# Sampling methods and bias

> IB Mathematics: Applications and Interpretation SL · IB AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u4-sampling-and-bias/

This module covers how statisticians collect data in IB AI SL: the difference between a population and a sample, discrete versus continuous data, the five sampling methods you must know (simple random, convenience, systematic, quota and stratified), how to judge whether a data source is reliable, and how sampling bias and outliers can distort conclusions.

## Learning objectives

- Distinguish between a population and a sample, and between discrete and continuous data
- Describe and select appropriate sampling techniques: simple random, convenience, systematic, quota and stratified sampling
- Evaluate the reliability of data sources and recognise how sampling bias arises
- Identify outliers in a data set and reason about their possible causes

## Populations, Samples and Types of Data

**Population and Sample** — The population is the entire group you want to draw conclusions about. A sample is a subset of that population that you actually collect data from, because measuring the whole population is usually too expensive or impractical.

*Notation:* Population = whole group; Sample = subset actually measured

*Example:* To find the average height of all 1200 students in a school (the population), you might measure only 60 of them (the sample).

We collect a sample because studying an entire population is often impossible. The key requirement is that the sample should be representative: it should reflect the whole population so that conclusions we make from the sample are trustworthy. A poorly chosen sample can mislead us even if all the arithmetic afterwards is correct.

**Discrete and Continuous Data** — Discrete data can only take separate, countable values (often whole numbers). Continuous data can take any value within a range and comes from measuring on a scale.

*Notation:* Discrete = countable values; Continuous = measured values on a scale

*Example:* Number of siblings (0, 1, 2, ...) is discrete; the time a runner takes to finish a race (12.47 s) is continuous.

Knowing whether data is discrete or continuous matters later on: it affects how you group data and which graphs are appropriate. For example, continuous data such as mass or time is usually grouped into class intervals, while discrete counts can be listed value by value.

> **Exam tip:** A quick test for discrete versus continuous: if it makes sense to have a value 'in between' two others (like 1.5 minutes), the data is continuous; if only separate values are possible (like 1.5 children is meaningless), it is discrete.

## The Five Sampling Methods

IB AI SL requires you to know five named sampling techniques. For each, you should be able to describe how it works and comment on its advantages and disadvantages. Only simple random sampling is truly random in the strict sense; the others trade some fairness for convenience or better representation.

**Simple Random Sampling** — Every member of the population has an equal and independent chance of being selected, for example by assigning each member a number and using a random number generator.

*Notation:* Every member equally likely; e.g. names from a hat, random number generator

*Example:* Assign each of 1200 students a number 1-1200, then use a calculator's random number generator to pick 60 distinct numbers.

**Convenience Sampling** — The sample is made up of the members who are easiest to access. It is quick and cheap but often unrepresentative and highly prone to bias.

*Notation:* Choose whoever is easiest to reach

*Example:* Surveying only the students who happen to be in the library at lunchtime.

**Systematic Sampling** — Order the population in a list, choose a random starting point, then select every k-th member, where k = population size divided by the desired sample size.

*Notation:* Every k-th member from an ordered list

*Example:* From a list of 1200 students, to get 60 you take every 20th student (since 1200 / 60 = 20), starting from a random position in the first 20.

**Quota Sampling** — The population is split into groups and the researcher selects members non-randomly until a set quota for each group is filled.

*Notation:* Fill fixed 'quotas' for each group, non-randomly

*Example:* An interviewer is told to survey 20 teenagers and 20 adults, and simply approaches people until each quota is reached.

**Stratified Sampling** — The population is divided into distinct subgroups (strata) and a number is randomly sampled from each stratum in proportion to the stratum's size in the population.

*Notation:* Sample size per stratum = (stratum size / population) x sample size

*Example:* If a school is 60% girls and 40% boys, a stratified sample of 60 would contain 36 girls and 24 boys, each chosen randomly within their group.

**Worked example:** A school has 1200 students: 500 in the junior years and 700 in the senior years. A stratified sample of 60 students is to be taken. How many should be selected from each group?

1. Find the sampling fraction: the sample size divided by the population size.
2. $$\frac{60}{1200} = \frac{1}{20}$$
3. Apply this fraction to each stratum to keep the sample proportional:
4. $$\text{Junior: } 500 \times \tfrac{1}{20} = 25; \quad \text{Senior: } 700 \times \tfrac{1}{20} = 35$$
5. Check the parts add to the total sample: 25 + 35 = 60. So select 25 juniors and 35 seniors, each at random within their group.

> **info**
>
> Stratified sampling is the method most often tested with a calculation, because it gives a clean numerical answer. Always check that your stratum sizes add back up to the required total sample size.

> **Exam tip:** When asked to name the sampling method from a description, look for the trigger phrase: 'every 10th' means systematic, 'proportional to group size' means stratified, 'until we had enough of each type' means quota, and 'whoever was nearby' means convenience.

## Reliability, Sampling Bias and Outliers

**Sampling Bias** — Sampling bias occurs when the way a sample is chosen makes some members of the population more likely to be included than others, so the sample does not represent the population. Biased samples lead to misleading conclusions.

*Notation:* Systematic non-representation of the population

*Example:* Asking only gym members whether people exercise regularly will overestimate exercise levels in the whole town.

Closely linked to bias is the reliability of a data source. When judging whether data can be trusted, ask: Who collected it and might they be biased? How large was the sample? How was the sample selected? A large, well-chosen sample from a neutral source is far more reliable than a small, self-selected one from an interested party.

**Outlier** — An outlier is a data value that lies an abnormal distance from the other values in the data set. Outliers can arise from genuine extreme cases or from measurement and recording errors.

*Notation:* A value abnormally far from the rest of the data

*Example:* In the ages 12, 13, 12, 14, 13, 65 for a class of students, the value 65 is an outlier and is likely a recording error.

When you spot an outlier, do not automatically delete it. First consider its cause: if it is clearly an error (a typo, a faulty instrument) it may be reasonable to remove or correct it, but if it is a genuine observation it should be kept, since it may carry important information. Always state your reasoning rather than removing values silently.

> **Exam tip:** If a question asks you to 'comment' on a survey, mention the sampling method, whether the sample size is adequate, and any reason the source might be biased. One clear sentence on each usually earns the marks.

## Common pitfalls

- **Wrong:** Confusing systematic sampling with simple random sampling
  - Why it fails: Both use a random element, so students assume they are the same, but systematic sampling only randomises the starting point, not every choice
  - Correct: State that systematic sampling picks every k-th member from an ordered list, with only the start chosen randomly
- **Wrong:** Rounding stratum sizes so the parts no longer add to the total sample size
  - Why it fails: Careless rounding of each stratum independently can make the totals disagree with the required sample size
  - Correct: Use the sampling fraction consistently and check that all stratum sizes add up to the required total
- **Wrong:** Assuming a large sample automatically removes bias
  - Why it fails: Size and bias are different issues: a huge but self-selected sample is still biased
  - Correct: Judge bias by how the sample was selected, not just by how big it is
- **Wrong:** Deleting any outlier automatically
  - Why it fails: Genuine extreme values carry real information and should not be discarded without reason
  - Correct: Investigate the cause first; only remove or correct an outlier if it is clearly an error, and state your reasoning

## Cheatsheet

| Sampling Method | How It Works | Main Advantage | Main Drawback |
| --- | --- | --- | --- |
| Simple random | Every member equally likely (random numbers) | Unbiased, fair | Needs a full list of the population |
| Convenience | Choose whoever is easiest to reach | Fast and cheap | Highly prone to bias |
| Systematic | Every k-th member of an ordered list | Simple and quick | Bias if the list has a repeating pattern |
| Quota | Fill fixed group quotas non-randomly | Ensures each group is present | Selection within groups is not random |
| Stratified | Random sample from each group in proportion | Representative of every subgroup | Needs known group sizes |

## What's next

Sampling is the first step in any statistical study, so this sub-topic underpins everything else in Unit 4. Once you can collect a representative sample, the next stage is to summarise and display that data using measures of central tendency and spread and the standard data representations, and later to analyse relationships between variables through correlation and regression. Understanding where data comes from also helps you judge how far you can trust any statistical conclusion.

- [Data representation: histograms, box plots, cumulative frequency](https://www.owlsprep.com/study/ib-math-ai-sl-u4-data-representation-histograms-box-plots/)
- [Measures of central tendency](https://www.owlsprep.com/study/ib-math-ai-sl-u4-measures-of-central-tendency/)
- [Correlation and linear regression](https://www.owlsprep.com/study/ib-math-ai-sl-u4-correlation-and-linear-regression/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-sl-u4-sampling-and-bias/