IGCSE Maths · Area 9 · Statistics

Classifying and Interpreting Data

Statistics starts before any calculation: you have to know what kind of data you are holding. This page covers discrete and continuous data, quantitative and qualitative data, primary and secondary sources, and the difference between a population and a sample. It finishes with why a sample has to be representative, and with the limits Cambridge expects you to state when you are asked what a set of data can and cannot show.

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Syllabus coverage

  • Classify and tabulate statistical data, for example using tally tables and two-way tables (Core C9.1, Extended E9.1)
  • Read, interpret and draw inferences from tables and statistical diagrams (C9.2.1, E9.2.1)
  • Compare sets of data using tables, graphs and statistical measures, for example comparing averages and ranges between two data sets (C9.2.2; Extended E9.2.2 compares averages and measures of spread)
  • Appreciate the restrictions on drawing conclusions from given data (C9.2.3, E9.2.3)
  • Distinguish between discrete and continuous data, and between quantitative and qualitative data, as part of classifying data
  • Recognise primary and secondary data, and identify a population and a sample, explaining why a sample must be representative (Extended: recognise bias in a sample and the limits on conclusions)

Part of the IGCSE Maths syllabus 0580.

Classifying statistical data: what each type of data allowsdataquantitative (numerical)qualitative (categories)discrete: counted, for example thenumber of goals in a matchcontinuous: measured, for example thetime to run 100 mcategories, for example eye colour ormake of car: only the mode can be foundcountedmeasureda fixed list of valuesno numbers, so no meanclassify the data first, then use only the statistics that are allowed
Classifying data as quantitative or qualitative, and what each type allows you to calculate.

Counting gives discrete data, measuring gives continuous data

Discrete data is data you count. It can take only particular values, usually whole numbers, with gaps between them: the number of goals in a match can be 0, 1, 2 or 3, but never 2.5. Continuous data is data you measure. It can take any value in a range, limited only by how precisely you can read the instrument: a time of 12.34 seconds, a mass of 1.375 kg, a temperature of 19.4 °C.

The test is one question: did you count it or measure it? Shoe size looks continuous because 6.5 is a real shoe size, but sizes come from a fixed list, so 6.37 is impossible, and that makes shoe size discrete. A length written as 12 cm because you measured to the nearest centimetre is still continuous data: the rounding is in your measurement, not in the data, and the true length could have been 11.6 cm.

DataTypeWhy
The number of goals scored in a matchDiscreteYou count them, so 2.5 goals is impossible
The time taken to run 100 mContinuousYou measure it, so 12.34 s is possible
Shoe sizeDiscreteA label from a fixed list: 5, 5.5, 6, 6.5, 7
The mass of a bag of flourContinuousMeasured on a scale, so any value in a range is possible
The number of cars passing a school gate in an hourDiscreteYou count them
The temperature of a classroomContinuousMeasured, so 19.4 °C is possible
Counting gives discrete data, measuring gives continuous data. If a value in between is possible because the quantity can be split, the data is continuous; if it is possible only because it is another label on the list, the data is discrete.

Quantitative and qualitative data

Quantitative data is numerical, and it is either discrete or continuous. Qualitative data describes a quality or a category rather than a quantity: eye colour, make of car, favourite subject, the letter grade in a test.

The difference decides which statistics you are allowed to work out. You can add quantitative data up, so a mean, a median and a range all exist. Qualitative data cannot be added at all, so the only average that means anything is the mode: the category that occurs most often. You can also count how many items fall in each category, and it is those counts that a tally table or a bar chart shows.

  • Quantitative: the number of pages in a book (discrete), the mass of the book in grams (continuous)
  • Qualitative: the colour of the book's cover, the language it is written in, its genre
  • The mean of qualitative data does not exist. 'The average colour of the cars is blue' is not a statistic
  • For qualitative data, report the mode or the counts in each category
Cambridge will offer you a mean of something qualitative as a wrong answer to spot. Colours, names and categories have no numerical value, so nothing can be added or divided.

Primary data and secondary data

Primary data is data you collect yourself, at first hand, for the question you are answering: measuring the heights of your class, counting the cars that pass the school gate, timing how long each student takes to solve a puzzle. Secondary data was collected by somebody else, for a different purpose: a table in a textbook, a figure on a website, last year's attendance records.

Primary data has one large advantage: you know exactly what was measured, how, and when, so you can be confident it answers your question. Its cost is time and effort. Secondary data is quick and often far larger, but you cannot check how it was collected, what definitions were used, or whether it is still current.

SourceAdvantagesLimitations
Primary: you measure the heights of 30 students yourselfYou know the instrument, the units and when it was doneSlow, and limited to the people you can reach
Secondary: a table of average monthly rainfall from a weather websiteQuick, free, and covers many years and many placesYou cannot check how it was measured, and the years may not match your question
Secondary: last year's school attendance figuresCovers the whole school and is already tabulatedThe classes and the recording rules have changed since
When a question asks why secondary data may be unreliable, the mark is for saying that it was collected for a different purpose, so you cannot check how it was measured or whether it is still up to date.

The population, the sample and the census

The population is every member of the group you are studying: every student in the school, every light bulb made on one day at a factory, every fish in a lake. A census collects data from every member of the population. A sample is part of the population, chosen so that what you find in the sample tells you something about the whole.

Samples are used because a census is usually impossible or wasteful. Testing every light bulb for how long it lasts destroys all of them, and asking all 1200 students in a school takes weeks. A sample of 100, chosen properly, answers the question in an afternoon.

For a sample to be useful it must be representative: it must contain the same mix of the population in every characteristic that could affect the answer. If the population is the whole school, a sample drawn only from the sixth form is not representative, because age affects almost everything you might ask about.

PopulationA sample of itRepresentative?
All 1200 students in a school40 students chosen at random from the full registerYes, provided the random choice really could pick any student
All the fish in a lakeThe fish caught in one net in one corner of the lakeNo: the corner and the net size leave most of the lake out
All the bulbs from a production line50 bulbs taken from the line every hour through the dayYes, provided the line does not change during the day
A large sample is not automatically a good one. A sample of 500 students taken from one after-school club is more biased than a random sample of 50 taken from the whole school.

Tally tables and two-way tables

To tabulate data means to put it into a table. A tally table is the quickest way to record data as you collect it. Draw one stroke for each item, and when you have four upright strokes draw the fifth across them, so a completed group of five can be counted at a glance. Add a frequency column holding the counts, and check that the frequencies add up to the number of items you collected.

A two-way table classifies each item twice at once, for example by year group and by how the student travels to school. Every row has a total, every column has a total, and the two grand totals must agree. Filling in the totals is not an optional extra: it is how you find a missing entry and how you check that the table makes sense.

Year groupWalks to schoolTravels by busTotal
Year 10243660
Year 11213960
Total4575120
Add the rows and add the columns. If the two grand totals disagree, the table is wrong, and any inference you draw from it will be wrong too.

Bias, and the limits of what the data can show

A sample is biased if some members of the population are more likely to be chosen than others. Asking people in a sports centre how much exercise they take gives you a sample of people who exercise; a phone survey run at midday misses everyone who is at work; a survey of your own tutor group about screen time is not a survey of the school. The question to ask of any sample is simple: who has been left out, and does it matter for the thing I am measuring?

The syllabus also asks you to appreciate the restrictions on drawing conclusions. Four limits cover almost every case. The sample may not be representative of the population you want to talk about. It may be too small for the difference you are claiming. The data may be secondary, so you cannot check how it was collected. And even a strong pattern in the data does not prove that one thing caused the other, because some third factor may explain both.

  • Not representative: the sample came from one group, so the conclusion only applies to that group
  • Too small: 20 students cannot show a small difference between two year groups
  • Out of date or secondary: the definitions, the classes or the year may not match your question
  • Cause: two quantities that rise together may both be following a third thing, such as the weather or the time of year
The answer Cambridge rewards is a full sentence: 'The sample was taken from the athletics team, so it is not representative of all Year 11 students and the conclusion may not apply to them.' Naming the group that was left out is what earns the mark.

Key terms you must be able to define

Data
Information collected about a group of people or things, as numbers, words or categories.
Discrete data
Data that can take only particular values, usually whole numbers, with gaps between them. The number of goals in a match is discrete.
Continuous data
Data that can take any value in a range, limited only by the accuracy of the measuring instrument. A time of 12.34 seconds is continuous.
Quantitative data
Numerical data, which can be counted or measured, and can therefore be added up and averaged.
Qualitative data
Data that describes a category or a quality rather than a quantity, such as eye colour or make of car. It cannot be averaged.
Primary data
Data collected at first hand, by the person who is going to use it.
Secondary data
Data collected by somebody else, for a different purpose, such as a table in a book or a figure on a website.
Population
Every member of the group being studied.
Census
A survey that collects data from every member of the population.
Sample
A part of the population, chosen so that the results can be used to make inferences about the whole population.
Representative sample
A sample that has the same mix of characteristics as the population it was drawn from.
Bias
A feature of the way a sample is chosen that makes some members of the population more likely to be included than others.
Tally table
A table that records each item as a stroke while it is counted, with every fifth stroke drawn across the other four, and a frequency column for the totals.
Frequency
The number of times a value or a category occurs in a set of data.
Two-way table
A table that classifies each item by two characteristics at once, with a total for every row and every column.
Inference
A conclusion drawn from data. It is only as trustworthy as the sample the data came from.

In context

Cambridge sets 0580 in real situations - a recipe, a bank statement, a map, a garden, a graph of a car's journey. These show how the mathematics appears in a question, then you write your own working for the numbers you are given.

Counting or measuring

Decide the type of each set of data. The number of pages in a book is discrete, because you count the pages and 350.4 pages is impossible. The mass of the same book in grams is continuous, because it is measured and 348.6 g is a possible reading. The colour of the cover is qualitative, because it is a category and there is nothing to add up.

A tally table

Twenty-five students name their favourite colour: red 7, blue 9, green 4, other 5. The tally column holds one stroke per student, with every fifth stroke drawn across the other four, and the frequency column holds the counts. Check the total: 7+9+4+5=25, which matches the number of students asked. The mode is blue, the category with the highest frequency.

A two-way table

A school records how 120 students travel to school. In Year 10, 24 walk and 36 go by bus, a total of 60. In Year 11, 21 walk, so 60−21=39 go by bus. The column totals are 24+21=45 who walk and 36+39=75 who go by bus, and 45+75=120 matches the grand total. Every row and column has to be checked this way before you read anything from the table.

A sample that is not representative

A student wants the mean time Year 11 students spend on homework. She asks the 15 members of the school orchestra. The sample is biased: students who chose orchestra are not typical of the year group, and 15 students is a small sample. A better plan is 40 students chosen at random from the whole Year 11 register, which gives every Year 11 student the same chance of being included.

Flashcards

Click a card to flip it. Mark honestly — the tracker uses this.

Q1What is the difference between discrete and continuous data?
Discrete data is counted and takes particular values, usually whole numbers. Continuous data is measured and can take any value in a range.
Q2Is shoe size discrete or continuous data?
Discrete. Sizes come from a fixed list, so 6.37 is impossible even though 6.5 exists.
Q3A length is recorded as 12 cm, measured to the nearest centimetre. Is the data discrete or continuous?
Continuous. The rounding is in the measurement, not in the data: the true length could have been 11.6 cm.
Q4What is qualitative data, and which average can you use?
Data that describes a category, such as eye colour or make of car. Only the mode, or the counts in each category, makes sense: a mean does not exist.
Q5What is primary data?
Data you collect yourself, at first hand, for the question you are answering.
Q6Give one advantage and one limitation of secondary data.
It is quick and often covers far more data than you could collect, but you cannot check how it was collected or whether it is still current.
Q7What is the difference between a census and a sample?
A census collects data from every member of the population. A sample collects data from part of the population.
Q8What makes a sample representative?
It has the same mix of characteristics as the population, so that every member of the population had a fair chance of being chosen.
Q9What is bias in a sample?
A feature of the way the sample was chosen that makes some members of the population more likely to be included than others.
Q10Give two reasons why a conclusion drawn from a sample may be unsafe.
The sample may not be representative of the population, it may be too small, the data may be secondary or out of date, or the pattern may not prove cause.
0 of 10 marked as known

Exam-style questions

6 questions, 15 marks. Write your answer, then reveal the mark scheme and tick each point you actually made. Answering honestly here is what makes the rest of the site useful.

StateAO11 mark
A hockey team records the number of goals it scores in each match. State whether this data is discrete or continuous.
Write a short answer in words.

Mark scheme — tick each point you actually made

  • B1discrete, because the number of goals is counted and cannot be 2.5not checked yet
Examiner tip: Give the reason as well as the word. Cambridge awards the mark for identifying the data as discrete, and 'it is counted' is the reason that shows you know why.
Write downAO12 marks
A survey records two things for each student in a class: (i) the time taken to run 800 m, (ii) the make of the student's mobile phone. Write down whether each set of data is quantitative or qualitative.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • B1time taken: quantitative, because it is a measurement given as a numbernot checked yet
  • B1make of phone: qualitative, because it is a category and not a numbernot checked yet
Examiner tip: Answer in the order the question asks. The time is a number, so it is quantitative; the make of the phone is a label, so it is qualitative and no mean of it could be found.
Work outAO23 marks
A two-way table shows how 120 students travel to school. In Year 10, 24 students walk and 36 travel by bus, a total of 60. In Year 11, 21 students walk and the rest travel by bus, a total of 60. Work out the number of Year 11 students who travel by bus, and the total number of students who walk to school.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • M1
  • M1
  • A139 by bus in Year 11 and 45 who walk, and 45+75=120 checks against the grand totalnot checked yet
Examiner tip: Every row total is given, so a missing entry is always found by subtracting. Finish by adding the column totals and checking that they come to 120.
Work outAO12 marks
The tally table shows the number of pets owned by the 25 students in a class. No pets: 4 students. One pet: 9 students. Two pets: 6 students. Three pets: 4 students. Four or more pets: 2 students. Work out the number of students who own at least one pet.
Write a number, a fraction such as 3/4, or an expression such as 3x+2.

Mark scheme — tick each point you actually made

  • M1
  • A121 students, and the whole table checks because 4+21=25not checked yet
Examiner tip: 'At least one' includes everyone except the students with no pets, so the quickest route is to subtract: 25−4=21. Either method scores, and the check 4+21=25 proves the table is complete.
DescribeAO23 marks
Two classes sit the same mathematics test, marked out of 80. For Class A the mean mark is 62 and the range is 30. For Class B the mean mark is 68 and the range is 12. Describe what these two measures tell you about the marks of the two classes.

Mark scheme — tick each point you actually made

  • B1
  • B1
  • B1
Examiner tip: Quote both numbers and say what each one means. 'Class B is better' scores nothing, because a comparison needs the average and the spread together.
ExplainAO24 marks
A student wants to find the mean number of hours of sleep that Year 11 students get on a school night. He asks the 20 members of the school athletics team. Explain why this sample may not be representative of all Year 11 students, and state one way he could improve the sample.

Mark scheme — tick each point you actually made

  • B1
  • B1
  • B1
  • B1
Examiner tip: Two things are wanted: why the sample is a problem, and how to improve it. The improvement mark is for changing who is chosen, such as choosing at random from the whole year group, not for asking more questions of the same 20 people.

Exam technique for this topic

  • Decide discrete or continuous by asking whether you counted it or measured it, and put that reason in the same sentence as your answer: 'discrete, because the number of goals is counted'.
  • When a question asks for one reason why a sample may not be representative, name the group that was left out. 'It might be biased' scores nothing; 'the sample was taken from the athletics team, so students outside the team had no chance of being chosen' scores the mark.
  • In any table question, add the totals before you answer anything. The row totals and the column totals of a two-way table must give the same grand total, and the frequency column of a tally table must add up to the number of items collected.
  • If a question asks what the data cannot show, use one of the four limits: the sample is not representative, it is too small, the data is secondary or out of date, or the pattern does not prove cause. One clear sentence per limit is enough.

Why students lose marks here

A student wroteShoe size is continuous data, because 6.5 is a possible shoe size.
Why it loses marksContinuous means any value in a range is possible. Shoe sizes come from a fixed list of half sizes, so 6.37 is impossible, and a value you cannot have in between makes the data discrete.
The answer that scoresShoe size is discrete: you take a label from a fixed list. A length measured in centimetres is continuous, because 24.6 cm and 24.61 cm are both possible readings.
A student wroteThe average eye colour of the class is blue.
Why it loses marksEye colour is qualitative data. It has no numerical value, so it cannot be added up or divided by the number of students, and no mean exists.
The answer that scoresFor qualitative data quote the mode or the counts: blue 14, brown 8, green 2. A mean only exists for quantitative data.
A student wroteThe sample was large, so the results must be reliable.
Why it loses marksSize does not remove bias. A large sample drawn from one group is still a sample of that group, and asking more people from the same group does not widen it.
The answer that scoresA sample must be representative as well as large. 500 students asked in the school gym is more biased than 50 students chosen at random from the whole school.
A student wroteMore people visited the beach on the sunniest days, so the sunshine caused the increase in visitors.
Why it loses marksThe data shows that the two quantities rise together; it does not show that one caused the other. Sunny days are also holidays and weekends, and those could explain the visitors.
The answer that scoresThe table shows a link between the hours of sunshine and the number of visitors. It does not prove that sunshine caused the visitors to come, because other factors such as the day of the week were not measured.

Where this fits in IGCSE Maths

This page is one topic of Cambridge IGCSE Maths 0580. IGCSE Maths is the course home: every area and every topic in Cambridge's own order, the papers you sit and what each one tests, and the exam-practice tools. The full topic list, with each topic linked to its revision page, is on the IGCSE Maths syllabus page.