Averages, Ranges & Comparing Data

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IGCSE Extended Maths

Averages, Ranges & Comparing Data

The big idea: An "average" is just one number trying to represent a whole pile of data — and the "range" or "IQR" tells you how honest that single number actually is, because it shows how spread out the real data was.

Summary Everything in this chapter, at a glance

Discrete vs continuous data — discrete data is counted (whole steps), continuous data is measured (any value on a scale).
Mode, median, mean — three different ways to describe "the typical value," each with strengths and weaknesses.
Calculations with the mean — rearranging mean = total ÷ number of values to solve backwards problems.
Averages from frequency tables — finding mode, median, and mean when data is summarised in a table rather than listed out.
Averages from grouped data — when exact values are hidden in groups, you can only estimate the mean using midpoints, and you find the modal class / median class instead of exact values.
Calculators — using one-variable statistics mode to get mean, median, and quartiles instantly (for both raw and grouped data).
Range & Interquartile Range (IQR) — two measures of spread; IQR is more resistant to extreme values.
Comparing data sets — always compare an average AND a spread, and explain what each comparison means in real-world terms.

1. Discrete & Continuous Data

What's the difference?

Numerical data splits into two families, and knowing which one you're dealing with matters for how you group and analyse it later.

Continuous data can take any value on a scale — there are no "gaps" between possible values. Think height, weight, length, time, temperature. If you zoom in with a more precise instrument, you always find more decimal places are possible.
Discrete data can only take particular, separate values — there are gaps. Usually this means counting something (number of pets, number of cars) so it's whole numbers, but it doesn't have to be — shoe sizes include "halves" (6, 6.5, 7) yet are still discrete because you can never get a shoe size of 6.3.

Quick test

Ask yourself: "Do I need a measuring scale to get this value, and could it in theory be any decimal?" If yes → continuous. If the data is really a count, or restricted to specific "steps" — discrete. Watch out for time recorded "to the nearest hour" — even though time is naturally continuous, rounding it to fixed steps like that makes the recorded data discrete!

Q1. State whether the following is discrete or continuous, and explain why: "the number of goals scored by a football team in each match this season."

Q2. A scientist records the exact mass, in grams, of chemical samples using a precision balance. Is this discrete or continuous?

2. Mean, Median & Mode

Three ways to find "the average"

In everyday speech we say "the average," but in maths there are three distinct averages, and picking the wrong one for a data set can genuinely mislead people. Think of them like three different photographers trying to capture "what this data looks like" — each one focuses on a different feature.

Mode most often bimodal
Median middle
Mean

Worked example

Data: 1, 2, 2, 5, 6

Mode:

Median:

Mean:

Core Formulas

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Also in the full note

Which average should I use?
Working backwards from a known mean
Reading averages from a table instead of a list
Finding the mode
Finding the median
Finding the mean
Why "estimate"? What are we missing?
Step-by-step method for estimating the mean

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Averages, Ranges & Comparing Data