What you'll learn
This guide covers the complete Statistics and Probability content for WJEC GCSE Mathematics. You'll learn to collect, represent, analyse and interpret data using statistical measures and diagrams. You'll also master probability calculations including independent and dependent events, probability trees, and experimental versus theoretical probability.
Key terms and definitions
Mean — the sum of all values divided by the number of values; sensitive to outliers and extreme values.
Median — the middle value when data is arranged in ascending order; if there's an even number of values, the median is the mean of the two middle values.
Mode — the most frequently occurring value in a dataset; the only average that can be used with non-numerical data.
Range — the difference between the highest and lowest values; a measure of spread or dispersion.
Quartiles — values that divide ordered data into four equal parts: lower quartile (Q1), median (Q2), and upper quartile (Q3).
Interquartile range (IQR) — Q3 minus Q1; measures the spread of the middle 50% of data and is resistant to outliers.
Frequency density — frequency divided by class width; used on the vertical axis of histograms to ensure area represents frequency.
Mutually exclusive events — events that cannot occur at the same time; for example, rolling a 3 and rolling a 5 on a single dice roll.
Core concepts
Collecting and representing data
Types of data
Qualitative data describes characteristics (e.g., colour, nationality) and cannot be measured numerically.
Quantitative data can be measured:
- Discrete data: counted in whole numbers (e.g., number of students)
- Continuous data: measured and can take any value within a range (e.g., height, time)
Sampling methods
Random sampling ensures every member of the population has an equal chance of selection. Use random number generators or draw names from a hat.
Stratified sampling divides the population into groups (strata) and samples proportionally from each:
- Calculate the sampling fraction: sample size ÷ population size
- Multiply each stratum size by the sampling fraction
Systematic sampling selects every nth item from a list after a random start.
Data representation
Pie charts: Each sector angle = (frequency ÷ total frequency) × 360°
Bar charts: Height represents frequency; bars are separated for discrete data.
Histograms: Area represents frequency; used for continuous grouped data. Vertical axis shows frequency density, not frequency.
Stem-and-leaf diagrams: Preserve original data values while showing distribution shape. Include a key.
Scatter graphs: Show correlation between two variables. Draw a line of best fit through the mean point.
Measures of central tendency and spread
Calculating from raw data
For the dataset: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29
- Mean = sum ÷ count = 330 ÷ 15 = 22
- Median: Order the data first. Position = (n+1)÷2 = 8th value = 23
- Mode = 23 (appears 4 times)
- Range = 56 - 3 = 53
Calculating from frequency tables
For grouped frequency tables, use midpoints:
- Estimated mean = Σ(midpoint × frequency) ÷ Σfrequency
- Modal class = the class with highest frequency
Calculating from cumulative frequency
Plot cumulative frequency against upper class boundaries to create a cumulative frequency curve.
From the curve, read off:
- Median at n÷2
- Lower quartile (Q1) at n÷4
- Upper quartile (Q3) at 3n÷4
- IQR = Q3 - Q1
Box plots and comparing distributions
A box plot (box-and-whisker diagram) displays:
- Minimum value (lower whisker)
- Lower quartile Q1 (left edge of box)
- Median (line inside box)
- Upper quartile Q3 (right edge of box)
- Maximum value (upper whisker)
Outliers lie beyond 1.5 × IQR from the quartiles and are shown as crosses.
When comparing distributions, comment on:
- Average: Compare medians or means
- Spread: Compare ranges or IQRs
- Context: Relate statements to the situation
Basic probability
Probability measures the likelihood of an event occurring, expressed as a fraction, decimal or percentage between 0 and 1.
P(event) = number of favourable outcomes ÷ total number of possible outcomes
Key probability rules:
- P(not A) = 1 - P(A)
- For mutually exclusive events: P(A or B) = P(A) + P(B)
- Sum of all probabilities = 1
Probability from two-way tables
Use tables to organise information about two characteristics. Read probabilities directly from the table:
P(event) = frequency of event ÷ total frequency
Expected frequency
Expected frequency = probability × number of trials
Compare expected with actual results to test fairness or bias.
Combined events and tree diagrams
Sample space diagrams
List all possible outcomes systematically (e.g., for two dice, create a 6×6 grid).
Independent events
Events where one outcome doesn't affect the other.
P(A and B) = P(A) × P(B)
Dependent events
Events where the first outcome affects probabilities for the second (e.g., selecting without replacement).
Probability tree diagrams
Draw branches for each stage:
- Label branches with probabilities
- Probabilities on branches from the same point sum to 1
- Multiply along branches for "and"
- Add between branches for "or"
For dependent events, adjust probabilities on second branches based on the first outcome.
Venn diagrams
Use overlapping circles to represent sets and intersections.
For events A and B:
- n(A ∪ B) = total in either A or B or both
- n(A ∩ B) = total in both A and B
- n(A') = total not in A
Calculate probabilities by dividing frequencies by the total.
Worked examples
Example 1: Histogram and frequency density
A sports centre records the time (in minutes) members spend exercising:
| Time (t minutes) | Frequency |
|---|---|
| 0 < t ≤ 20 | 24 |
| 20 < t ≤ 40 | 36 |
| 40 < t ≤ 50 | 15 |
| 50 < t ≤ 80 | 45 |
Question: Calculate the frequency density for each class and draw a histogram.
Solution:
Frequency density = frequency ÷ class width
| Time | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0 < t ≤ 20 | 24 | 20 | 24 ÷ 20 = 1.2 |
| 20 < t ≤ 40 | 36 | 20 | 36 ÷ 20 = 1.8 |
| 40 < t ≤ 50 | 15 | 10 | 15 ÷ 10 = 1.5 |
| 50 < t ≤ 80 | 45 | 30 | 45 ÷ 30 = 1.5 |
Draw bars with heights equal to frequency density, ensuring no gaps between bars. Label axes appropriately: "Time (minutes)" on horizontal axis, "Frequency density" on vertical axis.
Example 2: Probability tree diagram (dependent events)
A box contains 5 red marbles and 3 blue marbles. Two marbles are selected without replacement.
Question: Calculate the probability that both marbles are red.
Solution:
Draw a tree diagram:
First marble:
- P(Red) = 5/8
- P(Blue) = 3/8
Second marble (after selecting red first):
- P(Red|Red first) = 4/7 (only 4 red left from 7 total)
- P(Blue|Red first) = 3/7
P(Both red) = P(1st red AND 2nd red) = 5/8 × 4/7 = 20/56 = 5/14
Mark allocation: Method (1 mark), correct probabilities on branches (1 mark), final answer (1 mark)
Example 3: Comparing distributions using box plots
Two groups of students took the same test. Their results are summarised:
Group A: Min = 34, Q1 = 48, Median = 62, Q3 = 71, Max = 89 Group B: Min = 41, Q1 = 55, Median = 58, Q3 = 64, Max = 78
Question: Compare the distributions and make two statements about the test results.
Solution:
Statement 1: Group A had a higher median score (62) compared to Group B (58), suggesting Group A performed better overall on average.
Statement 2: Group A had a larger interquartile range (71 - 48 = 23) compared to Group B (64 - 55 = 9), showing Group A's results were more spread out and less consistent.
Alternative valid statement: Group A had a larger range (55 compared to 37), indicating more variation in performance.
Mark allocation: Valid comparison of average with context (1 mark), valid comparison of spread with context (1 mark)
Common mistakes and how to avoid them
Confusing median position with median value: For n values, the median position is (n+1)÷2, but you must find the value at that position after ordering the data.
Incorrect frequency density calculations: Always divide frequency by class width. Students often plot frequency directly on histograms, which is wrong for unequal class widths.
Adding probabilities when you should multiply: Use "and" = multiply along branches, "or" = add between branches. The word "and" in a question signals multiplication.
Forgetting to adjust probabilities for dependent events: When sampling without replacement, both the numerator and denominator change for the second selection.
Misreading cumulative frequency graphs: Read across from the cumulative frequency axis to the curve, then down to the data value. Common error: reading the wrong way around.
Not giving probability answers in simplest form: Always cancel fractions fully unless the question specifies a decimal or percentage. Check if the answer can be simplified.
Exam technique for "Statistics and Probability"
Command word "estimate": For grouped data, use midpoints. Your answer won't be exact, so showing the method earns marks even if you make arithmetic errors.
Show full working for probability: Write out the multiplication or addition before calculating. This earns method marks if your final answer is incorrect. For tree diagrams, label all branches clearly.
Two-part comparisons: When asked to compare distributions, you must mention both average (median/mean) and spread (range/IQR) with reference to context. One comparison only earns half marks.
Check answers are sensible: Probabilities must be between 0 and 1. Frequency density cannot be negative. Mean should lie between the minimum and maximum values.
Quick revision summary
Statistics involves collecting, representing and analysing data. Calculate mean (sum÷count), median (middle value), mode (most frequent) and range (highest-lowest) from raw data or frequency tables. Use frequency density for histograms. Cumulative frequency curves give quartiles for box plots. Probability ranges from 0 to 1. For combined events, multiply along branches of probability trees (and) and add between paths (or). Adjust probabilities for dependent events. Compare distributions using measures of average and spread with context.