15.1 Populations, Samples and Data
SL 4.1
- the words data, variable, qualitative and quantitative
- the difference between a population and a sample, and what a census is
- five sampling techniques — simple random, systematic, convenience, stratified and quota — and how effective each one is
- what bias is, and how to judge the reliability of data
- the difference between discrete and continuous data
- how to organise raw data in a frequency table and a grouped frequency table
Introduction
Statistics is the study of how to collect, organise and analyse numerical information. The pieces of information are called data. Good data lets us draw conclusions.
To collect data we observe or measure some property, for example the height of a student or the colour of a car. This property is called a variable.
Types of variable Not in the formula booklet
- A qualitative (categorical) variable cannot take a numerical value. Examples: eye colour, favourite sport.
- A quantitative (numerical) variable takes a numerical value. Examples: number of siblings, mass.
Descriptive statistics are the methods we use to collect, tabulate and describe data, and to calculate single numbers (such as the mean) that summarise it.
Samples and populations
Population and sample Not in the formula booklet
- A population is the set of all the members (or all the measurements) we are interested in.
- A sample is a subset of the population, chosen so that we can learn about the population.
- A survey of every member of a population is a census.
| Population | A possible sample |
|---|---|
| The masses of all newborn babies in Singapore this year | The masses of newborn babies in one hospital this year |
| All the light bulbs made by a factory today | 50 bulbs taken from today’s production |
| All the students at a school | The students in one Grade 11 class |
Whether a group is a “population” or a “sample” depends on the question we ask. The students of one school are a sample of all students in the country, but they are the whole population if the question is only about that school.
A census gives the most accurate information, but it is often too expensive or slow, and sometimes impossible (for example, testing every light bulb until it fails would destroy them all). So we usually take a sample and use it to estimate facts about the population.
The size of the sample matters. A very large sample costs a lot of time and money. A very small sample may not be representative of the population.
Bias
A sample is biased if some groups of the population are over-represented or under-represented. For example, if half the students in a school are in the junior school, but a sample contains mostly senior students, the sample is biased. Conclusions from a biased sample can be misleading.
Reducing bias
Using randomness when choosing the sample is the main way to reduce bias. Making sure each important subgroup is represented in proportion also helps.
Sampling techniques
There are five main methods. Each one has advantages and disadvantages.
Simple random sampling
Every member of the population has an equal chance of being chosen. For a small population we could draw names from a hat; usually we number the members and use a random number generator.
It is simple and free from selection bias, but it needs a full list of the population, and by chance it may not represent every subgroup well.
Systematic sampling
Choose a random starting point, then select members at regular intervals from an ordered list (for example every 20th member). Factories often use it to check items on a production line.
It spreads the sample evenly through the population and is easy to do, but it can be biased if the list has a repeating pattern that matches the interval.
Systematic sample of size \(n\) from a population of size \(N\) Not in the formula booklet
Interval \(k = \dfrac{N}{n}\). Choose a random start \(r\) with \(1 \le r \le k\), then take members \(r,\; r + k,\; r + 2k,\; \ldots\)
- \(n\): the sample size
- \(N\): the population size
- \(k\): the sampling interval
- \(r\): the random starting member (\(1 \le r \le k\))
Convenience sampling
The sample is made of members who are easy to reach, for example the first people who walk past an interviewer in a shopping centre.
It is quick and cheap, but it is unlikely to be representative of the whole population.
Stratified sampling
The population is divided into non-overlapping subgroups called strata (for example year groups). A random sample is taken from each stratum, and the number from each stratum is proportional to the size of that stratum.
Number chosen from one stratum Not in the formula booklet
\[ \text{number from the stratum} = \frac{\text{size of stratum}}{\text{size of population}} \times \text{sample size} \]
It usually gives the most representative sample, but it takes more time, and it can be hard to decide on the strata.
Example 1 A school has 48 teachers and 912 students. The principal wants a stratified sample of 40 people.
(a) How many teachers should be in the sample?
(b) How many students should be in the sample?
The population size is \(48 + 912 = 960\).
(a) \(\dfrac{48}{960} \times 40 = 2\) teachers.
(b) \(\dfrac{912}{960} \times 40 = 38\) students. (Check: \(2 + 38 = 40\) ✓.)
Quota sampling
Quota sampling is like stratified sampling, but it is not random. The researcher decides how many people from each group are needed (the quota), then fills each quota, often by convenience sampling. When a group’s quota is full, further people from that group are not asked.
It is fast, cheap and does not need a list of the population, but the interviewer chooses who to ask, so the sample can be biased.
Example 2 A sports centre has four clubs. The table shows the number of members in each club.
| Club | Chess | Drama | Football | Music |
|---|---|---|---|---|
| Members | 210 | 345 | 480 | 165 |
(a) Find the total number of members.
(b) A stratified sample of 1 in 15 members is taken. Find the size of the sample.
(c) Find the number of members chosen from each club.
(a) \(210 + 345 + 480 + 165 = 1200\) members.
(b) \(\dfrac{1200}{15} = 80\) members.
(c) Divide each club size by 15:
| Club | Chess | Drama | Football | Music |
|---|---|---|---|---|
| In sample | \(\frac{210}{15} = 14\) | \(\frac{345}{15} = 23\) | \(\frac{480}{15} = 32\) | \(\frac{165}{15} = 11\) |
Check: \(14 + 23 + 32 + 11 = 80\) ✓.
Example 3 Explain how a stratified sample of 200 staff could be taken from a company in which the staff are in these age groups.
| Age | Under 30 | 30 to 50 | Over 50 |
|---|---|---|---|
| Percentage of staff | 32% | 45% | 23% |
The number from each age group must be in proportion:
\[ 0.32 \times 200 = 64, \qquad 0.45 \times 200 = 90, \qquad 0.23 \times 200 = 46 . \]
Make a list of the staff in each age group. Number the people in each list, then use a random number generator to choose 64 staff under 30, 90 staff aged 30 to 50 and 46 staff over 50. (\(64 + 90 + 46 = 200\) ✓.)
Summary of sampling methods
| Method | How it works | Advantages | Disadvantages |
|---|---|---|---|
| Simple random | Every member has an equal chance | Free from selection bias; easy to analyse | Needs a full list; may miss small subgroups |
| Systematic | Random start, then every \(k\)th member | Evenly spread; easy for large populations | Biased if the list has a pattern |
| Convenience | Members who are easy to reach | Quick and cheap | Very unlikely to be representative |
| Stratified | Random sample from each stratum, in proportion | Usually most representative; strata can be studied separately | Slow; strata can be hard to define |
| Quota | Fixed numbers from each group, not random | Fast, cheap, no list needed | Interviewer can introduce bias |
Which methods reduce bias? The key question is: who chooses the members of the sample? If chance chooses them, there is no selection bias. If a person chooses them, bias can come in.
- Stratified — least bias. Members are chosen at random, and every group is represented in the correct proportion.
- Simple random — no selection bias. But by bad luck a small group may be over- or under-represented.
- Systematic — almost as good as simple random, unless the list has a repeating pattern.
- Quota — the group sizes are correct, but the interviewer chooses who to ask. This can bring in bias.
- Convenience — most bias. Only people who are easy to reach are asked.
A larger sample makes results less variable, but it does not remove bias. A large convenience sample is still biased.
Example 4 Name the sampling method in each case, and comment on how reliable it is.
(a) A café owner asks the first 30 customers on Monday morning about a new menu.
(b) A quality inspector tests every 25th phone case from a production line, starting from the 9th.
(c) A researcher needs 50 men and 50 women. She stops people in the street until each group is full.
(a) Convenience sampling. It is biased: Monday-morning customers may differ from customers at other times.
(b) Systematic sampling. It is reliable, unless faults repeat in a pattern that matches the interval of 25.
(c) Quota sampling. It is not random, so the people she chooses to stop may not be representative.
Reliability of data
Before analysing data, ask: Can I trust it?
- Source. Who collected the data, and why? Is the source independent, or does it want a particular result?
- Sampling. Was the sample random and large enough, or was it biased?
- Missing data. Some people may not answer, or some values may not be recorded. If many values are missing, the results may not represent the population.
- Recording errors. A value such as a student’s age written as \(162\) is clearly an error. Errors should be checked and corrected (or removed) before any calculation.
Types of data
Quantitative data falls into two categories.
Discrete and continuous data Not in the formula booklet
- Discrete data can take only exact, separate values, usually found by counting. Examples: the number of goals in a match, the number of apps on a phone.
- Continuous data can take any value in an interval, and is found by measuring. It can only be given to a certain accuracy. Examples: time, height, mass, speed, temperature.
A length of \(23\) cm, measured to the nearest cm, could be any value \(L\) with
\[ 22.5 \le L < 23.5 . \]
Shoe sizes and exam marks are discrete, even though they measure something: only certain values are possible.
Tabulating data
Raw data is data that has not been organised. We organise it in a frequency table, which shows how often each value occurs.
Discrete data
Example 5 The marks (out of 10) of 20 students in a quiz are:
\[ 7,\ 4,\ 8,\ 6,\ 9,\ 5,\ 7,\ 3,\ 8,\ 7,\ 6,\ 10,\ 5,\ 7,\ 8,\ 4,\ 6,\ 9,\ 7,\ 2 \]
(a) Make a frequency table of the individual marks.
(b) Make a grouped frequency table with the groups 1–2, 3–4, 5–6, 7–8, 9–10.
(a) Count each mark (a tally helps):
| Mark | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Frequency | 0 | 0 | 1 | 1 | 2 | 2 | 3 | 5 | 3 | 2 | 1 |
(b)
| Mark | 1–2 | 3–4 | 5–6 | 7–8 | 9–10 |
|---|---|---|---|---|---|
| Frequency | 1 | 3 | 5 | 8 | 3 |
Both tables have a total frequency of \(20\) ✓. The grouped table is shorter, but we lose the exact marks.
Continuous data
Continuous data is usually put into classes. In IB questions the classes are written as inequalities without gaps.
| Length, \(L\) (cm) | Frequency |
|---|---|
| \(4.5 \le L < 9.5\) | 4 |
| \(9.5 \le L < 14.5\) | 11 |
| \(14.5 \le L < 19.5\) | 10 |
| \(19.5 \le L < 24.5\) | 5 |
Words for grouped data Not in the formula booklet
- The numbers \(4.5, 9.5, 14.5, \ldots\) are the class boundaries. The upper boundary of one class is the lower boundary of the next.
- The class width is upper boundary \(-\) lower boundary. Here \(9.5 - 4.5 = 5\).
- The mid-interval value (class mark) is the middle of the class. Here \(\dfrac{4.5 + 9.5}{2} = 7\), then \(12\), \(17\), \(22\).
These leaves were measured to the nearest cm. A leaf recorded as “5 to 9 cm” really has \(4.5 \le L < 9.5\).
Common errors
| ✗ Wrong | ✓ Right | Why |
|---|---|---|
| “Shoe size is continuous because it is a measurement” | Shoe size is discrete | Only certain sizes exist. |
| Stratified sample with equal numbers from each group | Numbers proportional to group size | Equal numbers over-represent small groups. |
| “Quota sampling is random” | Quota sampling is not random | The interviewer chooses who fills the quota. |
| 23 cm (nearest cm) means \(23 \le L < 24\) | \(22.5 \le L < 23.5\) | Rounding goes both up and down. |
| Calculating with a recorded age of 162 | Check and correct errors first | Recording errors make results unreliable. |
Exercises
1 State whether each variable is qualitative or quantitative. If it is quantitative, state whether it is discrete or continuous.
(a) Eye colour
(b) Number of siblings
(c) Time to run 400 m
(d) Shoe size
(e) Mass of a backpack
(a) Qualitative.
(b) Quantitative, discrete.
(c) Quantitative, continuous.
(d) Quantitative, discrete. (e) Quantitative, continuous.
2 A survey of the ages of members of a gym gives these values: \(34, 27, 41, \text{(blank)}, 29, 255, 38\). Comment on the reliability of the data and say what should be done before calculating the mean age.
One value is missing and \(255\) is clearly a recording error (perhaps \(25\)). The error should be checked with the member and corrected, or removed. The missing value cannot be used. Only then should the mean be calculated, from the reliable values.
3 Name the sampling method in each case.
(a) Every student in a school is given a number, and a computer chooses 40 numbers at random.
(b) A teacher asks the students sitting in the front row.
(c) A bank checks every 50th transaction, starting from the 13th.
(d) A market researcher interviews 30 adults under 40 and 30 adults over 40, choosing people in a mall.
(a) Simple random. (b) Convenience. (c) Systematic. (d) Quota.
4 A school has 360 students in Grade 10, 300 in Grade 11 and 240 in Grade 12. A stratified sample of 60 students is taken. How many students are chosen from each grade?
Total \(= 900\). Grade 10: \(\dfrac{360}{900} \times 60 = 24\); Grade 11: \(\dfrac{300}{900} \times 60 = 20\); Grade 12: \(\dfrac{240}{900} \times 60 = 16\). (\(24 + 20 + 16 = 60\) ✓)
5 In a town, 28% of residents are under 25, 52% are aged 25 to 60 and 20% are over 60. A stratified sample of 250 residents is needed. Find the number from each age group.
\(0.28 \times 250 = 70\), \(\;0.52 \times 250 = 130\), \(\;0.20 \times 250 = 50\). (\(70 + 130 + 50 = 250\) ✓)
6 A club has 600 members on a numbered list. A systematic sample of 40 members is taken.
(a) Find the sampling interval.
(b) The random starting number is 7. Write down the numbers of the first three members chosen and of the last member chosen.
(a) \(k = \dfrac{600}{40} = 15\).
(b) \(7, 22, 37\). The 40th member chosen is \(7 + 39 \times 15 = 592\).
7 Explain the main difference between stratified sampling and quota sampling. Give one advantage of quota sampling.
Both choose a fixed number from each group, but in stratified sampling the members of each group are chosen at random, while in quota sampling they are not (the interviewer chooses). Advantage of quota sampling: it is quick and cheap, and it does not need a list of the whole population.
8 Write down the interval of possible true values for each measurement.
(a) A mass of \(3.6\) kg, to the nearest \(0.1\) kg.
(b) A time of \(48\) s, to the nearest second.
(a) \(3.55 \le m < 3.65\). \(\qquad\) (b) \(47.5 \le t < 48.5\).
9 The numbers of goals scored in 24 football matches are:
\[ 0,\ 2,\ 1,\ 3,\ 1,\ 0,\ 2,\ 4,\ 1,\ 2,\ 2,\ 3,\ 0,\ 1,\ 5,\ 2,\ 1,\ 3,\ 2,\ 1,\ 0,\ 2,\ 4,\ 1 \]
(a) Make a frequency table.
(b) In how many matches were at least 3 goals scored?
(a)
| Goals | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | 4 | 7 | 7 | 3 | 2 | 1 |
Total \(= 24\) ✓.
(b) \(3 + 2 + 1 = 6\) matches.
10 The lengths of some fish, measured to the nearest cm, are grouped in the classes 20–24, 25–29, 30–34 and 35–39 cm.
(a) Write the classes as inequalities without gaps.
(b) State the class width.
(c) Find the mid-interval value of each class.
(a) \(19.5 \le L < 24.5\), \(\;24.5 \le L < 29.5\), \(\;29.5 \le L < 34.5\), \(\;34.5 \le L < 39.5\).
(b) \(24.5 - 19.5 = 5\) cm.
(c) \(22, 27, 32, 37\) cm.