Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Statistics and Probability

Data Collection and Sampling

Plan a statistical investigation: write related questions and predictions, decide whether data are categorical, discrete or continuous, write and trial good questionnaires, and choose and justify random, systematic, stratified or convenience samples.

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In the curriculum

Choose, try out (trial) and justify a way of collecting data and a sampling method to test predictions for a set of related statistical questions, deciding what data to collect: categorical, discrete or continuous (8Ss.01)
Understand the advantages and disadvantages of different sampling methods (8Ss.02)

Statistics starts before any chart is drawn. A good investigation begins with clear questions and a sensible prediction, then collects the right data from the right people in a fair way. In Checkpoint-style tests you are often asked to criticise a question, name a type of data, choose a sampling method and explain why — so this page gives you the words to use.

Key words

Word / phrase	Meaning / use
statistical question	سؤال إحصائي — a question answered by collecting data that varies, e.g. “How long do Year 8 students sleep?”
prediction (hypothesis)	توقع / فرضية — what you think the data will show, written so it can be checked
categorical data	بيانات وصفية (نوعية) — words or groups: colour, favourite sport, type of phone
discrete data	بيانات منفصلة — numbers you count; only certain values: number of brothers, goals, shoe size
continuous data	بيانات متصلة — numbers you measure; any value in a range: height, mass, time
primary / secondary data	بيانات أولية / ثانوية — collected by you / collected by someone else
population	المجتمع الإحصائي — everyone (or everything) you want to find out about
sample	العينة — the part of the population you actually collect data from
biased	متحيز — not fair; some answers or people are more likely to be included than others
trial (pilot)	تجربة مبدئية — testing your questionnaire or method on a few people first, then improving it

1. Statistical questions and predictions

A statistical question expects answers that vary. “How old is Omar?” has one answer, so it is not statistical. “How much time do Year 8 students spend on homework each night?” has many different answers — it is statistical. A real investigation usually has a set of related questions about one big theme, and each question has a prediction that the data can show to be right or wrong.

Big theme: sleep and school	Prediction	Data needed (type)
How long do Year 8 students sleep on a school night?	Most sleep less than 8 hours.	hours of sleep (continuous)
Do students who use a phone after 10 p.m. sleep less?	Phone users after 10 p.m. sleep less, on average.	uses phone after 10 p.m.: yes/no (categorical) + hours of sleep (continuous)
Do older students sleep less than younger ones?	Year 9 students sleep less than Year 7 students.	year group (categorical) + hours of sleep (continuous)
How many times a week do students feel tired in class?	Students who sleep less feel tired more often.	number of times (discrete) + hours of sleep (continuous)

A good prediction is specific and checkable: “Year 9 students sleep less than Year 7 students, on average” can be tested by comparing two averages. “Sleep is important” cannot be tested with data.

The statistical enquiry cycle. After interpreting the data you may find new questions and go round again.

2. Types of data

Categorical data are words or groups. Numerical data are numbers: discrete (counted) or continuous (measured).

Type	How you get it	Examples	Quick test
Categorical	choose a group or a word	eye colour, favourite subject, way of travelling to school, yes/no answers	Can you do arithmetic with it? No.
Discrete	count	number of pets, people in a family, goals scored, shoe size (only 38, 38½, 39 …)	Only certain separate values; no values in between.
Continuous	measure	height, mass, time to run 100 m, temperature, length of a leaf	Any value is possible in a range; it is rounded when recorded.

Tricky cases. Age is continuous (you get older every second), even though we usually record it in whole years. Shoe size is discrete because only certain sizes exist. Money is usually treated as discrete because amounts go up in steps (1 piastre). A number used as a label (a bus number, a football shirt number) is categorical — you cannot do sensible calculations with it.

The type of data matters because it decides how you record it and which chart to use: categorical and discrete data go in tally charts and bar charts; continuous data are put into class intervals such as 150 ≤ h < 155 and shown on frequency diagrams with no gaps between the bars.

3. Ways of collecting data

Method	Good for	Advantages	Disadvantages
Questionnaire / survey	opinions, habits	many people quickly; easy to compare answers	people may not answer, may not tell the truth, or may misunderstand
Observation	counting what happens (cars passing, birds visiting)	you see what really happens	takes time; you must be there at the right moment
Experiment / measuring	times, lengths, masses, results of trials	accurate numbers you control	needs equipment; measuring errors
Secondary data (internet, school records, books)	large or old data sets	quick and cheap; lots of data	may be out of date, unreliable or not quite what you need

Primary data are collected by you for your investigation. Secondary data were collected by someone else. Always say where secondary data came from, and check that the source is reliable.

Writing good questionnaire questions. A question should be clear, not personal, not leading, and every person must be able to tick exactly one box.

Poor question	What is wrong	Better question
“How much do you sleep?” □ a little □ a lot	vague; “a lot” means different things to different people; no time frame	“On a school night, how many hours do you usually sleep?”
Hours: □ 5–6 □ 6–7 □ 7–8 □ 8–9	overlapping boxes (where does 6 go?); no box for less than 5 or more than 9	□ less than 6 □ 6 or more but less than 7 □ 7 or more but less than 8 □ 8 or more
“Don’t you agree that phones are bad for sleep?”	leading — it pushes people to say yes	“Do you think using a phone before bed affects sleep? □ Yes □ No □ Not sure”

Trial your method. Before the real survey, try the questionnaire on a few people (a pilot). You may find a confusing question, a missing box or that it takes too long. Improve it, then collect the real data. In a test, “trial” answers sound like: “I would test my questionnaire on five students first to check the questions are clear.”

4. Populations and samples

The population is everyone the question is about (for example, all 480 students in a school). Asking everyone is called a census. It gives complete information but takes a long time and costs a lot, so we usually ask a sample. A sample must represent the population: it should contain the same kinds of people in about the same proportions. A sample that leaves out or favours some groups is biased, and conclusions from it can be wrong.

Bigger sample → results are more reliable (less affected by a few unusual people), but it takes more time and effort.
Too small (for example 5 people) → results can change a lot by chance.
Two different samples from the same population usually give slightly different results. This is sampling variation — it is normal, and it is why we should not trust a tiny sample.

5. Sampling methods

Random sampling. Every member of the population has the same chance of being chosen — for example, numbers in a bag, or a random number generator on a calculator or computer, using a numbered list of the population.

Systematic sampling. Put the population in a list, work out the step k = population size ÷ sample size, choose a random start between 1 and k, then take every k-th member.

A systematic sample: 60 students, sample of 10, so every 6th name starting from number 4 (4, 10, 16, …, 58).

Stratified sampling. Split the population into groups (strata) such as year groups, then choose from each group in proportion to its size. Number from a group = (group size ÷ population size) × sample size. Inside each group, choose people at random.

Convenience (opportunity) sampling. Ask whoever is easiest to reach — your friends, the first 20 people at the gate. It is quick, but it is usually biased.

Method	Advantages	Disadvantages
Random	fair: everyone has an equal chance; avoids the collector’s own bias	needs a full list of the population; by chance it may miss a group
Systematic	quick and easy from a list; spread evenly through the list	needs a list; can be biased if the list has a pattern (e.g. every 10th house is a corner house)
Stratified	every group is fairly represented, in the right proportion	needs to know the size of each group; more calculation and work
Convenience	very quick, cheap and easy	likely to be biased — the people are not typical (friends think alike; people at a gym exercise more)

بالعربي: العينة الكويسة

المجتمع (population) = كل الناس اللي عايز تعرف عنهم. العينة (sample) = الجزء اللي هتسألهم فعلًا.
العينة لازم تمثّل المجتمع: نفس أنواع الناس وبنفس النِّسَب تقريبًا، وإلا تبقى متحيزة (biased).
عشوائية (random): كل واحد له نفس الفرصة. منتظمة (systematic): كل k-th اسم في القائمة. طبقية (stratified): من كل مجموعة حسب حجمها. ملائمة (convenience): أسهل ناس — سريعة بس غالبًا متحيزة.
عينة أكبر = نتيجة أدق، بس محتاجة وقت ومجهود أكتر.

Worked examples

Worked example 1 — Types of data [2 marks] Write whether each is categorical, discrete or continuous: (a) the number of books a student read last month, (b) the time a student takes to walk to school, (c) a student’s favourite type of book, (d) the mass of a school bag.

(a) Books are counted, only whole numbers: discrete.
(b) Time is measured; it can be any value, e.g. 12.4 minutes: continuous.
(c) Answers are words (adventure, science fiction…): categorical.
(d) Mass is measured: continuous.

Answer: discrete, continuous, categorical, continuous (2 marks all correct; 1 mark for 2 or 3 correct). Common mistake: calling every number “discrete”. Ask yourself: is it counted or measured?

Worked example 2 — Criticising a questionnaire [2 marks] Laila wants to know how much time students spend reading. She writes: “How long do you read for? □ 0–1 hour □ 1–2 hours □ 2–3 hours”. Write two different criticisms of her question.

There is no time frame — per day? per week?
The boxes overlap: a student who reads exactly 1 hour could tick two boxes.
There is no box for more than 3 hours.

Answer: any two of the three points (1 mark each). A good improved version: “How long did you spend reading yesterday? □ less than 1 hour □ 1 hour or more but less than 2 hours □ 2 hours or more but less than 3 hours □ 3 hours or more”. Common mistake: writing two criticisms that are really the same point, e.g. “the boxes overlap” and “1 is in two boxes”.

Worked example 3 — Systematic sample [2 marks] A sports club has 450 members on a numbered list. The manager wants a systematic sample of 30 members. Explain how to choose the sample, and write the first three member numbers if she starts at member 7.

Step size: 450 ÷ 30 = 15, so choose every 15th member.
Choose a random start between 1 and 15 — here 7.
Members 7, 7 + 15 = 22, 22 + 15 = 37, … (the last is 7 + 29 × 15 = 442).

Answer: every 15th member after a random start; 7, 22, 37. Common mistake: dividing the wrong way round (30 ÷ 450) or choosing every 30th member, which gives only 15 people.

Worked example 4 — Stratified sample [2 marks] A school has 220 students in Year 7, 200 in Year 8 and 180 in Year 9. A stratified sample of 60 students is taken. How many students should be chosen from each year?

Total population = 220 + 200 + 180 = 600.
The sample is 60 out of 600, which is 1/10 of each group.
Year 7: 220 ÷ 10 = 22. Year 8: 200 ÷ 10 = 20. Year 9: 180 ÷ 10 = 18.
Check: 22 + 20 + 18 = 60 ✓

Answer: 22, 20 and 18. Common mistake: taking 20 from each year. That ignores the different sizes of the groups.

Worked example 5 — Choosing and justifying a sample [2 marks] Youssef wants to find out whether students at his school (Years 7 to 9) think the school day should start later. He plans to ask the 25 students in his own Year 8 class. (a) Give one reason why his sample may be biased. (b) Describe a better sampling method.

(a) Only Year 8 is asked, so the views of Years 7 and 9 are missing; his classmates may also share his opinion.
(b) Use a stratified sample: choose students from each year group in proportion to the number of students in that year, picking names at random within each year (from the school register). This represents all years fairly.

Answer: as above (1 mark each). A random sample from a list of the whole school also earns the mark for (b) if explained. Common mistake: answering “the sample is too small” for (a). 25 might be small, but the main problem is that it is not representative.

Worked example 6 — Planning related questions [2 marks] Nour predicts: “Students who eat breakfast run 100 m faster than students who do not.” (a) What two pieces of data must she collect from each student? (b) State the type of each.

Answer: (a) whether the student eats breakfast, and the student’s 100 m time. (b) Eats breakfast (yes/no) is categorical; 100 m time is continuous. To check the prediction she can compare the mean (or median) times of the two groups. Common mistake: collecting only the times. Without the breakfast data she cannot split the students into two groups.

Common mistakes

Calling all numerical data discrete. Measured quantities (time, mass, length, temperature) are continuous.
Response boxes that overlap (1–2, 2–3) or leave gaps (1–2, 3–4). Use “1 or more but less than 2” or inequalities.
Saying a sample is biased only because it is “small”. Bias is about who is chosen, not just how many.
In systematic sampling, dividing sample size by population. The step is population ÷ sample.
In stratified sampling, taking the same number from each group regardless of size.
Forgetting to explain a choice. “Random sampling” alone is not a justification — say why (everyone has an equal chance, so it is not biased).
Using a convenience sample (friends, people outside a shop) and claiming it represents everyone.

How Checkpoint asks this

“Is this data categorical, discrete or continuous?” — often a table to complete or words to ring.
“Write one criticism of this question” or “Write a better question” — look for vague words, overlapping or missing boxes, no time frame, leading wording.
“Explain why this sample may be biased” — name the group that is left out or over-represented.
“Which sampling method …?” and “Give one advantage and one disadvantage” — use the words fair, representative, quick, needs a list.
Short calculations: the step for a systematic sample, or how many from each group in a stratified sample.
Explanations get the marks when they mention the context (the students, the club, the shop…) — not just a general rule.

Practice

Remember

Start with related statistical questions and clear, checkable predictions; decide what data you need for each.
Categorical = words/groups; discrete = counted; continuous = measured.
Good questions: clear, with a time frame, not leading, and response boxes that do not overlap and cover every answer. Trial the questionnaire first.
A sample must represent the population. Random = equal chance; systematic = every k-th (k = population ÷ sample); stratified = in proportion to group sizes; convenience = quick but often biased.
Bigger samples are more reliable; different samples give slightly different results (sampling variation).

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