Interpreting statistical data
Reading values from tables and statistical diagrams and drawing sensible conclusions from them, comparing two sets of data using an average and a measure of spread, and knowing the limits of what a set of data can show. Extended: comparing with the median and interquartile range.
What the syllabus says
C9.2 (Core)
- 1 Read, interpret and draw inferences from tables and statistical diagrams.
- 2 Compare sets of data using tables, graphs and statistical measures.
- 3 Appreciate restrictions on drawing conclusions from given data.
Notes and examples
- e.g. compare averages and ranges between two data sets.
E9.2 (Extended)
- 1 Read, interpret and draw inferences from tables and statistical diagrams.
- 2 Compare sets of data using tables, graphs and statistical measures.
- 3 Appreciate restrictions on drawing conclusions from given data.
Notes and examples
- e.g. compare averages and measures of spread between two data sets.
Quoted from the Cambridge IGCSE Mathematics 0580 syllabus for 2025 to 2027.
Reading tables and charts
Before you read a single value from a table or chart, spend a few seconds finding out what it shows:
- the title: what was measured, of whom, and when;
- the axis labels and units: hours or minutes, dollars or thousands of dollars;
- the scale: how much each small square or gridline is worth. On a bar chart going up in 10s, a bar halfway between 40 and 50 shows 45;
- any key: which shading or symbol stands for what.
Then read carefully. Take the value from the top of the bar, along to the axis, using a ruler on paper. Read tables along the row and down the column, and check you are in the right cell.
Drawing inferences. An inference is a conclusion that the data supports. Good inferences:
- are about what the data actually shows: "Saturday had the most visitors", "sales rose every month from April to July";
- use numbers from the data to back them up: "Saturday had 120 visitors, three times as many as Monday";
- look at proportions as well as amounts: "over a quarter of the week's visitors came on Saturday".
A trend is the general direction of the data over time: rising, falling, or rising then falling. Describe the overall pattern, not every single change.
Worked example 1
The bar chart shows the number of visitors to a small museum on each day of one week.
(a) Write down the busiest day. (b) How many more visitors came on Saturday than on Tuesday? (c) What fraction of the week's visitors came at the weekend (Saturday and Sunday)? (d) Write down one conclusion the manager could draw.
(a) The tallest bar is Saturday.
(b) Saturday: 120. Tuesday: 35 (halfway between 30 and 40). more.
(c) Total for the week: . Weekend: . Fraction .
(d) Nearly half the week's visitors ( is just under ) come on the two weekend days, so the museum needs the most staff at weekends.
Answer: (a) Saturday (b) 85 (c) (d) nearly half the visitors come at the weekend
Comparing two sets of data
To compare two sets of data properly, you need two kinds of statistic:
- an average (mean, median or mode), which says what is typical, which group is generally higher or lower;
- a measure of spread, which says how varied or consistent the data is. At Core level this is the range (largest value minus smallest value).
Two groups can have the same average and still be very different. A footballer who scores 2 goals every match and one who scores 0 in most matches and 8 in a few could both average 2 goals a match. Only the spread tells them apart.
How to write a comparison. Make two separate statements, and for each:
- say which group is higher or lower, quoting the statistic;
- say what that means in the context of the question.
For example, comparing the marks of two classes:
- "The mean for Class A (64) is higher than the mean for Class B (58), so on average Class A did better."
- "The range for Class A (38) is larger than the range for Class B (21), so Class B's marks were more consistent."
A smaller spread means the values are more consistent (closer together). Statements such as "Class A's range is bigger" earn little on their own: say what it means. And compare like with like: a mean with a mean, a range with a range.
Worked example 2
Amir and Bella play in a basketball team. Their points in six games are:
Amir:
Bella:
Compare the points scored by the two players.
Amir: mean ; range .
Bella: mean ; range .
Average: both means are 15, so on average they score the same number of points.
Spread: Bella's range (4) is much smaller than Amir's (11), so Bella's scoring is more consistent. Amir has better and worse games.
Answer: Same mean (15); Bella is more consistent (range 4 compared with 11)
Check yourself
Two machines fill bags of rice. The masses, in grams, of five bags from each machine are:
Machine A:
Machine B:
Work out the range for each machine.
Answer it in the portal, where your answer is checked as you type →
Comparing with the median and interquartile range Extended
Extended candidates compare data sets using the median and the interquartile range (IQR) as well. The IQR is the upper quartile minus the lower quartile: the spread of the middle half of the data. Section 9.3 shows how to find the quartiles.
The IQR is often a better measure of spread than the range, because it ignores the top quarter and the bottom quarter of the data. One unusually large or small value (an outlier) can make the range huge, but it hardly moves the IQR. For the same reason, the median is often a better average than the mean when there are outliers.
Comparisons are written exactly as before, one statement about the average and one about the spread, each in context:
- "The median time for the bus (22 minutes) is lower than for the bike (26 minutes), so the bus is usually quicker."
- "The IQR for the bike (4 minutes) is smaller than for the bus (6 minutes), so the bike times are more consistent."
Worked example 3 Extended
Kofi recorded his journey times to school, in minutes, on seven days by bus and seven days by bike.
Bus:
Bike:
(a) Find the median and the interquartile range for each method. (b) Compare the two methods. (c) Explain why the IQR is a better measure of spread than the range for the bus times.
(a) Bus, in order: . Median (4th value) . Lower quartile (2nd) , upper quartile (6th) . IQR .
Bike, in order: . Median . LQ , UQ . IQR .
(b) The median for the bus (22 min) is lower than for the bike (26 min), so the bus is usually quicker. The IQR for the bike (4 min) is smaller than for the bus (6 min), so the bike times are more consistent.
(c) The bus range is minutes, made huge by the single 40-minute journey. The IQR uses only the middle half of the data, so that one unusual day does not distort it.
Answer: Bus: median 22, IQR 6. Bike: median 26, IQR 4. Bus quicker on average; bike more consistent.
Check yourself Extended
The heights, in cm, of 11 seedlings are . Find the interquartile range.
Answer it in the portal, where your answer is checked as you type →
What the data can and cannot tell you
Data only supports conclusions about what it actually measured. Before you accept (or write) a conclusion, ask these questions.
- Is the sample big enough? Ten people is too few to say anything firm about a whole town. A larger sample gives more reliable results.
- Is the sample representative? People asked outside a gym, or at a train station at 8 am, are not typical of everyone. A survey only tells you about the kind of people it reached.
- When and where was the data collected? Ice cream sales in July say little about December; results from one school may not apply to another.
- Does it compare like with like? 56 passes out of 80 students is a lower pass rate than 45 out of 60, even though 56 is more. With different totals, compare fractions or percentages. Two pie charts can show the same angle for very different numbers of people.
- Is the diagram misleading? A vertical axis that does not start at zero makes small differences look large. Check the scale before believing your eyes.
- Is it about a group or an individual? "Class A has the higher mean" does not mean every student in Class A did better than every student in Class B, or that the top mark was in Class A.
- Does one thing cause the other? Two quantities rising together does not prove that one causes the other (more of this in 9.5).
- Is it beyond the data? A trend over six months may not continue next year.
When a question asks why a conclusion "may not be valid" or "may not be reliable", give a specific reason about that survey or diagram, not a general phrase.
Worked example 4
A newspaper asked 50 people at a train station at 8 am on a Monday how they travel to work. 40 of them said they use public transport. The newspaper reported: "80% of people in the city use public transport to get to work."
Give three reasons why this conclusion may not be reliable.
The figure itself is correct for the sample: . The question is whether it applies to the whole city.
- The sample was taken at a train station, where most people are using public transport. It is not representative of the city.
- It was taken at one time on one day (8 am on a Monday); people travelling at other times may travel differently.
- 50 people is a small sample for a whole city.
Answer: Biased place (a station), one time only, small sample
Check yourself
Group A: mean sleep 7.2 hours, range 3 hours. Group B: mean sleep 6.5 hours, range 5 hours. Which statement is correct?
Answer it in the portal, where your answer is checked as you type →
Key points
Before reading values, check the title, units, scale and key.
An inference is a conclusion supported by the data: back it up with numbers.
Compare two data sets with an average and a measure of spread, each in context.
A smaller range (or IQR) means the data is more consistent.
Extended: the median and IQR are not distorted by outliers the way the mean and range can be.
Conclusions are limited by sample size, how representative the sample is, when and where data was collected, and misleading scales.
With different totals, compare proportions (fractions or percentages), not raw counts.
Where marks are lost
Misreading the scale: taking each gridline as 1 when it is worth 10, or 2.
Comparing only averages, with no comment on spread.
Stating numbers ("A has a mean of 64, B has a mean of 58") without saying which is higher and what that means in context.
Saying a larger range means more consistent. It means less consistent.
Comparing counts from groups of different sizes, such as two pie charts with different totals.
Giving vague limits such as "it might be wrong" instead of a specific reason (the sample was taken outside a gym).
Claiming that a statement about an average applies to every individual in the group.
Related IGCSE Mathematics 0580 lessons
An original lesson written to the Cambridge IGCSE Mathematics 0580 syllabus (2025 to 2027). The worked examples and questions are our own, not Cambridge questions. Back to every IGCSE Mathematics 0580 lesson.