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IB Mathematics: Analysis and Approaches HL — Topic 4 Statistics and Probability

Data, Sampling and Reliability

Learn how data are collected, judge whether a sample is trustworthy and handle bias, missing values, errors and outliers.

AA HL · SL 4.1

Learning goal

Understand how data are obtained, choose an effective sampling method and evaluate whether a statistical conclusion is trustworthy.

Syllabus link

AA HL: SL 4.1. Population, samples, data types, reliability, bias, data quality, outliers and sampling methods.

Big idea

A calculation can be correct but the conclusion can still be poor if the data were collected badly.

Key relationship

Good statistics follows the whole journey from a clearly defined population to a justified conclusion.

IB focus: This lesson covers all five required sampling methods and the syllabus definition of an outlier using the \(1.5\times\mathrm{IQR}\) rule.
1

Why good statistics begins before calculation

Suppose a headline says: “82% of students want the school day to start later.” Before accepting the claim, a careful statistician asks where the data came from and who the result genuinely represents.

  • Which students were meant to be represented?
  • Who actually answered the survey?
  • How were those students chosen?
  • Was the question worded neutrally?
  • Were any responses missing, impossible or unusual?
Key idea

The reliability of a conclusion depends on the whole data journey.

Question
Population
Sample
Collection
Analysis
Conclusion

A weakness at an early stage can remain hidden even when every later calculation is correct.

IB exam note

AA HL students study this common SL 4.1 foundation in full. For questions assessing this content, a data set is treated as the population unless the question states otherwise.

2

Population, sample and random sample

Population

The complete group about which we want information.

Sample

A subset of the population from which data are actually collected.

Census

Data are collected from every member of the population.

Random sample

Chance is used to choose members, so personal choice does not control who is selected.

Representative sample

The sample reflects the important characteristics of the population closely enough for the intended conclusion.

Diagram showing a sample selected from a larger population.
A sample is selected from the population so that it can be used to learn about the whole group.

A sample is useful because a census may be too expensive, too slow or impossible. For example, testing the lifetime of every light bulb would destroy the whole population of bulbs being tested.

Common trap

A large sample is not automatically a good sample. Surveying 10,000 people from the wrong part of the population can be more misleading than surveying 200 people chosen carefully.

Worked example 1 — identifying the population and sample

Question

A school has 780 students. To estimate the average time students spend travelling to school, 60 students who arrive by the main gate between 08:15 and 08:25 are questioned.

  1. Identify the population.
  2. Identify the sample.
  3. State whether the sample is random. Explain one way in which the selection method could bias the result.

Solution

  1. The population is all 780 students in the school.
  2. The sample is the 60 students who were questioned.
  3. The sample is not random. It may under-represent students who arrive earlier, arrive later, use another entrance or are absent.

The conclusion may be biased because the method does not give all members of the population a fair chance of selection.

3

Discrete and continuous data

A numerical variable can be discrete or continuous.

FeatureDiscrete dataContinuous data
MeaningSeparate, countable values. There are gaps between possible values.Any value in an interval is possible, at least in principle. The variable is measured.
Typical question“How many?”“How much?”, “How long?” or “How heavy?”
ExamplesNumber of siblings; goals scored; shoe size; number of messagesHeight; mass; time; temperature; distance
A useful test

Ask whether a value between two recorded values would make sense.

  • Between 3 and 4 siblings, 3.5 siblings is not possible: the variable is discrete.
  • Between 1.72 m and 1.73 m, a height of 1.725 m is possible: the variable is continuous.
Rounding does not change the underlying type

Heights recorded to the nearest centimetre may appear as whole numbers, but height is still continuous. The measuring process has rounded a continuous quantity. Similarly, age in completed years is discrete, while exact age measured in years is continuous.

Worked example 2 — classifying variables

Question

Classify each variable as discrete or continuous. Give a brief reason for each answer.

  1. the number of customers entering a shop in one hour;
  2. the time a customer waits to be served;
  3. the mass of a parcel, recorded to the nearest gram;
  4. the number of correct answers on a 20-question quiz.

Solution

  1. Discrete: customers are counted in whole numbers.
  2. Continuous: any waiting time in an interval is possible.
  3. Continuous: mass is measured; recording to the nearest gram is only rounding.
  4. Discrete: only the integers from 0 to 20 are possible.
4

The key questions to ask when you see a data set

The syllabus expects you to question a data set or statistical analysis, not merely calculate from it.

Who?

What is the intended population? Who was actually included or excluded?

Where did the data come from?

Is the source identifiable, current and transparent about its method? Does it have a reason to influence the result?

How was the sample chosen?

Was chance used? Could people volunteer? Could the collector choose convenient respondents?

What was measured?

Are variables, units and categories defined clearly and consistently?

What is missing?

Are blanks, non-responses or unavailable measurements concentrated in a particular group?

What looks wrong?

Are there impossible values, duplicated records, inconsistent units or data-entry errors?

What looks unusual?

Are there outliers? Are they valid unusual cases or possible errors?

What conclusion is made?

Does the conclusion apply only to the sampled group, or is it being extended to a wider population without justification?

Reliability of a data source

A source is more convincing when it clearly states:

  • who collected the data and for what purpose;
  • when and where the data were collected;
  • how the population, sample and variables were defined;
  • the sample size, sampling method and non-response level;
  • how missing values, errors and outliers were handled.

A respected organisation can still produce a biased result if the sampling method is poor. Source reputation and sample quality are separate questions.

Worked example 3 — evaluating a published claim

Question

A website reports that “91% of teenagers support replacing printed textbooks with tablets”. The result comes from 2,400 visitors who clicked on an online poll advertised on the website’s technology page.

Evaluate the reliability of this claim. Identify at least two concerns about the data collection and state whether the result can reasonably be generalised to all teenagers.

Solution

  • Participation is voluntary, so people with strong opinions may be more likely to respond.
  • Visitors to a technology page may be more positive about tablets than teenagers generally.
  • The population “teenagers” is not defined by country, age range or school status.
  • The sample size is large, but the selection method is biased.

The poll describes the website respondents, but it does not provide strong evidence about all teenagers.

5

Reliability and bias in sampling

Bias is a systematic tendency for a method to favour some outcomes or groups over others. Bias does not disappear simply by repeating the same flawed method many times.

Diagram showing a convenience sample taken from only one accessible part of a population, producing a biased sample.
Sampling only one convenient part of a population can produce a systematically biased sample.

Undercoverage

Some parts of the population have little or no chance of selection.

Voluntary response

People choose whether to participate; strong opinions may be over-represented.

Non-response

Selected people do not answer, and non-responders may differ from responders.

Interviewer or selection bias

The collector chooses people who are easy to approach or likely to agree.

Response or wording bias

The question, setting or desire to give a socially acceptable answer affects responses.

Why “more data” may not help

A larger sample reduces random variation, but it does not repair a systematic selection problem. A very large biased sample can produce a very precise estimate of the wrong group.

Worked example 4 — spotting bias in a school survey

Question

The student council asks the first 100 students leaving the cafeteria whether the cafeteria food is good value for money.

  1. Identify the sampling method.
  2. Explain two ways in which this method could produce a biased result.
  3. Suggest a more reliable way to choose the students.

Solution

  1. This is a convenience sample: the council questions students who are easiest to reach.
  2. It excludes students who do not use the cafeteria and may over-represent frequent cafeteria customers. The chosen time and exit may also favour particular year groups.
  3. Define the population, divide students into relevant groups such as year levels, and randomly select students within each group.
6

The five required sampling methods

The effectiveness of a method depends on the context. The best method is the one that produces useful, representative information with acceptable time and cost.

1. Simple random sampling

Method. Number every member of the population from \(1\) to \(N\), then use a random-number generator to choose \(n\) distinct labels.

Strength. Personal choice is removed; every member has an equal chance of selection.

Limitation. A complete and accurate population list is needed. Small subgroups can still be missed by chance.

2. Convenience sampling

Method. Select people or items that are easiest to reach.

Strength. Fast, inexpensive and sometimes useful for a preliminary investigation.

Limitation. Often strongly biased because easy-to-reach members may differ from the population.

3. Systematic sampling

Method. Put the population in an ordered list, calculate a sampling interval \(k\approx N/n\), choose a random starting position from \(1\) to \(k\), then select every \(k\)th member.

Strength. Simple to carry out and spreads the sample across the list.

Limitation. A repeating pattern in the list can create bias. The first position must be random.

4. Quota sampling

Key point. A quota fixes how many people are needed from each group; it does not randomly determine which individuals are chosen.

Method. Set group targets, then approach available or willing people until every target is filled.

Strength. Chosen group proportions can be represented without a complete population list.

Limitation. Selection depends on who is present, willing or approached, so not every member has a known or equal chance of inclusion.

5. Stratified sampling

Method. Split the population into non-overlapping groups called strata, then take a random sample from every stratum.

Strength. Important groups are represented in appropriate proportions.

Limitation. Strata and their sizes must be known, and random selection is still needed inside each stratum.

Proportional stratified sampling

Let \(N_h\) be the size of one stratum, \(N\) the population size and \(n\) the total sample size. The number selected from that stratum is

\[ n_h=\frac{N_h}{N}n. \]

The actual individuals must still be chosen randomly inside each stratum.

Worked example 5 — constructing a systematic sample

Question

A company has an ordered list of 600 customers and wants a systematic sample of 50 customers.

  1. Calculate the sampling interval \(k\).
  2. A random starting position of 7 is chosen. Write down the first five customer numbers selected.
  3. State the rule used to complete the sample.

Solution

  1. \[k=\frac{600}{50}=12.\]
  2. Starting at 7 and adding 12 each time gives \[7,\ 19,\ 31,\ 43,\ 55,\ldots\]
  3. Continue choosing every 12th customer until 50 customers have been selected.

Choose every 12th customer after a random start between 1 and 12.

Worked example 6 — how a repeating pattern can create bias

Question

A café has daily sales records for 364 consecutive days, arranged in chronological order. The manager wants a systematic sample of 52 days to estimate the mean daily sales.

  1. Calculate the sampling interval.
  2. A random starting position of 2 is chosen. Describe the pattern of days that will be selected.
  3. Explain why this sample may be biased if sales depend on the day of the week.

Solution

  1. \[k=\frac{364}{52}=7.\]
  2. The selected positions are \[2,\ 9,\ 16,\ 23,\ldots\] Because the list repeats in seven-day weeks, all selected records represent the same weekday. If position 2 is a Tuesday, every selected record is a Tuesday.
  3. A random start changes which weekday is selected, but the interval of 7 still produces a sample containing only one weekday. If sales differ across weekdays and weekends, this sample will not represent all days of the year.
Fourteen-day sequence showing that a systematic interval of seven repeatedly selects Tuesday.
When the interval matches a repeating pattern, systematic sampling can repeatedly select the same type of record.

Systematic sampling should not use an interval that matches a relevant repeating pattern in the ordered list.

Worked example 7 — forming a quota sample

Question

A college has 500 students: 300 study humanities, 125 study sciences and 75 study arts. An interviewer wants a proportional quota sample of 40 students.

  1. Calculate the quota for each programme.
  2. Describe how the interviewer could collect the sample using quota sampling.
  3. Explain why the resulting sample is not a random sample and identify one possible source of bias.

Solution

  1. \[ \frac{300}{500}\times40=24,\qquad \frac{125}{500}\times40=10,\qquad \frac{75}{500}\times40=6. \] The quotas are 24 humanities, 10 sciences and 6 arts students.
  2. The interviewer could stand in the student centre and approach students who pass by, continuing until each programme target is full.
  3. The quotas determine how many students are needed from each programme, but not which students must be chosen. No programme list or random-number generator is used. Selection depends on who visits the centre, who is available, who agrees to answer and whom the interviewer approaches.

The group totals are controlled, but the individuals are chosen non-randomly. Random selection from separate programme lists would instead be stratified random sampling.

Worked example 8 — a proportional stratified sample

Question

A sixth-form college has 140 Year 12 students and 210 Year 13 students. A proportional stratified sample of 50 students is required.

  1. Calculate how many students should be selected from each year group.
  2. State how the students should be chosen within each year group.

Solution

  1. The population size is \(140+210=350\). Hence \[ \frac{140}{350}\times50=20, \qquad \frac{210}{350}\times50=30. \]
  2. Choose 20 Year 12 students and 30 Year 13 students randomly from the two separate year-group lists.

Year 12: 20 students; Year 13: 30 students, chosen randomly within their strata.

Quota and stratified sampling are not the same
  • Stratified: the target number is selected randomly from a complete list for each stratum.
  • Quota: the target number is filled using available or willing people; the quota controls the number selected, not the identities of those selected.

A quota sample may match selected population proportions but still be biased. Once random selection is used within every group, the method is stratified sampling rather than quota sampling.

MethodChance used?Main advantageMain risk
Simple randomYesFair chance for every memberNeeds a complete list; subgroups may be missed
ConvenienceNoFast and cheapOften unrepresentative
SystematicRandom startEasy and well spread through a listA periodic pattern in the list
QuotaNo: targets fix numbers, not individualsGroup targets are filled quicklyInterviewer or convenience bias
StratifiedYes, within groupsImportant groups are representedRequires group information and lists
How to choose a method
  • Use simple random sampling when a complete population list exists and no subgroup needs guaranteed representation.
  • Use stratified sampling when subgroup representation is important.
  • Use systematic sampling for a well-ordered list with no relevant repeating pattern.
  • Treat convenience and quota samples cautiously and explain their limitations clearly.
7

Missing data and recording errors

Data sets are rarely perfect. A blank entry, an impossible value or an unusual value should be investigated before analysis.

IssueExamplesSensible response
Missing dataBlank cell; a respondent skipped a question; a sensor failedFind out why the value is missing. Do not silently replace it with 0. State how it is treated.
Recording error170 cm entered as 1700 cm; 3.5 entered as 35; kg mixed with lbCheck the original source. Correct only when there is evidence for the intended value.
Impossible valueNegative age; 14 on a 10-point scaleTreat it as an error unless the coding system explains it.
Duplicate recordThe same person or measurement entered twiceVerify identifiers and remove only a genuine duplicate.
Inconsistent coding“M”, “Male” and “1” all used for the same categoryStandardise labels before analysis and document the rule.
Zero is a value; missing is an absence of a value

Replacing every blank with 0 can seriously distort a mean, total or graph.

Worked example 9 — auditing a small data set

Question

A survey records daily travel time to school, in minutes:

18, 22, [blank], 0, 25, 240, 19

Identify the entries that require investigation and explain what should be checked before calculating a mean.

Solution

  • The blank value is missing and must not automatically be replaced by 0.
  • A travel time of 0 could be valid for a student living on campus, but it could also be a coding value for “missing”. The data definition must be checked.
  • A value of 240 minutes is unusual. It may be a genuine long journey, a return-trip time, or an error such as 24.0 entered as 240.
  • The units and the wording of the question should be confirmed.

Investigate first. Correct or remove a value only when there is a defensible reason, and record the decision.

A practical data-cleaning sequence
Data-cleaning flow diagram: identify unusual values, check the source and definitions, correct only with evidence, and document changes.
A defensible correction must be supported by evidence and recorded clearly.
8

Outliers and the \(1.5\times\mathrm{IQR}\) rule

An outlier is a data value that lies unusually far from the main body of the data. In this course, a value is defined as an outlier if it is more than \(1.5\) interquartile ranges from the nearest quartile.

First calculate

\[ \mathrm{IQR}=Q_3-Q_1. \]

Then calculate the two fences:

\[ Q_1-1.5(\mathrm{IQR}) \qquad Q_3+1.5(\mathrm{IQR}). \]

A value \(x\) is an outlier when

\[ x<Q_1-1.5(\mathrm{IQR}) \quad\lor\quad x>Q_3+1.5(\mathrm{IQR}). \]
Number line showing the lower fence, first quartile, third quartile, upper fence and outliers beyond the fences.
Values strictly beyond the lower or upper fence are classified as outliers.
The phrase “more than” matters

A value exactly on a fence is not an outlier under this definition. Only values strictly below the lower fence or strictly above the upper fence are outliers.

Worked example 10 — identifying outliers

Question

For a data set, \(Q_1=18\) and \(Q_3=26\).

  1. Calculate the interquartile range and the lower and upper outlier fences.
  2. Decide whether 5, 6, 38 and 44 are outliers.

Solution

  1. \[ \mathrm{IQR}=26-18=8. \] Therefore \[ 18-1.5(8)=6, \qquad 26+1.5(8)=38. \]
    • \(5<6\), so 5 is an outlier.
    • 6 is exactly on the lower fence, so it is not an outlier.
    • 38 is exactly on the upper fence, so it is not an outlier.
    • \(44>38\), so 44 is an outlier.

The outliers are 5 and 44.

An outlier is a question, not an automatic deletion

An outlier may be:

  • a genuine but unusual member of the population;
  • a measurement or recording error;
  • a value from a different population or measured under different conditions;
  • evidence of an important event or pattern.

Check the context and source. If an outlier is removed, the reason must be explained. It is often useful to compare conclusions with and without the outlier.

9

A complete investigation workflow

Consider the question: How many hours do Diploma Programme students at a school sleep on a typical school night?

Worked example 11 — designing a trustworthy investigation

Question

Design a trustworthy investigation for the question above. Include the population, variable, sampling method, data-collection and checking procedures, and one limitation.

Solution

  1. Define the population. All Diploma Programme students currently enrolled at the school.
  2. Define the variable. Hours of sleep on a typical school night. This is continuous data.
  3. Choose a sampling method. Use a proportional stratified sample by year group so that both DP years are represented.
  4. Select randomly within strata. Use student ID numbers and a random-number generator, without replacement.
  5. Ask a neutral, precise question. For example: “During the last five school nights, how many hours did you sleep each night?” This is clearer than asking whether students “sleep enough”.
  6. Record units and non-response. Keep hours consistent, distinguish 0 from missing, and note students who do not respond.
  7. Check the data. Investigate impossible entries, duplicates and outliers before analysis.
  8. Report limitations. The results apply most directly to the school’s DP students and may be affected by self-reported sleep times.
Statistical investigation workflow from defining the question and population through sampling, collection, checking, analysis and reporting.
A reliable conclusion depends on every stage of the investigation, not only the final calculation.
10

Summary checklist

Before trusting a data-based conclusion, ask:

  1. Is the population clearly defined?
  2. Is the sample clearly identified and large enough for the purpose?
  3. Was the sample selected by an appropriate method?
  4. Could any group be under-represented or over-represented?
  5. Are variables, units and categories clear?
  6. Are missing values distinguished from genuine zeros?
  7. Have impossible values, duplicates and recording errors been checked?
  8. Have outliers been identified and interpreted rather than automatically removed?
  9. Does the conclusion stay within what the data can reasonably support?
Common mistakes to avoid
  • Calling a sample random merely because it was chosen “without a pattern”.
  • Assuming a large sample removes bias.
  • Confusing quota sampling with stratified random sampling.
  • Starting a systematic sample at the first item instead of using a random start.
  • Treating rounded continuous data as discrete.
  • Replacing missing values by 0 without justification.
  • Deleting every outlier automatically.
11

Practice questions

How to use this practice

Try each question before opening the answer key. In explanation questions, name the statistical issue and connect it directly to the context.

A. Foundations

  1. A town has 18,400 households. A researcher surveys 460 households about recycling habits. State the population and the sample.
  2. Classify each variable as discrete or continuous.
    1. number of books borrowed from a library in one month;
    2. mass of a watermelon;
    3. temperature at noon;
    4. number of goals scored in a match;
    5. exact time taken to complete a race;
    6. shoe size.
  3. Explain the difference between a random sample and a sample chosen by taking “whoever happens to be nearby”.
  4. A school asks every student to complete a wellbeing survey. Is this a sample or a census? Explain.
  5. Explain why increasing a convenience sample from 100 people to 10,000 people does not necessarily remove bias.
  6. A company cites a survey on remote working but does not state the date, population, sample size or sampling method. Give four questions you should ask before trusting the result.

B. Sampling methods

  1. Name the sampling method used in each situation.
    1. Every 25th name is selected from an employee list after a random start.
    2. A researcher interviews people leaving the nearest shopping centre.
    3. Twenty students are chosen randomly from each year group.
    4. A pollster keeps interviewing convenient respondents until age-group targets are filled.
    5. Fifty distinct ID numbers are selected at random from a complete database.
  2. A database contains 960 customer records. A systematic sample of 80 records is required.
    1. Find the sampling interval.
    2. If the random starting position is 9, write the first five selected record numbers.
  3. Explain why selecting every 10th passenger from a list ordered by a repeating ten-day travel cycle could produce a biased systematic sample.
  4. A school has 84 Grade 11 students and 126 Grade 12 students. A proportional stratified sample of 35 students is required. Find the number selected from each year group.
  5. State one similarity and one important difference between quota sampling and stratified sampling.
  6. A city wants to estimate average bicycle commuting time. The city has a complete list of registered bicycle commuters together with their district. Choose a suitable sampling method and justify your choice.

C. Data quality and outliers

  1. Explain why a blank value and a recorded value of 0 should not automatically be treated as the same thing.
  2. A data column records students’ heights in centimetres: \[168,\ 172,\ 165,\ 1690,\ 171,\ \mathrm{blank},\ 168.\] Identify three issues or checks that should be made before analysis.
  3. For a data set, \(Q_1=12.5\) and \(Q_3=19.5\).
    1. Find the IQR and the two outlier fences.
    2. Decide whether 1.5, 2, 30 and 31 are outliers.
  4. For another data set, \(Q_1=10\) and \(Q_3=18\). Show that a value of 30 is not an outlier under the \(1.5\times\mathrm{IQR}\) rule.
  5. A very high household income is identified as an outlier in a survey. Explain why it should not automatically be removed.
  6. An online newspaper asks readers, “Do you agree that the wasteful city council should finally reduce its unnecessary spending?” Readers may choose whether to respond. Identify two different sources of bias.

D. Integrated questions

  1. A sports club has 620 members in three age groups:
    Age groupUnder 1818–3940 and over
    Number of members210260150
    The club wants a proportional stratified sample of 62 members to investigate weekly training time.
    1. Find the number to select from each age group.
    2. Describe how the members should be chosen after the group sizes have been calculated.
    3. State whether weekly training time is discrete or continuous.
    4. State one possible non-response concern.
  2. A school wants to estimate the proportion of students who use public transport. A teacher surveys 150 students in the bus queue after school and finds that 94% use public transport.
    1. Identify the sampling method.
    2. Explain why the estimate is likely to be biased.
    3. Suggest a more reliable sampling plan.
    4. Explain why surveying more students from the same bus queue would not solve the main problem.
12

Answer key and solution outlines

A. Foundations — answers 1–6

1. Population: all 18,400 households in the town. Sample: the 460 surveyed households.

2. (a) discrete; (b) continuous; (c) continuous; (d) discrete; (e) continuous; (f) discrete, because only fixed permitted sizes occur.

3. A random sample uses a chance process so the collector does not decide who is selected. Taking nearby people is convenience sampling and can over-represent one location, time or type of person.

4. Census, because every student in the defined school population is asked.

5. A larger convenience sample can estimate the same biased group more precisely without becoming representative. Sample size reduces random variation, not systematic selection bias.

6. Any four sensible questions, for example: Who is the intended population? When was the survey conducted? How many people responded? How were they selected? What proportion did not respond? How was remote working defined? Who funded the survey?

B. Sampling methods — answers 7–12

7. (a) systematic; (b) convenience; (c) stratified; (d) quota; (e) simple random.

8. \[k=\frac{960}{80}=12.\] Starting at 9 gives \[9,\ 21,\ 33,\ 45,\ 57,\ldots\]

9. If the list repeats every 10 positions, selecting every 10th item can repeatedly select the same stage of the cycle and miss the others. The sample would not represent the whole cycle.

10. Total population: \(84+126=210\).

\[ \frac{84}{210}\times35=14, \qquad \frac{126}{210}\times35=21. \]

Select 14 Grade 11 students and 21 Grade 12 students, choosing randomly within the two year groups.

11. Similarity: both set targets for groups. Difference: stratified sampling selects randomly within each group; quota sampling fixes the number required from each group but fills those targets using available or willing respondents, so the individuals are not randomly selected.

12. A proportional stratified sample by district is suitable because commuting conditions may differ by district. Choose each district’s sample size proportionally, then select randomly from the commuter list in each district. A simple random sample is also defensible if district representation is not an important concern.

C. Data quality and outliers — answers 13–18

13. Zero is a measured or reported value. A blank means no value was recorded. Replacing blanks with 0 can lower means and totals and may create a false pattern.

14. The value 1690 is probably a recording error and should be checked against the original record; the blank is missing and should not be replaced automatically; duplicate values such as 168 may be valid but should be checked if duplicate records are possible; units should be confirmed as centimetres.

15.

\[ \mathrm{IQR}=19.5-12.5=7, \] \[ 12.5-1.5(7)=2, \qquad 19.5+1.5(7)=30. \]

Thus 1.5 and 31 are outliers. Values 2 and 30 lie exactly on the fences and are not outliers.

16. \(\mathrm{IQR}=18-10=8\), so the upper fence is \(18+1.5(8)=30\). A value equal to the fence is not more than \(1.5\mathrm{IQR}\) above \(Q_3\), so 30 is not an outlier.

17. It may be a genuine high-income household and an important part of the population. Check the record, units and eligibility. If valid, it should normally remain, with its influence discussed. Comparing the analysis with and without it may be informative.

18. Voluntary-response bias: readers choose whether to participate, and people with strong views may respond more. Wording bias: “wasteful” and “unnecessary” lead respondents towards agreement. The newspaper’s readership may also be unrepresentative of all city residents.

D. Integrated questions — answers 19–20

19.

(a) Since the total is 620 and the sample is 62, select one tenth of each group:

\[ 21,\qquad 26,\qquad 15. \]

(b) Use a random process within each age-group list to select the required number, without replacement.

(c) Continuous, because training time is measured and any value in an interval is possible.

(d) For example, members who train very little may be less willing to answer, or busy members may not respond. If non-responders differ from responders, the estimate may be biased.

20.

(a) Convenience sampling.

(b) Students already waiting for a bus are far more likely to use public transport than the school population generally. Students who walk, cycle or travel by car are under-covered.

(c) Define all enrolled students as the population, then use a simple random sample from the student list or a proportional stratified sample by year group. Ask the selected students using a neutral question and record non-response.

(d) More respondents from the same bus queue produce a larger sample of the same over-represented group. The selection bias remains.

Final self-check

I can distinguish a population from a sample, classify discrete and continuous data, evaluate reliability and bias, choose among five sampling methods, deal cautiously with missing or incorrect values, and identify and interpret outliers using the \(1.5\times\mathrm{IQR}\) rule.