GCSE Maths Statistics and Probability: Complete Revision Guide

Statistics and probability form a substantial component of GCSE Maths across all exam boards. This guide is part of our complete GCSE Maths revision guide — visit there for an overview of all topics. These topics test your ability to interpret data, calculate averages, construct charts, and work with probability scenarios. Understanding the underlying concepts, not just learning procedures, is key to accessing higher-grade questions.

Measures of Average

Three measures of average appear at GCSE: mean, median, and mode. Each has different uses and characteristics that you need to understand.

The mean is calculated by adding all values and dividing by how many values there are. For the data set 3, 7, 8, 8, 12, the mean is (3 + 7 + 8 + 8 + 12) ÷ 5 = 38 ÷ 5 = 7.6. The mean uses all data values but can be distorted by extreme values (outliers).

The median is the middle value when data is arranged in order. For the same data set already in order, the median is 8 (the middle of five values). For even numbers of values, find the mean of the two middle values. The median isn’t affected by outliers, making it useful for data with extreme values.

The mode is the most frequently occurring value. In our example, 8 appears twice whilst others appear once, so the mode is 8. Data can have more than one mode, or no mode if all values appear equally often.

Measures of Spread

Range measures spread by subtracting the lowest value from the highest. For our data set, range = 12 - 3 = 9. Range is easy to calculate but affected by outliers.

Interquartile range (IQR) is more robust. First find quartiles: Q1 (lower quartile) is the median of the lower half, Q2 is the overall median, and Q3 (upper quartile) is the median of the upper half. IQR = Q3 - Q1. This shows the spread of the middle 50% of data and isn’t affected by outliers.

Charts and Graphs

Different charts suit different data types. Bar charts compare discrete categories with separated bars. Histograms represent continuous data using joined bars where area (not height alone) represents frequency. The vertical axis shows frequency density, calculated by frequency ÷ class width.

Pie charts represent proportions as sectors of a circle. To construct them, find the total frequency, divide 360° by the total to find degrees per item, then multiply each frequency by this value to find each sector angle.

Cumulative frequency graphs plot running totals and help estimate medians and quartiles. Plot cumulative frequency against upper class boundaries, join with a smooth curve, and read off values. The median is at n/2 on the cumulative frequency axis, Q1 at n/4, and Q3 at 3n/4.

Scatter graphs show correlation between two variables. Positive correlation means as one variable increases, so does the other. Negative correlation means as one increases, the other decreases. No correlation means no clear relationship exists. A line of best fit should pass through the mean point (mean of x values, mean of y values) and have roughly equal numbers of points either side.

Probability Basics

Probability measures how likely events are on a scale from 0 (impossible) to 1 (certain). Calculate probability using: P(event) = number of favourable outcomes ÷ total number of possible outcomes.

For a standard six-sided die, P(rolling a 3) = 1/6 because there’s one favourable outcome out of six equally likely outcomes. P(rolling an even number) = 3/6 = 1/2 because there are three even numbers (2, 4, 6).

Expected frequency applies probability to repeated trials: expected frequency = probability × number of trials. If you roll a die 60 times, expect to roll a 3 approximately 60 × 1/6 = 10 times.

Combined Probability

When events are independent (one doesn’t affect the other), multiply probabilities for consecutive events. The probability of flipping heads then rolling a 6 is 1/2 × 1/6 = 1/12.

For mutually exclusive events (that cannot happen simultaneously), add probabilities. The probability of rolling either a 3 or a 5 is 1/6 + 1/6 = 2/6 = 1/3.

Probability Trees

Tree diagrams visualise multiple events. Each branch represents a possible outcome with its probability written on the branch. Multiply along branches for “and” probabilities. Add outcomes for “or” probabilities.

For example, a bag contains 3 red and 2 blue balls. Drawing one ball then another without replacement creates a probability tree with four outcomes. The probability of red then blue is 3/5 × 2/4 = 6/20 = 3/10. Note the second probability changes because you don’t replace the first ball.

Venn Diagrams

Venn diagrams show relationships between sets. The universal set contains everything being considered. Overlapping circles represent different sets, with the overlap showing elements in both sets.

To solve Venn diagram problems, often start by filling in the intersection (elements in both sets), then work outwards. Questions might ask for unions (elements in set A or set B or both), intersections (elements in both sets), or complements (elements not in a particular set).

Two-Way Tables

Two-way tables organise data for two variables. Read carefully to extract correct values. Common questions involve calculating probabilities or percentages from the table data. Always check row and column totals to verify your entries are correct.

Sampling and Bias

Understanding sampling methods is important for Higher tier. Simple random sampling gives every member an equal chance. Stratified sampling maintains proportions of subgroups from the population. Systematic sampling selects every nth member.

Bias occurs when samples aren’t representative. Common sources include convenience sampling (choosing easy-to-access people), voluntary response bias (only those who want to respond do so), and leading questions in surveys.

Exam Technique for Statistics and Probability

Show working clearly, especially for multi-step problems. Check probabilities sum to 1 when finding all possible outcomes. Use appropriate notation: write probabilities as fractions, decimals, or percentages as requested. Draw clear, labelled diagrams for tree diagrams and Venn diagrams. For histogram questions, always calculate frequency density.

Related GCSE Maths Guides

• GCSE Maths formula sheet guide — which formulas are and aren’t provided in the exam
• Algebra revision guide — expressions, equations, and sequences

Ready to practise? Try GCSE Maths questions on UpGrades to find and fix your weak spots.

UpGrades provides adaptive statistics and probability practice with detailed explanations, helping you develop both calculation skills and the conceptual understanding needed for problem-solving questions.