Statistics — Edexcel GCSE Mathematics
Test yourself on Statistics with PEARSON EDEXCEL GCSE practice questions.
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Statistics explained
A population is the entire group of interest (e.g., all shoppers at a supermarket), while a sample is a smaller part you collect data from.
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We sample because measuring the whole population is often impractical. From a sample, we can infer properties of the population. This includes estimating totals: if 12 of 50 shoppers (24%) buy fruit, we estimate that 0.24 × 8000 = 1920 of the 8000 total shoppers will. We can also infer properties of distributions: if the median exam score of a sample from School A is higher than a sample from School B, we infer School A's students tend to have higher scores. However, inferences are limited by the sample. A small or biased sample (e.g., taken at one time of day) may not be representative, making the estimate unreliable.
interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data, tables and line graphs for time series data and know their appropriate use
To organise raw data, construct a frequency table by defining categories and using a tally chart to count items in each. For time series data, a table lists values against time points. Different displays suit different data. Bar charts (equal width bars, equal gaps) and pictograms (symbols with a key) show frequencies of categories. Vertical line charts do the same for ungrouped discrete data. Line graphs plot time series data to show trends over time. For a pie chart, the whole circle (360°) is the total. A category's angle is (frequency ÷ total) × 360°. For example, if 7 out of 30 people chose red, the angle is (7/30) × 360° = 84°. Questions require constructing tables, drawing charts accurately, or interpreting them by reading values.
construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use (Higher tier only)
A histogram shows frequency by area, not height. For unequal class intervals the vertical axis must be frequency density, calculated as frequency ÷ class width. The area of each bar (width × frequency density) then equals the frequency. For grouped discrete data, e.g. ages 1–10, 11–20, continuous class boundaries are needed so bars touch; use halfway points such as 0.5–10.5, 10.5–20.5. For continuous data grouped as 0 ≤ x < 10, 10 ≤ x < 20, the boundaries are simply 0, 10, 20. A cumulative frequency graph plots a running total of frequencies against the upper class boundary of each interval, and the points are joined by a smooth curve. This graph is used to estimate the median (at the n/2 value), quartiles (n/4 and 3n/4) and other percentiles. Use a histogram to see the shape of a distribution; use a cumulative frequency graph to estimate medians and quartiles.
interpret, analyse and compare the distributions of data sets from univariate empirical distributions through: • appropriate graphical representation involving discrete, continuous and grouped data • appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers) (Foundation tier)
To compare distributions, analyse their location (average) and spread (consistency) using graphs and calculations. Graphical representations like bar charts or frequency polygons show the distribution's shape. From a graph, you can identify the mode or modal class (the highest bar/point). For calculations, the mean is the total divided by the count; the median is the middle of an ordered list; the mode is the most frequent value. The range (largest value - smallest) measures spread. For the data 3, 5, 8, 8, 11: the mean is 7, median is 8, mode is 8, and range is 8. Outliers (extreme values) distort the mean and range, so the median can be a more representative average. A full comparison requires two contextual statements: one on the averages (e.g., 'on average, group A scored higher') and one on the spread (e.g., 'group B's scores were more consistent').
interpret, analyse and compare the distributions of data sets from univariate empirical distributions through: • appropriate graphical representation involving discrete, continuous and grouped data, including box plots • appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers, quartiles and inter-quartile range) (Higher tier)
This topic involves analysing and comparing univariate data distributions. You must use appropriate graphical representations: for discrete data, use bar charts or pictograms; for continuous or grouped data, use histograms or frequency polygons. Box plots are used to visualise the five-figure summary: minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum. Key calculations include measures of central tendency (mean, median, mode/modal class) and spread (range, inter-quartile range/IQR = Q₃ - Q₁). Outliers, extreme values that lie outside the main data distribution, can be identified using rules like being more than 1.5 × IQR below Q₁ or above Q₃. A core skill is comparing two distributions by commenting on both a measure of location (e.g., median) and a measure of spread (e.g., IQR), always interpreting your findings in the context of the data.
apply statistics to describe a population
Describing a population means turning a data set into a few figures and then saying what those figures mean about the group. Choose an average that suits the data: the mode for categories such as favourite sport, the median where a few very large or very small values would distort things, and the mean where you want every value to count. Pair the average with a measure of spread, because two groups can share a mean and still look nothing alike. For wages of £18000, £19000, £20000, £21000 and £90000 the mean is £33600, yet four of the five people earn less than that, so the median of £20000 describes a typical wage better. Questions give you a table or a chart and ask what it shows about the group, or ask which average is the fairest to use and why.
use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends while knowing the dangers of so doing
A scatter graph plots two measurements for each item, one against each axis, so you can see whether the two quantities are linked. Points rising from left to right show positive correlation, points falling show negative correlation, and points with no pattern show none. Describe strength as well as direction, since points lying close to a straight line show strong correlation. A line of best fit is one straight line following the trend with roughly as many points above it as below, and it need not pass through the origin. Predictions are read off that line, for example going up from 6 hours of revision to the line and across to about 48 marks. A reading taken inside the range of the plotted data is interpolation and is fairly safe; going beyond that range is extrapolation, where the trend may not continue. Correlation alone never shows that one quantity causes the other.
Your focus
- Distinguish between a population and a sample and explain why sampling is used.
- Estimate a population total by scaling up a sample proportion, or compare sample statistics to infer properties of population distributions.
- Explain why an inference may be unreliable, referring to the size or potential bias of the sample.
Show all 27 objectives
- Construct a frequency table from raw data using a tally system.
- Select and draw an appropriate chart for categorical, discrete, or time series data, including calculating pie chart angles.
- Interpret tables, charts and diagrams by reading values and describing what they show in context.
- Describe why unequal class widths require frequency density, so that a histogram shows frequency by area.
- Calculate frequency density as frequency divided by class width, and recover a frequency as width times height.
- Construct continuous class boundaries for grouped discrete data to ensure histogram bars touch.
- Estimate a median or a quartile from a cumulative frequency curve, showing the reading taken across and down.
- Explain why each cumulative frequency is plotted against the upper boundary of its class.
- Describe what the mean, median, mode and range each tell you about a set of data.
- Calculate an estimated mean for grouped data from midpoints times frequencies divided by the total frequency.
- Compare two distributions with one sentence on the averages and one on the spread, both written in context.
- Explain why an outlier drags the mean and stretches the range, so the median may describe the data better.
- Describe how the lower quartile, the median and the upper quartile are located in an ordered list of values.
- Calculate the interquartile range as upper quartile minus lower quartile, measuring the spread of the middle half.
- Draw a box plot showing the five figures against a given scale, or read them back off one.
- Explain why a single outlier changes the range far more than it changes the interquartile range.
- Describe a data set by choosing the mode, median or mean together with a measure of spread that suits the type of data collected.
- Calculate the mean and the median of a set containing one extreme value and state which of the two better describes a typical member.
- Explain which average is the fairest to use for a given data set, giving a reason that refers to an extreme value in the data.
- Compare two groups shown in a table or chart by quoting a figure from it and saying what that figure shows about the people counted.
- Describe the correlation on a scatter graph by its direction and its strength, and say what that means for the two quantities plotted.
- Draw a ruled line of best fit through plotted points and estimate a missing value by reading up to the line and across from it.
- Explain why a prediction found by extrapolating beyond the plotted data is less reliable than one interpolated inside that range.
- Give a reason why strong correlation between two quantities is not evidence that one of those quantities causes the other.
Statistics exam tips
Quick Revision Summary (Key Takeaway)
Statistics in Pearson Edexcel GCSE Mathematics covers collecting, representing, analysing and interpreting data using charts, averages, spread and scatter graphs. You must master calculating mean, median, mode and range from raw and grouped data, drawing and reading cumulative frequency and box plots, and describing correlation and trends in context.
Topic Overview
Statistics is the branch of mathematics concerned with collecting, organising, summarising and interpreting data. In Pearson Edexcel GCSE Mathematics, you will work with raw data lists, frequency tables, grouped frequency tables, bar charts, pie charts, histograms, cumulative frequency graphs, box plots and scatter graphs. You must be able to calculate and compare averages (mean, median, mode) and measures of spread (range, interquartile range), and use these to compare distributions in context.
This topic matters because statistical literacy is essential in science, business, journalism and everyday decision-making. Edexcel exam questions often place statistics in real-world contexts such as waiting times, heights, test scores or customer satisfaction, and they test both your calculation accuracy and your ability to interpret results in the original context. Statistics also connects to probability, as you may need to use probability to predict expected frequencies or assess risk.
Key Concepts
- →Averages and spread: mean, median, mode, range and interquartile range (IQR). The median and IQR are more resistant to outliers than the mean and range.
- →Grouped data: when data is grouped, you can only estimate the mean using midpoints, and you identify the modal class and median class rather than exact values.
- →Cumulative frequency: a running total of frequencies, plotted against the upper class boundary to find the median, quartiles and percentiles.
- →Box plots: a visual summary showing minimum, lower quartile (LQ), median, upper quartile (UQ) and maximum, used to compare distributions.
- →Scatter graphs and correlation: positive, negative or no correlation; line of best fit used to estimate values; correlation does not imply causation.
Marking Points
- writing the sample result as a fraction or proportion of the sample size
- multiplying that proportion by the stated population size to get an estimate
- a valid comparison of sample statistics (e.g., mean, median, range) to make an inference about populations
- a reason for unreliability related to sample size, bias, or timing, explaining its potential effect
- A correctly constructed frequency table with accurate tallies and frequencies from raw data.
- Correct calculation of pie chart angles, showing the method (e.g., freq/total × 360).
- Accurately drawn chart with appropriate scales, labels, and keys where needed.
- A correct interpretation that quotes a figure from the display and relates it to the context.
- Calculate at least one correct frequency density where the method (frequency ÷ class width) is clear.
- Draw histogram bars with correct widths corresponding to the continuous class boundaries, with no gaps between them.
- Draw bar heights corresponding to the calculated frequency densities.
- Plot cumulative frequency points accurately against the upper class boundary for each interval.
- Join the plotted cumulative frequency points with a smooth curve, starting from the lowest boundary on the horizontal axis.
- Clearly show reading lines on a cumulative frequency graph when estimating the median, quartiles or percentiles.
- Choose an appropriate diagram for a given purpose, e.g. a histogram to compare frequencies across classes or a cumulative frequency graph to estimate the median.
- Correctly identifying the mode or modal class from a list, frequency table or bar chart.
- Calculating the mean or an estimated mean, showing the sum of values (or fx) divided by the total frequency.
- Finding the median from a correctly ordered list of data.
- Making two separate, contextualised comparison statements: one for an appropriate average and one for the range.
- Calculate measures of central tendency (mean, median, mode, modal class) and spread (range, quartiles, IQR) from lists and frequency tables.
- Draw and interpret appropriate diagrams, including bar charts for discrete data, histograms and frequency polygons for grouped/continuous data, and box plots.
- Compare two distributions by commenting on a measure of central tendency (e.g., median) and a measure of spread (e.g., IQR), ensuring comments are contextualised.
- Identify outliers using a given rule (e.g., values more than 1.5 × IQR below Q₁ or above Q₃) and explain their effect on statistical measures like the mean and range.
- naming the average or measure of spread chosen, with a reason that refers to the data such as an extreme value
- working that statistic out correctly, even if the conclusion drawn from it is then wrong
- a statement about the group of people or items, not only about the numbers
- supporting that statement with a figure quoted from the table or diagram
- plotting every point correctly against both scales
- a ruled line of best fit that follows the trend of the points
- a reading taken from your own line of best fit, even if the line is not the expected one
- naming the type of correlation, and a further mark for saying what it means about the two quantities in context
- a comment on reliability that refers to the range of the data collected
Examiner Tips
- 💡For scaling questions, find the proportion first (sample result ÷ sample size), then multiply by the population size.
- 💡When comparing distributions, use comparative language like 'tends to be higher' or 'shows more variation', acknowledging it's an inference from a sample.
- 💡For a limitation, name who or what the sample excluded and suggest how this could affect the result.
- 💡When creating a frequency table from a list, use a tally system to count accurately.
- 💡Always label your axes and give your chart a title. For pictograms and pie charts, include a key.
- 💡Check that your pie chart angles add up to 360° before you start drawing.
- 💡Before drawing a histogram, add two columns to the table: one for class width and one for frequency density. This organises your calculations.
- 💡Always draw the lines on the cumulative frequency graph that you used to find an estimate. This allows you to be awarded method marks even if your reading is slightly inaccurate.
- 💡When asked to compare, always write two sentences. One compares the averages (e.g., mean/median) and one compares the spread (range). Both must be in the context of the question (e.g., 'higher average score', 'more consistent times').
- 💡For an estimated mean from a grouped table, methodically add columns for the midpoint (x) and frequency × midpoint (fx). Use the totals of the frequency and fx columns to calculate the mean.
- 💡When comparing distributions (e.g., from box plots), always compare a measure of average (like the median) and a measure of spread (like the IQR). Then, explain what each comparison means in the context of the problem. For example, a smaller IQR means the data is more consistent.
- 💡The mean is affected by outliers, whereas the median and IQR are resistant to them. Choose the median as the most appropriate average if the data contains extreme values.
- 💡Answer in a full sentence that names the group and the quantity, for example "most club members finished in under 30 minutes".
- 💡If asked which average to use, say what the other averages would get wrong on this particular data.
- 💡Keep the units on every figure you quote, and round money to the nearest penny.
- 💡Draw the line of best fit in pencil and check that the points are balanced on both sides of it before taking any reading.
- 💡Show a prediction with dashed lines from the axis to the line and back, so the method can be credited.
- 💡If the value you are asked to predict lies outside the range of the plotted data, say so and say why the answer may be unreliable.
- 💡Always read the question carefully to see whether it asks for an exact answer or an estimate. For grouped data, the mean is always an estimate because you use midpoints.
- 💡When comparing distributions, use the phrase 'on average' when comparing means or medians, and 'more consistent' or 'more spread out' when comparing ranges or IQRs. This matches Edexcel mark scheme wording.
- 💡Show all stages of working, especially for cumulative frequency and box plot questions. Even if your final answer is wrong, method marks are available for correct midpoints, cumulative frequencies or quartile readings.
Common Mistakes
- Giving the raw sample count as the population estimate.
- Scaling with the wrong pair of numbers, such as dividing population size by the sample result.
- Stating that a sample proves a population property, rather than suggesting it is an estimate or inference.
- Drawing bars of different widths or with no gaps on a bar chart.
- Rounding pie chart angles so the total is not 360°.
- Creating overlapping or non-exhaustive class intervals in a grouped frequency table.
- Plotting frequency on the vertical axis of a histogram when class widths are unequal. Correction: the vertical axis must be frequency density (frequency ÷ class width).
- Plotting cumulative frequency against class midpoints instead of upper class boundaries. Correction: cumulative frequency is a running total up to the end of an interval, so it must be plotted at the upper boundary.
- For grouped discrete data (e.g. 1–10, 11–20), drawing bars with gaps between them. Correction: a histogram's bars must touch; use continuous class boundaries halfway between the discrete intervals, e.g. 0.5–10.5, 10.5–20.5.
- Joining cumulative frequency points with straight line segments. Correction: join them with a smooth curve, as required for Edexcel GCSE.
- Misinterpreting graphs, for example reading the frequency from the vertical axis as the data value in a bar chart.
- Forgetting to put the data in numerical order before finding the median.
- Making non-contextual comparisons, e.g., stating 'The mean for A is 10 and the mean for B is 12' without explaining what this means for groups A and B.
- Forgetting to use midpoints when estimating the mean from a grouped frequency table. Correction: The estimated mean is calculated as Σfx / Σf, where 'x' is the midpoint of each class interval.
- Confusing the mode with the highest frequency. Correction: The mode is the data value or class interval with the highest frequency, not the frequency value itself.
- When comparing distributions, just stating calculated values (e.g., 'The median for A is 25 and for B is 30'). Correction: You must interpret this in context (e.g., 'On average, group B's scores were higher than group A's because their median was higher').
- Quoting a mean and a range with no sentence saying what they show about the group.
- Using the mean on data holding one very large value and then calling it the typical amount.
- Describing the sample that was measured and calling the description true of the whole population.
- Giving a modal category as its frequency instead of naming the category itself.
- Joining the points in order instead of drawing a single straight line of best fit.
- Forcing the line of best fit through the origin, or through the first and last points only.
- Writing that the correlation proves one quantity causes the other.
- Reading a prediction off a nearby plotted point rather than off the line of best fit.
- Extending the line well past the data and giving the prediction with no comment on its reliability.
- Students often think the mean is always the best average to use. In fact, the median is better when data contains outliers or is skewed, because it is not affected by extreme values.
- Many students believe that a strong correlation between two variables proves that one causes the other. Correlation only shows a relationship; a third factor or coincidence may explain it.
- When drawing a cumulative frequency graph, students sometimes plot cumulative frequency against the midpoint of the class interval instead of the upper class boundary. This shifts the curve and leads to incorrect quartile readings.
Revision Plan
- 1Days 1-2: Revise averages and spread from raw data lists. Practise calculating mean, median, mode, range and IQR, and compare two data sets in context.
- 2Days 3-4: Move on to grouped frequency tables. Learn to find midpoints, estimate the mean, and identify modal and median classes. Complete at least five exam-style questions.
- 3Days 5-6: Study cumulative frequency graphs and box plots. Practise drawing these from grouped data and reading off medians, quartiles and percentiles. Compare box plots in context.
- 4Days 7-8: Cover scatter graphs and correlation. Practise describing correlation, drawing lines of best fit and using them to estimate values. Learn the difference between correlation and causation.
- 5Days 9-10: Complete a mixed set of past paper questions under timed conditions. Mark your work using the official mark scheme and note recurring errors for final revision.
Exam Question Types
- 📋Calculation questions: 'Calculate the mean from the grouped frequency table.' These test your ability to use midpoints and divide by total frequency. Always show your fx column.
- 📋Comparison questions: 'Compare the distributions of the two box plots.' You must compare an average (median) and a measure of spread (IQR or range) in context. Use 'on average' and 'more consistent'.
- 📋Graph interpretation questions: 'Use your cumulative frequency graph to find the median and interquartile range.' Read across from the correct cumulative frequency value and down to the axis, showing your lines.
- 📋Scatter graph questions: 'Describe the correlation and estimate the value of y when x = ...' Describe as positive, negative or no correlation, then draw or use the line of best fit to estimate.
Command Word Expectations (PEARSON EDEXCEL)
You must work out a numerical answer using the given data. Show all stages of working. For grouped data, the answer is an estimate, so write 'estimate' if asked. Units are often required.
You must state similarities or differences between two distributions. For full marks, compare one average (mean or median) and one measure of spread (range or IQR), and interpret these in the context of the question.
You must write a statement about the data or graph. For correlation, state whether it is positive, negative or no correlation and relate it to the variables. For distributions, describe the shape or key features.
How Students Lose Marks (Examiner Pitfalls)
Step-by-Step Worked Solutions
Question: The table shows the time, t minutes, taken by 20 students to complete a puzzle. 0 < t <= 10: frequency 4; 10 < t <= 20: frequency 7; 20 < t <= 30: frequency 9. Calculate an estimate for the mean time taken.
- 1.Step 1: Identify the midpoints of each class interval: 5, 15 and 25.
- 2.Step 2: Multiply each midpoint by its frequency: 5 x 4 = 20, 15 x 7 = 105, 25 x 9 = 225.
- 3.Step 3: Add these products: 20 + 105 + 225 = 350.
- 4.Step 4: Divide by the total frequency: 350 divided by 20 = 17.5.
- 5.Step 5: State the final answer with units: the estimated mean time is 17.5 minutes.
Question: A scatter graph shows the relationship between hours of revision (x) and test score (y) for 12 students. The line of best fit passes through (2, 40) and (8, 70). Describe the correlation and use the line to estimate the score for a student who revises for 5 hours.
- 1.Step 1: Describe the correlation: as hours of revision increase, test scores increase, so there is positive correlation.
- 2.Step 2: Find the equation of the line of best fit. Gradient = (70 - 40) divided by (8 - 2) = 30 divided by 6 = 5.
- 3.Step 3: Use y = mx + c with point (2, 40): 40 = 5(2) + c, so c = 30. Equation is y = 5x + 30.
- 4.Step 4: Substitute x = 5: y = 5(5) + 30 = 25 + 30 = 55.
- 5.Step 5: State the estimate in context: the estimated test score is 55 marks.