Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 1) Find the median of 5, 11, 8, 3, 10.
Answer
8. In order: 3, 5, 8, 10, 11, and the middle one is 8. - (Year 7, Unit 5, lesson 2) Write 27/4 as a decimal.
Answer
6.75. 27 ÷ 4 = 6 remainder 3, and 3/4 = 0.75. - (Year 7, Unit 6, lesson 3) In a frequency table the value 3 has frequency 5. What do those five 3s add up to?
Answer
15. Five lots of 3, which is 5 × 3.
Words used in this lesson
- mean
- the total shared out equally, found by total ÷ number of values. For example: mean of 2, 4, 9 is 15 ÷ 3 = 5.
- total
- all the values added together. For example: 3 + 8 + 4 = 15.
- frequency
- how many times a value appears. For example: 5 students have 2 siblings.
- fair share
- what each would get if everything were shared equally. For example: 12 sweets among 4 friends: 3 each.
- balance
- the level where the amount above equals the gaps below. For example: for 2, 4 and 9 the level 5 balances: 4 above, and gaps of 3 and 1 below.
Sharing out equally
Six students in a reading club read 3, 8, 4, 9, 5 and 7 books over a term. Some read a lot and some read few. Suppose every book is put in one pile and shared out so that every student gets the same number. Predict first: roughly how many books would each student get? Then move the level on the bars and see where above and below balance.
Level 4: there are 13 books above it and 1 book missing below it. The level is too low.
Show me the steps
- Level 4: the bars for 8, 9, 5 and 7 stick up by 4, 5, 1 and 3, which is 13 books above the level. Only the bar for 3 falls short, by 1. The level is too low.
- Level 6: above the level are 2, 3 and 1, which is 6 books. Below it are gaps of 3, 2 and 1, which is 6 books. The two totals match: 6 is the mean.
- Level 9: nothing is above it, and the gaps below are 6, 1, 5, 0, 4 and 2, which is 18 books. The level is too high.
- Share the 36 books equally between 6 students: 36 ÷ 6 = 6 each, the same value as the balanced level.
The mean is the fair share. Add all the values to get the total, then divide by how many values there are:
mean = total ÷ number of values.
Why it works: sharing the total equally between the values gives each the same amount, and that amount is the mean. It is also the level where the amount above exactly fills the gaps below, which is why the figure balances there.
From a frequency table, the total is the sum of (value × frequency) and the number of values is the sum of the frequencies.
A common mistake with a frequency table: dividing the total by the number of rows. The siblings table below has 4 rows but 20 students. A total of 25 divided by 4 is 6.25, which is bigger than the biggest value, 3. A mean always lies between the smallest and largest value, so that answer cannot be right. Divide by the frequencies' total, 20.
Worked examples
Five test scores out of 20: 12, 18, 11, 17, 17. Find the mean.
- Total: 12 + 18 + 11 + 17 + 17 = 75.
- Number of values: 5.
- Mean = 75 ÷ 5 = 15. (The median and the mode are both 17, so here the mean is lower than both.)
Number of siblings for 20 students is shown in the table. Find the mean.
| Siblings | Frequency | Siblings × frequency |
|---|---|---|
| 0 | 4 | 0 |
| 1 | 9 | 9 |
| 2 | 5 | 10 |
| 3 | 2 | 6 |
| Total | 20 | 25 |
- The frequencies add to 20 students, and the third column adds to 25 siblings in all.
- Mean = 25 ÷ 20 = 1.25 siblings. Nobody has 1.25 siblings; the mean is a fair share, not a real student.
Your go. The first step is given. A cyclist rides a mean of 30 km a day over 4 days. The first three days were 25, 35 and 28 km. How far was day 4?
- Four days with a mean of 30 km means a total of 4 × 30 = 120 km.
- Now find how much of the 120 km is already ridden, and what is left for day 4, then check.
Check the answer
25 + 35 + 28 = 88 km so far, and 120 − 88 = 32 km on day 4.
Check before you go on
1. In a table of pets per family, 1 pet has frequency 2, 2 pets has frequency 5 and 3 pets has frequency 3. What is the mean number of pets?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Find the mean of 3, 5, 7, 12 and 18.
3. A hockey team scored 3, 0, 2 and 3 goals in four matches. What was the mean number of goals per match?
Level 2 · Building
4. Cups of tea drunk in a day by 20 people: 1 cup by 3 people, 2 cups by 6, 3 cups by 8 and 4 cups by 3. What is the mean number of cups?
5. The mean of five numbers is 14. Four of them are 10, 18, 12 and 20. What is the fifth number?
6. Which of these lists has a mean of 10?
Level 3 · Stretch
7. A class of 20 students read a mean of 6 books each. Another class of 30 students read a mean of 5 books each. What is the mean for all 50 students?
8. The mean of 8 numbers is 7. A ninth number is added and the mean of all nine becomes 8. What is the ninth number?
Something to think about
Is it possible for every value in a data set to be below the mean? Can the mean be a number that nobody in the data set has?
One way to think about it
No to the first: if every value were below the mean, the total would be less than the mean times the number of values, so the mean could not be the fair share. The mean always lies between the smallest and the largest value. Yes to the second: two values, 2 and 8, have mean 5, which is neither of them.
Exit ticket
9. The mean of four numbers is 10. Which statement must be true?