What you'll learn
Statistics at GCSE level focuses on collecting, presenting, analysing and interpreting data. You'll master methods for calculating averages and measures of spread, understand how to represent data through various diagrams, and learn how to draw valid conclusions from statistical information. This topic appears in both Foundation and Higher tier papers, with more complex applications reserved for Higher tier.
Key terms and definitions
Mean — the sum of all values divided by the number of values; sensitive to extreme values
Median — the middle value when data is arranged in order; for an even number of values, it's the mean of the two middle values
Mode — the value that appears most frequently in a dataset
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 — a value that lies significantly outside the main pattern of data; typically defined as below Q₁ - 1.5 × IQR or above Q₃ + 1.5 × IQR
Frequency density — frequency divided by class width; used as the vertical axis in histograms
Correlation — a relationship between two variables; can be positive, negative or zero
Core concepts
Averages and measures of spread
The three main averages serve different purposes:
Calculating the mean:
- Add all values together
- Divide by the total number of values
- For grouped data, use midpoints: mean ≈ Σ(f × x) ÷ Σf where f is frequency and x is midpoint
Finding the median:
- Arrange data in ascending order
- Use position = (n + 1) ÷ 2 for discrete data
- If n is even, find the mean of the two middle values
- For grouped data, use interpolation from a cumulative frequency graph
Identifying the mode:
- The most frequently occurring value
- Can have more than one mode (bimodal or multimodal)
- For grouped data, identify the modal class (the class with highest frequency)
Measures of spread:
The range provides a basic measure but is affected by extreme values. The interquartile range is more robust:
- Lower quartile (Q₁): value at position ¼(n + 1)
- Upper quartile (Q₃): value at position ¾(n + 1)
- IQR = Q₃ - Q₁
The standard deviation (Higher tier only) measures how spread out data is from the mean. A larger standard deviation indicates more variation.
Data representation
Pie charts:
- Each sector angle = (frequency ÷ total frequency) × 360°
- Used for categorical data showing proportions of a whole
- Best for comparing parts to the whole
Bar charts and vertical line graphs:
- Bar charts: used for discrete or categorical data with gaps between bars
- Heights represent frequencies
- Dual bar charts compare two datasets
Histograms:
- Used for continuous grouped data
- No gaps between bars
- Vertical axis shows frequency density
- Frequency = frequency density × class width
- Area of each bar represents frequency
Scatter diagrams:
- Plot paired data to show correlation
- Line of best fit should pass through the mean point (x̄, ȳ)
- Can be used for interpolation (within data range) but extrapolation (beyond data range) is unreliable
Cumulative frequency diagrams:
- Plot upper class boundary against cumulative frequency
- Join points with a smooth curve
- Read off median at ½n
- Read off Q₁ at ¼n and Q₃ at ¾n
- Can estimate the number or percentage of values above/below a given value
Box plots:
- Display five key values: minimum, Q₁, median, Q₃, maximum
- The box represents the IQR (middle 50% of data)
- Whiskers extend to minimum and maximum (or to outlier boundaries)
- Easy comparison of distributions between datasets
Time series
A time series shows how data varies over time:
- Plot time on the horizontal axis
- Look for trends (general direction) and seasonal variation
- Moving averages smooth out fluctuations to show underlying trends
- For a 4-point moving average, calculate the mean of each consecutive set of 4 values
- Plot moving averages at the midpoint of the time period used
Sampling and data collection
Types of data:
- Discrete data: can only take specific values (e.g., number of students)
- Continuous data: can take any value within a range (e.g., height, mass)
- Categorical data: non-numerical categories (e.g., colours, preferences)
Sampling methods:
- Random sampling: every member of the population has an equal chance of selection
- Stratified sampling: population divided into groups (strata); sample size from each group proportional to group size
- Sample from stratum = (stratum size ÷ population size) × total sample size
- Systematic sampling: select every nth item
Data collection considerations:
- Questions must be clear, unbiased and unambiguous
- Response sections should cover all possibilities without overlap
- Time periods should be specified
- Inequalities for grouped data typically use ≤ for continuous data
Comparing distributions
When comparing two datasets, always comment on:
Averages (measure of location):
- Which dataset has a higher mean/median?
- What does this tell you about the typical value?
Spread (measure of variation):
- Which dataset has a larger range/IQR?
- What does this tell you about consistency or variation?
Make contextual statements linking to the real-world scenario. Never just state numbers without interpretation.
Two-way tables and Venn diagrams
Two-way tables:
- Organise data by two categories
- Row totals and column totals must be calculated
- Can find probabilities or frequencies
- Check totals add correctly
Venn diagrams (covered in probability but used with frequency data):
- Represent sets and overlaps
- Work from the intersection outwards
- Ensure all regions account for total frequency
Worked examples
Example 1: Histogram and frequency density (Higher tier)
The table shows the times, in minutes, that 60 students took to complete a puzzle.
| Time (t minutes) | Frequency |
|---|---|
| 0 < t ≤ 10 | 8 |
| 10 < t ≤ 15 | 12 |
| 15 < t ≤ 25 | 25 |
| 25 < t ≤ 40 | 15 |
Draw a histogram to represent this data.
Solution:
First, calculate frequency density for each class:
| Time (t) | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0 < t ≤ 10 | 8 | 10 | 8 ÷ 10 = 0.8 |
| 10 < t ≤ 15 | 12 | 5 | 12 ÷ 5 = 2.4 |
| 15 < t ≤ 25 | 25 | 10 | 25 ÷ 10 = 2.5 |
| 25 < t ≤ 40 | 15 | 15 | 15 ÷ 15 = 1 |
Draw histogram with these frequency densities on vertical axis and time on horizontal axis. Bars should have no gaps.
Mark scheme points: 1 mark for at least two correct frequency densities; 1 mark for all four correct; 1 mark for correct histogram drawn.
Example 2: Cumulative frequency and box plot
The cumulative frequency table shows the heights of 80 Year 11 students.
| Height (h cm) | h ≤ 150 | h ≤ 155 | h ≤ 160 | h ≤ 165 | h ≤ 170 | h ≤ 180 |
|---|---|---|---|---|---|---|
| Cumulative frequency | 5 | 18 | 38 | 60 | 72 | 80 |
(a) Draw a cumulative frequency diagram. (b) Use your diagram to find the median and interquartile range. (c) Draw a box plot for this data.
Solution:
(a) Plot points: (150, 5), (155, 18), (160, 38), (165, 60), (170, 72), (180, 80). Join with a smooth curve.
(b)
- Median at ½ × 80 = 40th value → read across from 40 on vertical axis → approximately 161 cm
- Q₁ at ¼ × 80 = 20th value → approximately 156 cm
- Q₃ at ¾ × 80 = 60th value = 165 cm (given in table)
- IQR = 165 - 156 = 9 cm
(c) Box plot shows:
- Minimum: approximately 145 cm (extrapolate curve)
- Q₁: 156 cm
- Median: 161 cm
- Q₃: 165 cm
- Maximum: 180 cm
Mark scheme: 1 mark for correct cumulative frequency curve; 1 mark for median; 1 mark for both quartiles; 1 mark for IQR; 2 marks for box plot with all five values correct.
Example 3: Stratified sampling
A school has 1200 students. A survey of 80 students is planned using stratified sampling.
| Year group | Number of students |
|---|---|
| Year 7 | 250 |
| Year 8 | 240 |
| Year 9 | 230 |
| Year 10 | 240 |
| Year 11 | 240 |
Calculate how many Year 9 students should be in the sample.
Solution:
Year 9 sample = (230 ÷ 1200) × 80 = 0.191666... × 80 = 15.333... = 15 students (round to nearest whole number)
Alternative method: 80 ÷ 1200 = 1/15 230 ÷ 15 = 15.333... = 15 students
Mark scheme: 1 mark for correct method; 1 mark for answer of 15.
Common mistakes and how to avoid them
Confusing mean and median calculations: Mean requires summing all values then dividing; median requires ordering data first. Always check which average is requested.
Forgetting to use frequency density in histograms: The vertical axis is NOT frequency when class widths vary. Always calculate frequency density = frequency ÷ class width.
Reading cumulative frequency incorrectly: The median is at position ½n, not at the value ½n. Draw a horizontal line from the position on the vertical axis to the curve, then read down.
Misinterpreting correlation: Correlation does not prove causation. Two variables can be correlated without one causing the other; there may be a third factor involved.
Poor comparisons of datasets: Never just state "the mean is 12" without comparing it to the other dataset and explaining what this means in context. Always compare both average AND spread.
Rounding too early: Keep full calculator values until the final answer, especially when calculating standard deviation or using grouped data. Only round the final answer to an appropriate degree of accuracy.
Exam technique for Statistics
"Estimate the mean" signals grouped data with class intervals. Use midpoints multiplied by frequencies, then divide by total frequency. The word "estimate" is essential because you don't know exact values.
Show all working clearly: Marks are awarded for method even if the final answer is incorrect. For stratified sampling, write the calculation explicitly; for frequency density, show the division.
Read scales carefully: On cumulative frequency diagrams and histograms, check the scale intervals. They may not increase by 1 or 10. Count gridlines to determine each division's value.
Interpret in context: Questions asking you to "compare" or "comment on" require sentences that relate to the scenario. For example, "Class A was more consistent because the IQR is smaller" rather than just stating "IQR = 5".
Quick revision summary
Statistics involves collecting, presenting and analysing data. Master the three averages (mean, median, mode) and measures of spread (range, IQR). Frequency density = frequency ÷ class width for histograms. Cumulative frequency curves help find median and quartiles at ½n, ¼n and ¾n positions. Stratified sampling uses proportional representation from each group. Always compare datasets using both location and spread, interpreting findings contextually. Draw scatter graphs to assess correlation and use box plots for visual comparison of distributions.