GCSE Maths · Statistics · Higher · Lesson 24.3

GCSE Maths: Mean, median, mode and range (Higher)

Part of the GCSE Maths statistics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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24.3

Mean, median, mode and range

Edexcel 1MA1: S4
Grade 1-5

An average is a single number that tells us what is typical for a set of data. There are three averages, and each one answers the question “what is typical?” in a different way. We also need a measure of how spread out the data is. That measure is the range.

The mean is the total of all the values divided by how many values there are. It is the amount each value would be if the total were shared out equally. The picture shows data as dots on a number line. The mean is the balance point: the number line would balance on a pivot placed at the mean.

The values 2, 3, 3, 4 and 8 have total 20 and there are 5 of them. The mean is 20 ÷ 5 = 4, the balance point.

The median is the middle value when the values are written in order. With an odd number of values it is the middle one. With an even number of values there are two middle ones and the median is halfway between them. The mode is the value that occurs most often. There can be more than one mode, or none if every value occurs once. The mode is the only average that works for categorical data (for example, the most popular colour).

The range is the biggest value minus the smallest value. A small range means the data is close together (consistent). A big range means the data is spread out.

Which average should we use? The mean uses every value, so it is affected by very big or very small values called outliers. If five people earn £18 000, £20 000, £21 000, £22 000 and £150 000, the mean is pulled up by the biggest salary and is not typical. The median is not affected by extreme values, so it is better when there are outliers. The mode is best for categorical data, or when we want the most popular choice.

Finding a missing value. If you know the mean of a set of numbers you know their total: total = mean × number of values. Then subtract the values you already know to find the missing one.

Key ideas

Mean = total of the values ÷ number of values.
Median = middle value when the values are in order. For an even number of values, halfway between the middle two.
Mode = most common value. There may be more than one.
Range = biggest value − smallest value.
Outliers pull the mean towards them, but they do not change the median much.
Total = mean × number of values (useful for missing values).

Worked example 24.3.1

Work out the mean of 12, 15, 9, 18, 14 and 11. Give your answer to 1 decimal place.

Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.

Method. To find the mean, add up all the values and divide by how many values there are.

Add up all the values. The mean uses the total of all the data.
12 + 15 + 9 + 18 + 14 + 11 = 79
Count how many values there are. The mean shares the total equally between all the values, so we need to know how many there are.
There are 6 values.
Divide the total by the number of values. mean = total ÷ number of values.
79 ÷ 6 = 13.1667… = 13.2 to 1 decimal place

Check. The mean must lie between the smallest (9) and the largest (18) value, and 13.2 does. Also 13.2 × 6 = 79.2, which is close to the total 79. ✓

Watch out. The mean is not the middle value. Add every value, and divide by how many values there are (not by the largest value).

Worked example 24.3.2

Work out the median of 7, 3, 9, 4, 12 and 6.

Method. The median is the middle value when the data is in order.

Write the values in order, smallest to largest. The median is the middle value, so the data must be in order first.
3, 4, 6, 7, 9, 12
Count the values. We need to know where the middle is.
There are 6 values.
Find the two middle positions. With an even number of values there are two middle values. They are at positions n ÷ 2 and n ÷ 2 + 1.
6 ÷ 2 = 3, so the 3rd and 4th values.
Pick out the two middle values. Count along the ordered list.
3, 4, 6, 7, 9, 12
Find the number halfway between them. Add the two middle values and divide by 2 (their mean).
(6 + 7) ÷ 2 = 13 ÷ 2 = 6.5

Check. 3 values are below 6.5 and 3 values are above it. ✓

Watch out. Put the numbers in order first. The middle of an unordered list is not the median.

Worked example 24.3.3

Find the mode and the range of 5, 8, 5, 9, 7, 5, 8.

Method. Find the mode by counting and the range by subtracting. Both are done from the same list.

Count how often each value occurs. The mode is the value that occurs most often.
5 occurs 3 times
7 occurs 1 time
8 occurs 2 times
9 occurs 1 time
State the mode. 5 occurs most often.
Mode = 5
Find the biggest and smallest values. We need them for the range.
Biggest = 9, smallest = 5
Subtract. Range = biggest − smallest.
9 − 5 = 4

Check. 5, 5, 5, 7, 8, 8, 9: the value 5 appears most. 5 + 4 = 9, the biggest value. ✓

Watch out. The mode is a value from the list (5), not the number of times it occurs (3).

Worked example 24.3.4

The mean of five numbers is 8. Four of the numbers are 6, 9, 11 and 7. Work out the fifth number.

Method. If we know the mean, we can find the total: total = mean × number of values. Then take away the values we already know.

Find how many values there are altogether. The mean is shared between all the values, including the missing one.
4 known values + 1 missing value = 5 values
Find the total the values must make. total = mean × number of values.
8 × 5 = 40
Add up the known values.
6 + 9 + 11 + 7 = 33
Take the known total away from the full total. What is left is the missing value.
40 − 33 = 7

Check. Add the missing value in: 33 + 7 = 40, and 40 ÷ 5 = 8. ✓

Watch out. Multiply the mean by the number of values including the missing one (not by the number you already know).

Worked example 24.3.5

Five people earn £18 000, £20 000, £21 000, £22 000 and £150 000 a year. The owner says: “The average wage is £46,200.” Is the mean a good average to use? Which average would be better?

Method. Work out the mean and the median, then decide which one is more typical.

Work out the mean. Add the salaries and divide by 5.
18 000 + 20 000 + 21 000 + 22 000 + 150 000 = 231,000
231,000 ÷ 5 = £46,200
Work out the median. The salaries are already in order. The middle (3rd) one is the median.
Median = £21,000
Compare. Four of the five people earn about £20 000, so £46,200 is far above what most people earn. The very large salary (an outlier) has pulled the mean up.
The mean is not typical.
Choose the better average. The median is not affected by the one very large value.
The median (£21,000) is a better average.

Check. Four of the five wages are within £4 000 of the median, so the median describes most of the people. ✓

Watch out. When there is an extreme value (an outlier), the median is usually a better average than the mean.

Watch out

Always put the data in order before finding the median. And the mode is the value that occurs most often, not how many times it occurs.

Exercise 24C Mean, median, mode and range

Non-calculator

FWork out the mean of each set of numbers.

(a)4, 7, 10

(b)12, 15, 9, 14

(d)21, 18, 24, 17, 20

(e)2.5, 3.5, 4.5

(f)£2.40, £3.60, £1.50, £4.50

Done

FWork out the median of each set of numbers.

(a)5, 9, 2, 7, 3

(b)11, 4, 8, 15, 6

(d)20, 14, 18, 22, 11, 16

(e)1.5, 0.8, 2.2, 1.1

(f)−3, 5, 0, −8, 2

Done

FWrite down the mode of each set of data.

(a)3, 5, 2, 5, 7, 5, 2

(b)8, 6, 8, 9, 6, 8, 7

(c)red, blue, red, green, blue, red

(d)4, 7, 4, 9, 7, 2 (two modes)

Done

FWork out the range of each set of numbers.

(a)4, 9, 3, 12, 7

(b)15, 22, 18, 30, 25

(d)−3, 5, 0, 8

Done

FHere are the numbers of minutes that nine students took to eat lunch: 6, 9, 4, 9, 7, 5, 9, 8, 6. Work out (a) the mean, (b) the median, (c) the mode, (d) the range.

(a)Mean

(b)Median

(c)Mode

(d)Range

Done

PSFour friends bought drinks costing £2.40, £1.80, £3.20 and £2.60. Work out the mean cost of a drink.

Done

PSWork out the missing number each time.

(a)The mean of four numbers is 7. Three of them are 5, 8 and 9. Find the fourth number.

(b)The mean of five numbers is 12. Four of them are 10, 15, 11 and 14. Find the fifth number.

(c)The mean mass of six parcels is 4.5 kg. Five parcels weigh 3, 5, 4, 6 and 4.5 kg. Find the mass of the sixth parcel.

Done

PSChloe sat five tests. Her scores were 14, 17, 12, 15 and 17 out of 20. She wants a mean of 15 after a sixth test. What score does she need in the sixth test?

Done

RFive people earn £18 000, £20 000, £21 000, £22 000 and £150 000 a year. Which average, the mean or the median, gives a better idea of what the people typically earn? Give a reason.

Done

RAisha has to find the median of 3, 9, 4, 7 and 5. She says the median is 4 because it is the middle number in the list. Explain her mistake and work out the correct median.

Done

RThe list 4, 5, 6, 7, 8 has mean 6 and median 6. The number 8 is changed to 28, giving 4, 5, 6, 7, 28.

(a)Work out the new mean.

(b)Work out the new median.

(c)Which average changed more?

Done

PSThe mean of 8 numbers is 12. A ninth number, 21, is added. Work out the mean of the nine numbers.

Done

PSClass A has 10 students and a mean mark of 62. Class B has 15 students and a mean mark of 70. Work out the mean mark of all 25 students.

Done

RWrite down five numbers with a mean of 6 and a range of 7.

Done

RFour whole numbers have a mean of 5, a median of 4.5 and a mode of 3. Find the four numbers, written in order.

Done