, 10, 7 and 12 is 8. Find x.
For the teacher
Work the first example aloud, writing each line. Then students solve its twin while you watch who orders the list first.
Do the same for each pair. The pets twin catches anyone dividing by 4 rows; the missing value catches anyone who never finds the total.
Open the answers slide once most have finished their twins.
Shorter: Do the first two pairs only (−3 min).
15 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- How many people are in the table? How many shoes did you add up?
Watch forDividing by the number of rows, or adding the sizes without their frequencies.
Show your working on paper: order the data first. The stretch is optional.
The question that shows who's stuck
- Shoe sizes of 20 people: size 4: 2 people, size 5: 5 people, size 6: 8 people, size 7: 5 people. What is the mean shoe size?
In your Classroom, as they answer (an example) - A8
- B9
- C3
- D5
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- Find the mean of 4, 9, 2, 7 and 8.
- Find the median of 14, 3, 8, 19, 6 and 10. Then drag the marker to it on the number line.They drag a marker on a number line from 0 to 20.
- Brothers and sisters of 25 students: 0: 5 students, 1: 9 students, 2: 7 students, 3: 3 students, 4: 1 student. What is the mode?
- The same 25 students: 0: 5 students, 1: 9 students, 2: 7 students, 3: 3 students, 4: 1 student. Find the mean number of brothers and sisters.
- The mean of 7, 9, x, 4 and 10 is 8. Solve the equation for x, one line at a time.They solve (7 + 9 + x + 4 + 10)/5 = 8 line by line, each line checked as they write it.
- Which average would you choose for each set of data? Sort each one into its group.They sort 6 items into Mean and Median and Mode.
- In a quiz, team A's scores had a mean of 62 and a range of 30. Team B's had a mean of 58 and a range of 12. Which is the best comparison?
- The mean of six numbers is 7. A seventh number is added, and the mean of all seven is 8. What is the seventh number?
- Stretch: journey times of 20 students. 0 < t ≤ 10 minutes: 4 students; 10 < t ≤ 20: 10 students; 20 < t ≤ 30: 6 students. Estimate the mean time in minutes.
Support · For a table: write one row out as a list (size 5, five times: 5, 5, 5, 5, 5). The total for that row is 5 × 5. Add every row's total, then divide by the number of people.
For the teacher
Everyone chooses before anyone explains. Ask a student for each answer to say how they got it, then ask the class: how many people are in the table, and what was divided by what?
Questions 1–2 build fluency; 3–4 change the look (a table); 5–6 ask for reasons; 7–8 are exam-style. The stretch estimates a mean from grouped data.
Students show their working on paper or the board; the quiz keeps their final answers.
If many chose a wrong answer, write the table out as its 20 values together before they practise.
Shorter: After the vote, questions 1–4 only (−6 min).
Longer: The stretch, then: why is a mean from grouped data only an estimate? (+10 min).
5 min 5 · Exit · students show
Mean or median?
Place a mean on a number line, then say which average is typical.
Say“Five values: 2, 3, 3, 8 and 9. Put the mean on the number line, then answer the second question.”
Watch forChoosing the mean because it is 'the average', without looking at the data.
Place the mean, then one or two sentences.
- Five values: 2, 3, 3, 8 and 9. Drag the marker to their mean on the number line.They drag a marker on a number line from 0 to 10.
- For 2, 3, 3, 8 and 9, the median is 3. Which would you give as the typical value, the mean or the median? Say why in a sentence or two.
For the teacher
Place the mean of 2, 3, 3, 8 and 9 on the number line.
Then say in a sentence or two which average you would give as the typical value, and why.
The second answers show who can choose an average with a reason: start the next lesson with two of them. Set the homework.
Longer: Make up five values whose mean is bigger than every value but one (+5 min).
One to one: ask the student to sketch every set as dots on a number line before working anything out, and to say each answer as a sentence about the data (the typical match had 2 goals).
Homework: lists, tables and choosing
Order the data first, and show your working. For a table, divide by the total frequency. Answer questions 4 and 5 in full sentences.
- Find the mean, median, mode and range of 5, 9, 3, 9, 4, 6.
- Books read last month by 20 students: 0 books: 2 students, 1: 8, 2: 6, 3: 3, 4: 1. Find the mean, the median and the mode.
- The mean of 8, 12, x and 15 is 11. Find x.
- Write five whole numbers with a mean of 6, a median of 5 and a range of 8.
- Class A's test marks: mean 64, range 35. Class B's: mean 71, range 18. Compare the two classes in two sentences.
- Heights of 20 plants, in cm. 140 < h ≤ 150: 3; 150 < h ≤ 160: 9; 160 < h ≤ 170: 6; 170 < h ≤ 180: 2. Estimate the mean height. Why is it only an estimate?
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching averages and range: common questions
How do you explain the mean visually?
Draw the data as dots on a number line. The mean is where a ruler would balance on one finger: the distances of the dots on the left add up to the distances on the right. Move one dot far out and the balance point has to follow it, while the middle dot (the median) stays where it was.
How do you find the mean from a frequency table?
Multiply each value by its frequency, add those up to get the total of all the data, then divide by the total frequency (how many people or things there are), never by the number of rows. Writing one row out as a list first (six matches with 1 goal: 1, 1, 1, 1, 1, 1) shows students why.
When should students use the mean, median or mode?
The mean uses every value, so it suits numbers with no extreme values. The median isn't moved by an extreme value, so it suits data such as pay or house prices. The mode suits data in words (colours, brands) and questions about what is most common. In an exam, the reason matters as much as the choice.
Is this averages lesson suitable for Foundation GCSE?
Yes. Averages and range from lists and frequency tables, missing values and comparing sets are on both tiers. The stretch, an estimate of the mean from grouped data, appears on both tiers' papers too, and is a good first step towards Higher work on grouped data.
Can I use the lesson without creating an account?
Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: S4 (median, mean, mode and modal class; range, including outliers) and S2 (frequency tables). Estimating a mean from grouped data is part of S4.
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