What you'll learn
Statistics is a fundamental strand of GCSE Mathematics that focuses on collecting, representing, analysing and interpreting data. You'll develop skills in calculating averages and measures of spread, understanding different sampling techniques, and working with various data representations including tables, charts and cumulative frequency diagrams. These techniques appear across both Foundation and Higher tier papers.
Key terms and definitions
Mean — the sum of all values divided by the number of values; sensitive to extreme values (outliers)
Median — the middle value when data is arranged in order; if there are two middle values, calculate their mean
Mode — the value that appears most frequently in a dataset; a dataset can be bimodal or have no mode
Range — the difference between the highest and lowest values; a simple measure of spread
Interquartile range (IQR) — the difference between the upper quartile (Q₃) and lower quartile (Q₁); measures the spread of the middle 50% of data
Outlier — an extreme value that lies outside the expected range, typically defined as values below Q₁ - 1.5 × IQR or above Q₃ + 1.5 × IQR
Correlation — the relationship between two variables shown on a scatter graph; can be positive, negative or zero
Stratified sampling — a sampling method where the population is divided into groups (strata) and samples are taken proportionally from each group
Core concepts
Averages and when to use them
The three main averages each have specific advantages and limitations.
Mean is calculated by dividing the sum of all values by the total number of values. Use the mean when data is evenly distributed without extreme outliers. For example, calculating average test scores across a class where marks range from 45% to 78%.
Median is the central value in ordered data. Use the median when data contains outliers or is skewed. For instance, house prices in an area are better represented by the median because a few expensive properties would distort the mean.
Mode identifies the most common value. Use the mode for categorical data (favourite colours, shoe sizes) or when you need to know the most popular item. In a survey of preferred mobile phone brands, mode is the appropriate average.
For grouped data, you can only estimate the mean using midpoints:
- Find the midpoint of each class interval
- Multiply each midpoint by its frequency
- Sum these products and divide by total frequency
Measures of spread
While averages tell you about the centre of data, measures of spread describe how data varies.
Range = highest value - lowest value
The range is easy to calculate but affected by outliers. A single extreme value makes the range unrepresentative of typical variation.
Quartiles divide ordered data into four equal parts:
- Lower quartile (Q₁): 25% of data falls below this value
- Median (Q₂): 50% of data falls below this value
- Upper quartile (Q₃): 75% of data falls below this value
Interquartile range (IQR) = Q₃ - Q₁
The IQR is more reliable than range because it ignores extreme values and focuses on the middle 50% of data.
To find quartiles from a list of n values:
- Q₁ position = (n + 1) ÷ 4
- Q₂ position = (n + 1) ÷ 2
- Q₃ position = 3(n + 1) ÷ 4
Sampling methods
A population is the entire group you want to study. A sample is a subset selected to represent the population.
Random sampling: Every member of the population has an equal chance of selection. Use random number generators or draw names from a hat. This eliminates bias but may not represent all groups.
Systematic sampling: Select every nth person (e.g., every 10th student entering school). This is practical for large populations but can introduce bias if there's a hidden pattern.
Stratified sampling: Divide the population into groups (by age, gender, year group) then sample proportionally from each stratum.
For example, a school has 400 students: 160 in Year 7, 140 in Year 8, and 100 in Year 9. For a stratified sample of 50 students:
- Year 7: (160 ÷ 400) × 50 = 20 students
- Year 8: (140 ÷ 400) × 50 = 17.5 → 18 students
- Year 9: (100 ÷ 400) × 50 = 12.5 → 12 students
Stratified sampling ensures all subgroups are fairly represented.
Representing data
Frequency tables organize raw data by showing how often each value or group occurs.
Two-way tables display two variables simultaneously, allowing you to analyse relationships. Read carefully to identify whether you need row totals, column totals or specific cell values.
Stem and leaf diagrams preserve individual data values while showing distribution shape. The stem shows the first digit(s), leaves show the final digit. Always include a key (e.g., 2|3 = 23).
Box plots (box and whisker diagrams) display five key statistics: minimum, Q₁, median, Q₃, and maximum. They're excellent for comparing distributions between two or more datasets.
Cumulative frequency diagrams show running totals. Plot cumulative frequency against the upper class boundary of each interval. Use these curves to estimate:
- Median (at 50% of total frequency)
- Q₁ (at 25% of total frequency)
- Q₃ (at 75% of total frequency)
Histograms differ from bar charts because the area of each bar represents frequency, not height. When class widths vary, calculate frequency density:
Frequency density = frequency ÷ class width
Scatter graphs and correlation
Scatter graphs plot pairs of values to show relationships between two variables. Plot the independent variable on the x-axis and dependent variable on the y-axis.
Positive correlation: as one variable increases, so does the other (e.g., hours revised and test score)
Negative correlation: as one variable increases, the other decreases (e.g., outside temperature and heating bills)
Zero correlation: no clear relationship between variables
A line of best fit is drawn to show the trend. It should:
- Pass through or near as many points as possible
- Have roughly equal numbers of points above and below the line
- Pass through the mean point (x̄, ȳ)
You can use the line of best fit for interpolation (estimating within the data range) but extrapolation (estimating beyond the data range) is unreliable.
Important: Correlation does not prove causation. Two variables may correlate due to coincidence or a third factor.
Comparing distributions
When comparing two datasets, structure your answer using:
- Average: State which measure you're using and compare values
- Spread: Compare ranges or IQRs
- Context: Relate your comparison to the real-world situation
For example, comparing test scores between two classes: "Class A has a higher median score (65 marks compared to 58 marks), suggesting better overall performance. Class A also has a smaller IQR (12 marks compared to 20 marks), indicating more consistent results with less variation between students."
Worked examples
Example 1: Calculating mean from a grouped frequency table
The table shows journey times to school for 60 students.
| Time (t minutes) | Frequency |
|---|---|
| 0 < t ≤ 10 | 8 |
| 10 < t ≤ 20 | 18 |
| 20 < t ≤ 30 | 22 |
| 30 < t ≤ 40 | 12 |
Calculate an estimate for the mean journey time.
Solution:
| Time (t minutes) | Midpoint | Frequency | Midpoint × Frequency |
|---|---|---|---|
| 0 < t ≤ 10 | 5 | 8 | 40 |
| 10 < t ≤ 20 | 15 | 18 | 270 |
| 20 < t ≤ 30 | 25 | 22 | 550 |
| 30 < t ≤ 40 | 35 | 12 | 420 |
| Total | 60 | 1280 |
Estimated mean = 1280 ÷ 60 = 21.3 minutes (1 d.p.)
Mark scheme notes: 1 mark for finding midpoints, 1 mark for multiplying by frequencies, 1 mark for correct final answer.
Example 2: Using cumulative frequency to find quartiles
The cumulative frequency diagram shows the masses of 80 bags of potatoes.
From the graph:
- Median is at 80 ÷ 2 = 40th value → 2.1 kg
- Q₁ is at 80 ÷ 4 = 20th value → 1.9 kg
- Q₃ is at 60th value (3 × 80 ÷ 4) → 2.3 kg
Calculate the interquartile range.
Solution:
IQR = Q₃ - Q₁ = 2.3 - 1.9 = 0.4 kg
Mark scheme notes: 1 mark for correct Q₃ value, 1 mark for correct Q₁ value, 1 mark for subtraction and answer with units.
Example 3: Scatter graph interpretation
A sports scientist records training hours per week and 100m sprint times for 12 athletes. The scatter graph shows negative correlation with the equation of the line of best fit: y = -0.15x + 12.5
Estimate the sprint time for an athlete who trains 25 hours per week.
Solution:
Substitute x = 25 into the equation: y = -0.15(25) + 12.5 y = -3.75 + 12.5 y = 8.75 seconds
Mark scheme notes: 1 mark for correct substitution, 1 mark for correct answer with appropriate units. Note: this is interpolation (25 is within the data range), so the estimate is reliable.
Common mistakes and how to avoid them
Confusing mean and median: Mean uses all values; median is the middle value when ordered. Don't add up values to find the median.
Incorrect quartile positions: For n values, Q₁ is at position (n+1)÷4, not n÷4. With cumulative frequency graphs, use n÷4 because you're reading from a continuous curve.
Wrong axis on histograms: The y-axis must show frequency density, not frequency, when class widths differ. Always calculate: frequency density = frequency ÷ class width.
Misreading grouped frequency tables: Pay attention to inequalities. "10 < t ≤ 20" includes 20 but not 10. Plot cumulative frequency at the upper boundary (20, not 10).
Assuming correlation means causation: Ice cream sales and drowning incidents correlate positively, but ice cream doesn't cause drowning. Both increase in summer due to weather (the third factor).
Extrapolating too far: Lines of best fit become unreliable outside the data range. An athlete training 0 hours won't run 12.5 seconds; the relationship breaks down.
Exam technique for Statistics
"Calculate the mean" (usually 2-3 marks): Show your method clearly. Write sum ÷ count even for simple lists. For grouped data, examiners want to see midpoints and the calculation table.
"Compare the distributions" (usually 2-4 marks): Make two comparisons using statistics. Comment on average (which dataset has higher typical value) AND spread (which is more consistent). Always relate to context for full marks.
"Draw a conclusion" or "Give a reason"**: Quote specific values from your calculations. "Class B performed better because their median was 67 compared to 59" scores marks; "Class B was better" doesn't.
Accuracy: Give answers to the same degree of accuracy as the data unless told otherwise. For "estimate," 1 or 2 decimal places is usually appropriate.
Quick revision summary
Statistics involves collecting, representing and analysing data. Master the three averages (mean, median, mode) and when each is appropriate. Understand measures of spread: range and interquartile range. Know sampling methods, especially stratified sampling which maintains proportional representation. Interpret scatter graphs to identify correlation types and use lines of best fit carefully. When comparing distributions, always comment on both average and spread, relating findings to context. Practice reading cumulative frequency curves and calculating frequency density for histograms. Show all working clearly for method marks.