# Data types, sampling and bias

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u4-data-types-sampling-and-bias/

This sub-topic covers foundational concepts for statistical analysis: how to classify different types of data, common methods to select samples from a population, and how to identify bias that invalidates statistical results.

**Prerequisites:** Basic numerical literacy

## Learning objectives

- Classify data as qualitative, discrete quantitative, or continuous quantitative
- Distinguish between common random and non-random sampling methods
- Identify different types of sampling bias in statistical studies
- Evaluate the validity of a sampling method for a given research question

## Classifying Data Types

**Data Classification** — Data is classified based on its nature and measurement scale, which determines what statistical analyses and graphs are valid for the dataset.

*Example:* Eye color is qualitative, while height of an adult is quantitative.

All data can first be split into two broad groups: qualitative (categorical) and quantitative (numerical). Qualitative data describes non-numerical qualities or attributes, while quantitative data represents numerical measurements.

- Discrete quantitative data: Counted values that can only take specific separated values (e.g. number of students in a classroom)
- Continuous quantitative data: Measured values that can take any value within a range (e.g. the weight of a cat)

**Worked example:** Classify each of the following as qualitative, discrete quantitative or continuous quantitative: (a) The mass of an apple, (b) The brand of a laptop, (c) The number of customers at a cafe per day.

1. For (a): Mass is a numerical measurement that can take any value within a range, even if it is rounded when recorded.
2. Conclusion: (a) is continuous quantitative.
3. For (b): Brand is a non-numerical category with no inherent numerical ordering.
4. Conclusion: (b) is qualitative.
5. For (c): Number of customers is a counted value that can only be a non-negative integer.
6. Conclusion: (c) is discrete quantitative.

## Common Sampling Methods

**Population and Sample** — The population is the entire group of interest in a statistical study. A sample is a smaller subset of the population selected for analysis.

*Notation:* $N$ = population size, $n$ = sample size

*Example:* If studying test scores of all IB SL students, the population is all IB SL students globally, and a sample is 100 students selected from 10 schools.

Random sampling methods give every member of the population a known non-zero chance of being selected, which helps produce a representative sample. The most common methods tested in IB are:

- Simple random sampling: Every sample of size $n$ has equal chance of being selected, usually generated with random numbers
- Systematic sampling: Select every $k$-th member of the population after a random starting point
- Stratified sampling: Split population into strata (groups sharing a characteristic), then sample proportionally from each stratum
- Cluster sampling: Split population into naturally occurring clusters, randomly select clusters, then sample all members within selected clusters

**Worked example:** A university has 800 first-year, 600 second-year, 400 third-year, and 200 fourth-year students. Calculate the number of students to sample from each year for a stratified sample of size 100.

1. First calculate total population size:
2. $$N = 800 + 600 + 400 + 200 = 2000$$
3. Calculate the sampling fraction, the proportion of the population to sample:
4. $$\frac{n}{N} = \frac{100}{2000} = 0.05$$
5. Multiply each year's size by the sampling fraction: First year: $800 \times 0.05 = 40$, Second year: $600 \times 0.05 = 30$, Third year: $400 \times 0.05 = 20$, Fourth year: $200 \times 0.05 = 10$
6. Final answer: 40, 30, 20, 10 students from each year respectively.

## Sampling Bias

**Sampling Bias** — Bias occurs when some members of the population are systematically more or less likely to be selected in the sample, leading to unrepresentative results that cannot be generalized to the full population.

Common types of bias you will be asked to identify include: selection bias (the sampling frame excludes part of the population), self-selection bias (participants choose to take part), and survivorship bias (only existing subjects are included).

> **tip**
>
> A large sample size does not fix systematic bias. A very large biased sample is still biased, it just gives more precise wrong results.

**Worked example:** A researcher wants to find the average amount of time high school students spend exercising per week. They post a survey on a gym's social media page and collect 200 responses. Explain why this sample is biased.

1. Identify the population of interest: All high school students. Identify who is systematically excluded: Students who do not follow this gym's social media, and students who do not go to the gym.
2. The sample will overrepresent students who exercise more than average, so the calculated average will be higher than the true average for all high school students.
3. This is selection bias, because the sampling frame (gym followers) does not represent the entire population of interest.

> **Exam tip:** When IB asks you to "comment on the validity" of a sampling method, you always need to discuss whether the method produces biased results.

## Common pitfalls

- **Wrong:** Calling rounded continuous data discrete (e.g. calling age in whole years discrete)
  - Why it fails: Data classification is based on the underlying measurement, not how it is recorded. Age is a continuous measurement even when rounded.
  - Correct: Always classify data based on the nature of the measurement, not how it is reported.
- **Wrong:** Confusing stratified sampling and cluster sampling
  - Why it fails: Both methods split the population into groups, but the sampling process is very different.
  - Correct: Stratified: sample proportionally from every group. Cluster: randomly select whole groups, sample all within selected groups.
- **Wrong:** Claiming a large sample size removes bias
  - Why it fails: Large sample size reduces random sampling error, but does not fix systematic bias.
  - Correct: Always check for systematic exclusion of population groups first, regardless of sample size.
- **Wrong:** Treating all qualitative data as the same type
  - Why it fails: While IB SL does not require you to distinguish ordinal vs nominal, you still need to recognize that qualitative data is never quantitative.
  - Correct: Even if categories are labeled with numbers (e.g. jersey numbers), they are still qualitative if the numbers do not represent measurements.

## Cheatsheet

| Category | Type | Key Summary |
| --- | --- | --- |
| Data Classification | Qualitative | Non-numerical categorical attributes |
| Data Classification | Discrete Quantitative | Counted, distinct values only |
| Data Classification | Continuous Quantitative | Measured, any value in range |
| Sampling | Simple Random | All $n$-sized samples equally likely |
| Sampling | Systematic | Every $k$-th member after random start |
| Sampling | Stratified | Proportional sample from every strata |
| Sampling | Cluster | Full sample from randomly selected clusters |
| Bias | Key Rule | Large sample size does not remove bias |

## What's next

Now that you understand foundational concepts of data collection, classification, and sampling bias, you are ready to move on to organizing, summarizing, and visualizing data. Data classification directly determines which summary measures and graphs are appropriate for a dataset. Understanding bias matters because unrepresentative samples lead to incorrect conclusions about a population. Note that confidence intervals and hypothesis testing, where this idea is formalized, belong to Math AI, not AA, and are not part of AA SL.

- [Data representation: histograms, box plots, cumulative frequency](https://www.owlsprep.com/study/ib-math-aa-sl-u4-data-representation-histograms-box-plots/)
- [Measures of central tendency and dispersion](https://www.owlsprep.com/study/ib-math-aa-sl-u4-measures-of-central-tendency-and/)
- [Cumulative distributions and percentiles](https://www.owlsprep.com/study/ib-math-aa-sl-u4-cumulative-distributions-and-percentiles/)

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