Using statistics to describe populations

This Edexcel skill ties measures of central tendency, spread, and inference together. Often examined as the final reasoning question on a Foundation paper.

Selecting an appropriate average

• Mean — symmetric data, no extreme values.
• Median — skewed data or outliers (extreme values won't drag the median).
• Mode — categorical data, or when the most "typical" value is wanted.

Example: salaries with one CEO at £500 000 and 99 employees at £25 000.

• Mean ≈ £29 750 (skewed by the CEO).
• Median = £25 000 (more representative of typical employee).
• Mode = £25 000.

For a representative description: use median.

Reporting describing a population

A complete description usually mentions:

• An average (with which one and why).
• A measure of spread (range or IQR).
• Relevant context (units, sample size).

Inference from sample to population

If a sample has mean m, we estimate population mean ≈ m. The estimate is more reliable for larger, more representative samples.

For categorical data: P(category) ≈ sample proportion. Then expected count = P × population size.

Edexcel exam tip

When asked "describe the data" or "what does this tell you about the population?", give two distinct facts:

1. Average + measure (e.g. "mean is 25 minutes").
2. Spread (e.g. "range is 18 minutes — fairly large variation").

Mention the sample size if relevant: "based on a sample of 50, the mean cannot be exact for the whole population".

Outliers

A value much smaller or larger than the rest. Edexcel often asks:

• "Identify any outliers" (sometimes using the rule: outlier = value > Q3 + 1.5 × IQR or < Q1 − 1.5 × IQR).
• "Should the outlier be included?" Depends on context (likely a data-entry error vs genuine extreme observation).

Common Edexcel question pattern

Comparing two samples: see S4. Description of one sample: median + IQR + a comment. Inference: from sample proportion to population estimate.