Edexcel IGCSE

Statistical measures

A statistical measure squeezes a whole set of data into one number — either a typical value, which is an average, or a description of how spread out the values are. It is section 6.2 of the Edexcel International GCSE Mathematics A specification, and almost every question on it reduces to the same move: work out which measure is being asked for, then either total the data or put it in order, because each measure needs one or the other.

Edexcel IGCSE 4MA1Specification 6.235 approved practice questions

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Before you start

What you need to be able to do already

If any of these is shaky, fix it first — every worked example below leans on all three.

Divide accurately and keep decimal answers exactDecimals
Work out $111 \div 8$ . It is $13.875$ . A mean is a division that rarely comes out whole, and rounding it part-way through is a misconception catalogued on this subtopic, so the arithmetic has to be comfortable before the statistics can be.
Order a list of numbers and find the value halfway between two of themFractions
Put $8, 15, 12, 21$ in order and find the number halfway between the middle two. Ordered it is $8, 12, 15, 21$ , and halfway between $12$ and $15$ is $13.5$ . That is exactly how a median of an even-sized data set is found, and how every class midpoint is found too.
Read a value off a cumulative frequency diagram
If a cumulative frequency curve is drawn for $200$ people, at what cumulative frequency do you read across to estimate the median? At $100$ , which is half of the total. Two of the eight objectives in this section are read off a cumulative frequency diagram rather than calculated, so for those the reading is the method.

The idea

What is statistical measures?

An average is a single number standing in for a whole data set, and you have three of them to keep apart: the mean, where you add every value and divide by how many there are; the median, the middle value once you have put the data in order; and the mode, the value that occurs most often. A measure of spread tells you how far apart the values are — the range is the largest minus the smallest, and the interquartile range covers only the middle half. Each is a fixed recipe, and your whole job is to match the recipe to the word the question used.

The notation, and how to read it

Every worked solution below is written in these five symbols.

SymbolReads asMeans
$\overset{x}{ˉ}$ x bar, the meanThe mean of a data set, worked out as $\overset{x}{ˉ} = \frac{\sum x}{n}$ : the total of the values divided by how many there are.
$\sum$ sigma, the sum ofAdd up everything that follows it. In a frequency table $\sum f$ is the total frequency — the number of data values, not the number of rows.
$\frac{\sum f x}{\sum f}$ the estimated mean for grouped dataMultiply each class midpoint $x$ by its frequency $f$ , total those products, and divide by the total frequency. The answer is an estimate.
$Q_{1}, Q_{2}, Q_{3}$ the lower quartile, median and upper quartileThe three values cutting ordered data into four equal parts. $Q_{2}$ is the median; $Q_{1}$ and $Q_{3}$ are the medians of the lower and upper halves, and when the count is odd the median itself belongs to neither half.
$Q_{3} - Q_{1}$ the interquartile rangeThe spread of the middle half of the data. Upper minus lower, always in that order, so a correct interquartile range is never negative.

The one idea underneath all of it

Every measure here is settled in one of two ways: by arithmetic on the values, or by where a value sits once you have ordered them. The mean and the range are arithmetic — total them, or take the smallest from the largest — and need no sorting at all. The median, the mode and the quartiles are all read off the data once it is in order, so ordering is not tidiness, it is the calculation. Deciding which kind you need is your first move, and confusing one measure for another is the most-recorded family of errors on this subtopic.

The smallest possible example

Find the mean of $3, 8, 6, 9, 4$ .

$6$

Add them: $3 + 8 + 6 + 9 + 4 = 30$ . There are $5$ values, so divide: $30 \div 5 = 6$ . Both halves are needed — stopping at $30$ gives the total and not the mean, and our question bank records exactly that as a misconception on this subtopic. Every mean question below is this, with the values buried in a table.

Key terms

The words a question will use

Exam wording is precise. These five carry the meaning a question depends on.

TermWhat it meansNotation
Mean
Add every value, then divide by how many there are. It uses all of them, so one unusually large or small value drags it — which is exactly when you should be quoting the median instead. $\overset{x}{ˉ} = \frac{\sum x}{n}$
Median
The middle value once you have put the data in order. With an even number of values there is no single middle, so take the mean of the middle two: $8, 10, 12, 21$ has median $11$ . $Q_{2}$
Mode and modal class
The value occurring most often. A grouped table does not reveal the exact mode, so you name the class with the highest frequency instead — and the answer is that class interval, not the frequency that made it modal. $20 \leq t < 30$
Range
Take the largest value and subtract the smallest, in that order, so your answer can never be negative. It measures spread rather than average, and because it uses only two of the values a single extreme one distorts it completely. $largest - smallest$
Interquartile range
The upper quartile minus the lower quartile: the spread of the middle half, with the extreme quarter at each end discarded. That is exactly why it is the more reliable measure of spread. $Q_{3} - Q_{1}$

The method

How to work out a statistical measures

Five steps, in this order. The worked examples below point back to these numbers.

Read which measure the question names, and whether the data is a list, a frequency table or a grouped table.Mean, median, mode, range and quartiles are five different recipes, and doing one correctly when another was asked scores nothing. Confusing the mean with the median, and the range with the mean, are catalogued as two separate misconceptions on this subtopic — this step is where both are avoided.
For the mean, total the values and divide by how many there are. $3 + 8 + 6 + 9 + 4 = 30$ , and $30 \div 5 = 6$ . Divide by the number of values, not by the number of different values and not by the largest one. In a frequency table the count is the total frequency $\sum f$ , so add that column up rather than counting rows.
For the median, mode or quartiles, write the data out in order first. $8, 15, 12, 21, 17, 10, 13$ becomes $8, 10, 12, 13, 15, 17, 21$ , and the median is the $4$ th of the seven values, $13$ . Picking the middle of an unordered list is not the median, and quartiles found without ordering are meaningless.
For grouped data, use the midpoint of each class as its value, and call the answer an estimate.The class $0 < m \leq 5$ has midpoint $2.5$ . Multiply each midpoint by its frequency, total the products, and divide by the total frequency: $\frac{\sum f x}{\sum f}$ . It is an estimate because the individual values inside each class are no longer known, and the word ‘estimate’ is part of the answer.
Check the answer sits inside the data, and that a spread has not come out negative.A mean or median must lie between the smallest and largest values. A range or interquartile range is a subtraction taken larger minus smaller, so a negative answer means it was done the wrong way round — and both of those reversals are recorded misconceptions here. Ten seconds of checking catches them.

Use it when

The question names an average or a measure of spread — mean, median, mode, modal class, range, quartile or interquartile range — and gives you a list, a frequency table, a stem-and-leaf diagram or a cumulative frequency curve.

Do not use it when

The question is about drawing or reading the diagram itself rather than measuring the data: pie charts, histograms and scatter graphs belong to section 6.1. Also, a grouped table never yields an exact mean — the answer to ‘calculate the mean’ from grouped data is an estimate of the mean, and saying so is part of the answer. And a modal class is a class interval such as $20 \leq t < 30$ , never the frequency that made it the modal one.

Worked examples

6 IGCSE Maths practice questions on statistical measures, fully worked

One per objective, so the range is covered rather than the same question repeated. Every step is written out, and the answer is shown — this is study material, not a test.

Example 1

Work out the mean of the following numbers: 12, 15, 18, 21, 24. [2 marks]

Mean from a list

Worked solution

Step 1: First, add up all the numbers: $12 + 15 = 27$ .
Step 2: Add the next number: $27 + 18 = 45$ .
Step 3: Add the next number: $45 + 21 = 66$ .
Step 4: Add the last number: $66 + 24 = 90$ .
Step 5: Now, count how many numbers there are: 5 numbers.
Step 6: Divide the total (90) by the number of values (5): $90 \div 5 = 18$ .

Answer18

Example 2

The scores of six students in a test are: 12, 15, 18, 20, 22, 24. What is the mean score? [2 marks]

Understand the concept of average

Worked solution

Step 1: Add all the scores together: $12 + 15 + 18 + 20 + 22 + 24 = 111$ .
Step 2: Count how many scores there are: there are 6 scores.
Step 3: Divide the total by the number of scores: $111 \div 6$ .
Step 4: $111 \div 6 = 18.5$ .

Answer18.5

Example 3

Here are the ages of 7 people: 22, 18, 29, 35, 22, 41, 31. Work out the median age. [2 marks]

Median from a list

Worked solution

Step 1: List all the ages: 22, 18, 29, 35, 22, 41, 31.
Step 2: Sort the ages in order from smallest to largest: 18, 22, 22, 29, 31, 35, 41.
Step 3: There are 7 numbers in total (odd), so the median is the middle number.
Step 4: The middle number is the 4th value in the sorted list, which is 29.

Answer29

Show 3 more worked examples

Example 4

The table shows the lengths, in cm, of 40 leaves sampled from a tree. Estimate the mean length of the leaves. [2 marks]
Length (cm) Frequency
0 < l ≤ 10 8
10 < l ≤ 20 16
20 < l ≤ 30 12
30 < l ≤ 40 4

Length (cm)	Frequency
0 < l ≤ 10	8
10 < l ≤ 20	16
20 < l ≤ 30	12
30 < l ≤ 40	4

Calculate an estimate for the mean for grouped data

Worked solution

Step 1: Find the midpoint of each class interval: 5, 15, 25, 35.
Step 2: Multiply the midpoint by the frequency: $(5 \times 8) = 40$ , $(15 \times 16) = 240$ , $(25 \times 12) = 300$ , $(35 \times 4) = 140$ .
Step 3: Add these products to get the estimated total length: $40 + 240 + 300 + 140 = 720$ .
Step 4: Divide by the total frequency (40): $720 \div 40 = 18$ .

Answer18 cm

Example 5

The frequency table shows the number of goals scored by a football team in 20 matches.
Goals scored Frequency
0 5
1 8
2 4
3 2
4 1

Work out the modal number of goals. [1 mark]

Goals scored	Frequency
0	5
1	8
2	4
3	2
4	1

Identify the modal class for grouped data

Worked solution

Step 1: The mode is the value that appears most often.
Step 2: Look at the frequency column: 5 teams scored 0 goals, 8 teams scored 1 goal, 4 scored 2, 2 scored 3, 1 scored 4.
Step 3: The highest frequency is 8, which corresponds to 1 goal.
Step 4: So the modal number of goals is 1.

Answer1

Example 6

A set of five numbers has a range of 10. The smallest number is 8. What is the largest number in the set? [1 mark]

Range

Worked solution

Step 1: Recall that the range of a data set is the largest number minus the smallest number.
Step 2: We know the range is 10 and the smallest number is 8.
Step 3: Let the largest number be x. Then x - $8 = 10$ .
Step 4: Solve for x by adding 8 to both sides: x = $10 + 8 = 18$ .

Answer18

Common mistakes

Where marks actually get lost

Learnly tags 12 distinct misconceptions on this subtopic. These are errors that show up in real student work — not generic revision advice.

Examiner traps

Adding instead of averaging
What goes wrong: Students may add all numbers but forget to divide by the count, giving the sum instead of the mean.
What to do instead: Remember: mean = total sum ÷ number of values. After adding, always divide by how many numbers you have.
Dividing by wrong number
What goes wrong: Students may divide by the wrong count, e.g., using the number of distinct values or the largest number instead of the total count.
What to do instead: Count all numbers, including repeats. Divide the total sum by that exact count, not by the biggest number or anything else.
Confusing mean with median
What goes wrong: Students may order the numbers and pick the middle one instead of calculating the sum divided by count.
What to do instead: Mean is the average: add all numbers then divide by how many. Median is the middle number when sorted. Don't mix them up.
Rounding too early
What goes wrong: Students may round intermediate steps, leading to an inaccurate final mean, especially when the sum is not a multiple of the count.
What to do instead: Keep exact values until the final step. Only round your answer at the end if the question asks for a specific format.
Confusing range with mean
What goes wrong: Students incorrectly calculate the mean (average) instead of the range. They add all values and divide by the count.
What to do instead: Remember: range is the difference between the highest and lowest values, not the average. Subtract the smallest from the largest.
Ignoring order when finding range
What goes wrong: Students subtract the first value from the last value without ordering the data set first.
What to do instead: Always sort the data from smallest to largest before finding the range. Then subtract the smallest from the largest.

See 6 more examiner traps

Getting a negative range
What goes wrong: Students subtract the largest value from the smallest, resulting in a negative number.
What to do instead: Range must be positive. Subtract the smallest value from the largest, not the other way around.
Using midpoint instead of range
What goes wrong: Students find the midpoint (average of highest and lowest) instead of the difference.
What to do instead: Range is the spread, not the center. Subtract the smallest from the largest, not add and divide by 2.
Including median in halves
What goes wrong: Students include the median when finding the lower and upper quartiles, instead of excluding it.
What to do instead: When finding quartiles, split the data into two halves but do not include the median in either half.
Subtracting Q1 from Q3 reversed
What goes wrong: Students mistakenly subtract Q3 from Q1 instead of Q1 from Q3, resulting in a negative value.
What to do instead: Always subtract the lower quartile (Q1) from the upper quartile (Q3): IQR = Q3 - Q1.
Wrong position for quartiles
What goes wrong: Students use incorrect formulas (e.g., (n+1)/4) for quartile positions in discrete data without ordering or handling odd/even counts.
What to do instead: First order the data. For Q1, find the median of the lower half; for Q3, median of the upper half.
Using mean instead of median
What goes wrong: Students calculate the mean of the lower or upper half instead of the median to find quartiles.
What to do instead: Quartiles are medians of the halves, not averages. Find the middle value of each half.

Exam technique

How the marks are actually awarded

Method marks are given for working a marker can follow. This is what to put on the page.

Where the marks are: Questions in this section run from $1$ to $4$ marks. Writing down a mode or a modal class is $1$ . A mean or a median from a list is $2$ : one mark for the total-and-divide, or for the correctly ordered list, and one for the value. Estimating the mean from a grouped table is $3$ or $4$ , and the marks sit on the midpoints and on $\sum f x$ — so add those two columns into the table, because they are earned before any dividing happens. Quartile and interquartile-range questions are $2$ or $3$ , split between the two quartiles and the subtraction. Reading a median off a cumulative frequency diagram carries a mark for the line you draw on the graph, so draw it.
Command words to expect: Work out the meanWrite down the modeFind the medianCalculate an estimate for the meanWrite down the modal classWork out the rangeFind the interquartile rangeUse the diagram to estimateShow thatCompare the two distributions
Accuracy and rounding: A mean rarely comes out whole. Give it exactly where the division terminates, as in $18.5$ , and otherwise to $3$ significant figures or $1$ decimal place unless the question says which. Never round part-way through — rounding $\sum f x$ before dividing is a misconception recorded on this subtopic. Carry the units into the answer: an estimated mean mass is $9.5$ kg, not $9.5$ . A modal class is written as the interval itself. And a median may legitimately land on a half, $13.5$ , even when every value in the list is a whole number.
Calculator: A calculator is allowed in every paper of this qualification, on both Foundation and Higher tiers, and it makes a mean a one-line sum. What it cannot do is put the data in order, which is the whole of the median and quartile work, and it will happily divide by the wrong number without complaint — so count the values, or total the frequency column, before you divide.

Check yourself

You should now be able to:

Say what an average is for, and choose sensibly between the mean, the median and the mode for a given data set.
Work out the mean of a list by totalling the values and dividing by how many there are.
Find the median by ordering the data and taking the middle value, or the mean of the middle two.
Write down the mode of a data set, and recognise when there is no mode or more than one.
Work out the range as largest minus smallest, and treat it as a measure of spread rather than an average.
Calculate an estimate of the mean for grouped data using class midpoints and $\frac{\sum f x}{\sum f}$ , and call it an estimate.
Identify the modal class of a grouped frequency table, giving the class interval and not its frequency.
Explain what a measure of spread tells you that an average on its own does not (Higher).
Find the lower quartile, the upper quartile and the interquartile range of a discrete data set (Higher).
Estimate the median from a cumulative frequency diagram by reading across at half the total frequency (Higher).
Estimate the quartiles and the interquartile range from a cumulative frequency diagram (Higher).

Specification coverage

The 11 things 4MA1 asks you to do

The assessable objectives for statistics and probability — statistical measures, in the order the specification lists them.

Mean from a list
Understand the concept of average
Median from a list
Calculate an estimate for the mean for grouped data
Range
Identify the modal class for grouped data

Show 5 more objectives

Mean from a frequency table
Estimate the median from a cumulative frequency diagram
Understand the concept of a measure of spread
Find the interquartile range from a discrete data set
Estimate the interquartile range from a cumulative frequency diagram

See who is stuck on statistical measures before you teach it.

Create a class, share the 8-character code, and set statistical measures as practice. Your students work through all 35 approved practice questions — not just the 6 on this page — and you see which objectives the class has not got yet. Free for individual students and teachers.

How Learnly works for teachers

Set up your class — free Practise on your own

What you need to be able to do already

Divide accurately and keep decimal answers exactDecimals

Order a list of numbers and find the value halfway between two of themFractions

Read a value off a cumulative frequency diagram

What is statistical measures?

The notation, and how to read it

The words a question will use

Mean

Median

Mode and modal class

Range

Interquartile range

How to work out a statistical measures

6 IGCSE Maths practice questions on statistical measures, fully worked

Work out the mean of the following numbers: 12, 15, 18, 21, 24. [2 marks]

The scores of six students in a test are: 12, 15, 18, 20, 22, 24. What is the mean score? [2 marks]

Here are the ages of 7 people: 22, 18, 29, 35, 22, 41, 31. Work out the median age. [2 marks]

The table shows the lengths, in cm, of 40 leaves sampled from a tree. Estimate the mean length of the leaves. [2 marks]Length (cm)Frequency0 < l ≤ 10810 < l ≤ 201620 < l ≤ 301230 < l ≤ 404

The frequency table shows the number of goals scored by a football team in 20 matches.Goals scoredFrequency0518243241Work out the modal number of goals. [1 mark]

A set of five numbers has a range of 10. The smallest number is 8. What is the largest number in the set? [1 mark]

Where marks actually get lost

Adding instead of averaging

Dividing by wrong number

Confusing mean with median

Rounding too early

Confusing range with mean

Ignoring order when finding range

Getting a negative range

Using midpoint instead of range

Including median in halves

Subtracting Q1 from Q3 reversed

Wrong position for quartiles

Using mean instead of median

How the marks are actually awarded

You should now be able to:

The 11 things 4MA1 asks you to do

See who is stuck on statistical measures before you teach it.

The table shows the lengths, in cm, of 40 leaves sampled from a tree. Estimate the mean length of the leaves. [2 marks]
Length (cm) Frequency
0 < l ≤ 10 8
10 < l ≤ 20 16
20 < l ≤ 30 12
30 < l ≤ 40 4

The frequency table shows the number of goals scored by a football team in 20 matches.
Goals scored Frequency
0 5
1 8
2 4
3 2
4 1

Work out the modal number of goals. [1 mark]