What you'll learn
Statistics forms a substantial component of both Foundation and Higher tier Edexcel GCSE Mathematics papers, typically accounting for 15-20% of the total marks. This topic examines your ability to collect, organise, analyse and interpret data using measures of central tendency, measures of spread, and various graphical representations. Questions appear across all three papers and often combine multiple statistical techniques in real-world contexts.
Key terms and definitions
Mean — the sum of all values divided by the number of values; sensitive to extreme values and can be calculated from frequency tables and grouped data.
Median — the middle value when data is arranged in ascending order; for even datasets, the mean of the two middle values.
Mode — the most frequently occurring value in a dataset; a dataset can be bimodal (two modes) or have no mode.
Range — the difference between the highest and lowest values; provides a simple measure of spread but is affected by outliers.
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 outside the expected range, typically defined as below Q₁ - 1.5 × IQR or above Q₃ + 1.5 × IQR.
Cumulative frequency — the running total of frequencies up to a particular value; used to find medians and quartiles for grouped data.
Correlation — the relationship between two variables, described as positive, negative, or zero; does not imply causation.
Core concepts
Measures of central tendency
The three main averages used in Edexcel GCSE Mathematics are mean, median and mode. Each has specific applications and limitations.
Calculating the mean from raw data:
- Add all values together
- Divide by the total number of values
- Formula: x̄ = Σx ÷ n
Calculating the mean from a frequency table:
- Multiply each value by its frequency
- Sum these products to find Σfx
- Divide by the total frequency (Σf)
- Formula: x̄ = Σfx ÷ Σf
Estimating the mean from grouped data:
- Find the midpoint of each class interval
- Multiply each midpoint by its frequency
- Calculate Σfx ÷ Σf
- This gives an estimate because exact values are unknown
Finding the median:
- Arrange data in ascending order
- For n values, the median position is (n + 1) ÷ 2
- If n is odd, the median is the middle value
- If n is even, calculate the mean of the two middle values
Finding the mode:
- Identify the value with the highest frequency
- For grouped data, identify the modal class (the class interval with the highest frequency)
Measures of spread
Measures of spread quantify how dispersed data values are around the average.
Range calculation:
- Range = highest value - lowest value
- Quick to calculate but heavily influenced by extreme values
- Often paired with mean or median in exam questions
Quartiles and interquartile range:
- Lower quartile (Q₁): the value 25% of the way through the ordered dataset
- Upper quartile (Q₃): the value 75% of the way through the ordered dataset
- Q₁ position = (n + 1) ÷ 4
- Q₃ position = 3(n + 1) ÷ 4
- IQR = Q₃ - Q₁
- IQR is more resistant to outliers than range
Identifying outliers:
- Lower boundary: Q₁ - 1.5 × IQR
- Upper boundary: Q₃ + 1.5 × IQR
- Any value outside these boundaries is classified as an outlier
- Outliers may be removed when calculating mean to give a more representative average
Statistical diagrams and graphs
Frequency polygons:
- Plot frequency against the midpoint of each class interval
- Join points with straight lines
- Used to compare distributions when multiple datasets are shown on the same axes
Cumulative frequency diagrams:
- Plot cumulative frequency against the upper class boundary
- Join points with a smooth curve
- Used to estimate median, quartiles and percentiles
- Median estimate: read across from ½n on the y-axis
- Q₁ estimate: read across from ¼n on the y-axis
- Q₃ estimate: read across from ¾n on the y-axis
Box plots:
- Visual representation showing five key values: minimum, Q₁, median, Q₃, maximum
- The box spans from Q₁ to Q₃ (containing the middle 50% of data)
- Whiskers extend to minimum and maximum (excluding outliers)
- Outliers are shown as individual points beyond the whiskers
- Useful for comparing distributions between datasets
Histograms:
- Bars represent class intervals of potentially different widths
- The frequency density determines the height of each bar
- Formula: frequency density = frequency ÷ class width
- Area of each bar represents the frequency
- Essential to label y-axis as "Frequency density" in exams
Scatter graphs:
- Display bivariate data (two variables) as coordinate points
- Used to identify correlation between variables
- Positive correlation: as one variable increases, the other increases
- Negative correlation: as one variable increases, the other decreases
- Zero/no correlation: no apparent relationship
- Line of best fit: drawn to pass through the mean point with roughly equal numbers of points on each side
Time series graphs
Time series graphs display how data changes over time, with time always on the horizontal axis.
Identifying trends:
- Overall upward trend: values generally increasing
- Overall downward trend: values generally decreasing
- Seasonal or cyclical patterns: regular fluctuations
Moving averages:
- Used to smooth out fluctuations and identify underlying trends
- For a 4-point moving average, calculate the mean of consecutive groups of 4 values
- Plot each moving average at the midpoint of the time period it covers
- Common in Higher tier questions involving seasonal data
Sampling and data collection
Types of data:
- Qualitative data: descriptive (categories, opinions)
- Quantitative data: numerical values
- Discrete data: can only take specific values (usually whole numbers)
- Continuous data: can take any value within a range (measurements)
Sampling methods:
- Simple random sample: every member has equal chance of selection
- Systematic sample: select every nth member from a list
- Stratified sample: population divided into groups (strata); sample size from each group proportional to its size in the population
- Formula for stratified sampling: (stratum size ÷ population size) × sample size
Bias in sampling:
- Samples must be representative of the population
- Convenience sampling (selecting easily accessible members) introduces bias
- Time and location of sampling can affect representativeness
Worked examples
Example 1: Calculating mean from grouped frequency table
The table shows the time taken by 50 students to complete a puzzle.
| Time (t seconds) | Frequency |
|---|---|
| 20 < t ≤ 30 | 8 |
| 30 < t ≤ 40 | 15 |
| 40 < t ≤ 50 | 18 |
| 50 < t ≤ 60 | 9 |
Estimate the mean time taken.
Solution:
Find midpoints: 25, 35, 45, 55
| Midpoint (x) | Frequency (f) | fx |
|---|---|---|
| 25 | 8 | 200 |
| 35 | 15 | 525 |
| 45 | 18 | 810 |
| 55 | 9 | 495 |
| Totals | Σf = 50 | Σfx = 2030 |
Mean = Σfx ÷ Σf = 2030 ÷ 50 = 40.6 seconds
Example 2: Drawing and interpreting a box plot
For the dataset: 12, 15, 18, 20, 22, 23, 25, 28, 30, 32, 35, 38, 42
(a) Find the median and quartiles (b) Calculate the interquartile range (c) Identify any outliers
Solution:
(a) n = 13 values (already ordered)
Median position = (13 + 1) ÷ 2 = 7th value = 25
Q₁ position = (13 + 1) ÷ 4 = 3.5, so Q₁ = mean of 3rd and 4th values = (18 + 20) ÷ 2 = 19
Q₃ position = 3(13 + 1) ÷ 4 = 10.5, so Q₃ = mean of 10th and 11th values = (32 + 35) ÷ 2 = 33.5
(b) IQR = Q₃ - Q₁ = 33.5 - 19 = 14.5
(c) Lower boundary = 19 - 1.5 × 14.5 = 19 - 21.75 = -2.75
Upper boundary = 33.5 + 1.5 × 14.5 = 33.5 + 21.75 = 55.25
All values fall within these boundaries, so no outliers.
Example 3: Stratified sampling
A school has 240 Year 10 students and 180 Year 11 students. A stratified sample of 70 students is required. How many students should be selected from each year group?
Solution:
Total population = 240 + 180 = 420
Year 10: (240 ÷ 420) × 70 = 0.5714... × 70 = 40 students
Year 11: (180 ÷ 420) × 70 = 0.4286... × 70 = 30 students
Check: 40 + 30 = 70 ✓
Common mistakes and how to avoid them
Confusing mean and median — Students often calculate mean when the question asks for median. Read the question carefully and understand that median requires ordered data and finding the middle position, while mean requires summing all values and dividing.
Using values instead of midpoints for grouped data — When estimating the mean from grouped data, always use the midpoint of each class interval, not the class boundaries. For 20 < t ≤ 30, the midpoint is 25, not 20 or 30.
Calculating range from frequency tables — Range is highest value minus lowest value. For grouped data, use the highest possible value (top of highest class) minus lowest possible value (bottom of lowest class), not the midpoints.
Forgetting frequency density for histograms — A major error is plotting frequency instead of frequency density on the y-axis. Always divide frequency by class width, and remember that unequal class widths make this essential.
Misinterpreting correlation as causation — Just because two variables are correlated does not mean one causes the other. Ice cream sales and drowning incidents may correlate (both increase in summer), but ice cream doesn't cause drowning.
Incorrect box plot construction — The box must extend from Q₁ to Q₃, with a line at the median inside the box. Whiskers extend to minimum and maximum (or to the boundary values if outliers exist). Outliers are plotted separately as crosses or dots beyond the whiskers.
Exam technique for Statistics
Command word recognition — "Calculate" or "work out" requires a numerical answer with working shown. "Estimate" typically indicates grouped data where the exact mean cannot be found. "Describe" for correlation requires both type (positive/negative/none) and strength (strong/weak) plus context from the question.
Show working clearly — Statistics questions often carry method marks. Write out formulas, show substitution, then calculate. For finding quartiles, explicitly state positions before identifying values. Examiners award marks for correct method even if the final answer is wrong.
Label axes and diagrams fully — Histograms must have "Frequency density" on the y-axis (1 mark often allocated). Box plots need a scale. Cumulative frequency curves need both axes labelled with units. Missing labels cost marks even when the diagram is otherwise correct.
Justify statistical choices — Higher tier questions may ask which average is most appropriate. Link your choice to the data characteristics: mean is affected by outliers so median is better for skewed data; mode is suitable for qualitative data; mean uses all data values so is representative when no extreme values exist.
Quick revision summary
Statistics in Edexcel GCSE Mathematics requires mastery of averages (mean, median, mode), measures of spread (range, IQR), and graphical representations (histograms, box plots, cumulative frequency curves, scatter graphs). Calculate mean from frequency tables using Σfx ÷ Σf and estimate from grouped data using midpoints. Find median and quartiles from ordered data or cumulative frequency graphs. Draw histograms using frequency density (frequency ÷ class width). Identify correlation type and strength from scatter graphs. Apply stratified sampling using proportional selection. Show all working, label diagrams correctly, and match statistical measures to data context.