Statistics — AQA GCSE Mathematics

Statistics in AQA GCSE Mathematics covers collecting, representing, analysing and interpreting data using tables, graphs, averages and measures of spread. You must be able to calculate mean, median, mode and range, construct and read charts such as bar charts, histograms and cumulative frequency graphs, and describe correlation and distributions in context.

Statistics is the branch of mathematics concerned with collecting, organising, summarising and interpreting data. In AQA GCSE Mathematics, you will learn to construct and interpret a wide range of statistical diagrams including bar charts, pie charts, histograms, box plots and cumulative frequency graphs. You will also calculate and compare averages (mean, median, mode) and measures of spread (range and interquartile range) to describe data sets.

This topic is essential for understanding real-world data in fields such as science, business and social research. It also connects to probability and algebra, and often appears in crossover questions with other areas of the GCSE. Mastering statistics helps you make informed decisions and spot misleading representations of data.

• →Averages: mean, median and mode, and when each is most appropriate to use.
• →Measures of spread: range and interquartile range, and how they describe consistency.
• →Constructing and interpreting statistical diagrams: bar charts, pie charts, histograms, cumulative frequency graphs and box plots.
• →Scatter graphs and correlation: positive, negative and no correlation, and drawing a line of best fit.
• →Sampling: understanding populations, samples, and the importance of random sampling to avoid bias.

• the correct sample proportion, written as a fraction or a decimal, even if the scaling that follows is wrong
• multiplying that proportion by the population size, and one for the resulting estimate
• a reason that names the sampling flaw, such as the sample being too small or drawn from only one place
• credit for presenting the figure as an estimate rather than an exact population count
• inferring a distributional property from a sample, such as using the sample mean to estimate the population mean, or the sample range or interquartile range to estimate spread
• A correctly constructed frequency table from raw data, including a tally column if required.
• A correct method for calculating a pie chart angle, such as (frequency ÷ total frequency) × 360°.
• Accurately drawn charts (e.g., bars of correct height, sectors of correct angle within tolerance) with labelled axes or sectors, and a key for a pictogram.
• A correct value read from a chart or table, or a correct interpretation of what the chart shows.
• Correctly complete a table of time series data from given information, matching each time period to its value.
• Plot all points at the correct coordinates from the table, within a small tolerance.
• Join consecutive points with straight line segments in the correct time order.
• Describe the overall trend by its direction over time, supporting the description with values read from the graph or table.
• Interpret a time series graph correctly when the vertical axis does not start at zero, avoiding false proportional comparisons.
• Correctly identifying class boundaries for grouped discrete or continuous data.
• A correct calculation of frequency density for at least one class where intervals are unequal.
• Bars drawn with no gaps, at the correct heights (frequency or frequency density) and spanning the correct class boundaries.
• Cumulative frequencies correctly calculated and plotted against the upper class boundary, with a sensible curve drawn through the points.
• a correct method for the mean, namely the total of the values divided by how many values there are
• using midpoints on grouped data, one for the total of midpoint multiplied by frequency, and one for dividing by the total frequency
• a comparison of averages and a separate mark for a comparison of spread, each quoting a value
• naming the modal class, or the class that contains the median, from a running total
• interpreting a back-to-back stem-and-leaf diagram or comparative bar chart, for example reading the median and range of each data set from the display
• comparing the shape of two distributions from their graphs, for example noting which is more symmetric or which has the longer tail
• the median and one for both quartiles, taken from ordered data or read off a cumulative frequency graph
• a box running from the lower quartile to the upper quartile with the median line inside it, drawn against a scale
• whiskers reaching the smallest and the largest values
• the inter-quartile range as upper quartile minus lower quartile, even if one quartile is wrong
• comparing medians and one for comparing spread, each in the context of the question
• choosing an appropriate average and a correct value for it
• a measure of spread quoted alongside that average
• a statement written in the context of the question, with units
• making clear that a figure based on a sample is an estimate rather than an exact value
• all points plotted correctly, within half a small square
• naming the type of correlation, and a further mark for stating its strength
• interpreting the correlation in the words of the question rather than in general terms
• identifying an outlier and saying why it does not fit the pattern
• a ruled straight line following the trend with points balanced on both sides of it
• the read-off drawn on the graph, and one for a predicted value inside the accepted range
• judging a prediction reliable or unreliable, with a reason that refers to the range of the data
• stating that a pattern does not show cause, supported by another factor that could explain both quantities

• 💡Write the sample proportion as a fraction of the sample size first, then multiply by the population size, so you avoid rounding early.
• 💡When a question asks for a reason a sample is unreliable, name the group that is missing or over-represented, rather than only saying that the sample is small.
• 💡For distributional inference, quote the sample statistic (mean, median, range or interquartile range) and state that it estimates the corresponding population value.
• 💡Check that your pie chart angles total 360° before you start drawing. If they are a degree out due to rounding, adjust the largest sector.
• 💡When constructing a frequency table from a long list of data, use a tally system (e.g., groups of five) to avoid errors in counting.
• 💡Before plotting, work out what one small square is worth on each axis and write it down.
• 💡When describing a trend, state both the direction and the period, e.g. 'a steady rise across the four years'.
• 💡For a histogram with unequal intervals, always add columns for class width and frequency density to the table before you start drawing.
• 💡On a cumulative frequency graph, mark the median (1/2 total), lower quartile (1/4 total) and upper quartile (3/4 total) positions on the vertical axis first, then read across to the curve and down to the answer.
• 💡Write each comparison as one sentence containing a figure, for example that the mean height is larger, so this group is taller on average.
• 💡A smaller range means the values are more consistent, so use the word consistent when you compare spread.
• 💡When a question shows a graph, quote values read from the graph in your comparison rather than recalculating from raw data.
• 💡Mark the five values lightly on the scale before drawing the box, so the median line lands in the right place.
• 💡Write one comparison sentence about the medians and one about the inter-quartile ranges, and say what each means for the situation described.
• 💡Answer in a full sentence that names what is being described, the statistic and the units.
• 💡If one value sits far from the rest, choose the median and say that the extreme value would distort the mean.
• 💡Give three things when you describe a scatter graph: strength, direction, and what it means for the two named quantities.
• 💡Plot with a sharp pencil and a small cross, because a thick dot can sit on the wrong side of a gridline.
• 💡Check whether the value you are asked about lies inside the plotted range, because that decides whether you call the prediction reliable.
• 💡Draw the read-off lines in pencil and leave them on the graph, since they can earn a method mark on their own.
• 💡Always show your working for calculations, especially for the mean from a frequency table. Method marks are often available even if the final answer is wrong.
• 💡When interpreting graphs, refer to the context of the question. For example, say 'the median time is 25 seconds' rather than just 'the median is 25'.
• 💡Check the scale on axes carefully. A common mistake is misreading the scale, leading to incorrect values for quartiles or medians.

• Dividing by the population size instead of multiplying, so the estimate comes out smaller than the sample itself.
• Treating the scaled-up figure as an exact count of the population rather than an estimate.
• Answering "the sample was biased" without saying which group was left out or over-represented.
• Treating the sample as though it were the whole population, so the answer describes only the people who were asked.
• Using the sample mean as if it were the exact population mean, ignoring that a different sample would give a different value.
• Drawing bars touching for categorical data. This is the convention for a histogram, not a bar chart, where gaps are required.
• Calculating pie chart angles incorrectly, for example as percentages of 100 instead of proportions of 360°.
• Using a bar chart for ungrouped discrete numerical data where a vertical line chart is more appropriate.
• In a pictogram, drawing a partial symbol that does not accurately represent the required fraction of the key's value.
• Swapping the axes, putting the measured quantity on the horizontal axis instead of time. Correction: time always goes on the x-axis.
• Describing a short-term fluctuation between two points as the overall trend for the whole graph. Correction: describe the direction across the full period, noting fluctuations separately.
• Misreading a vertical axis that does not start at zero, leading to incorrect calculations or comparisons. Correction: check the scale and starting value before reading or comparing.
• Plotting points at even spacing on the x-axis when the time intervals in the table are uneven. Correction: space points according to the actual time intervals.
• Plotting frequency instead of frequency density on the vertical axis when class widths are unequal.
• Leaving gaps between histogram bars or using incorrect class boundaries for discrete data (e.g., a bar for 10-19 drawn from 10 to 19).
• Plotting cumulative frequency at the class midpoint or lower boundary instead of the upper boundary.
• Forgetting that the area of a histogram bar represents the frequency, especially when asked to find the frequency of part of a group.
• Dividing by the number of classes instead of the total frequency when estimating a mean from a grouped table.
• Giving the position of the median rather than the value that sits in that position.
• Comparing averages only and saying nothing about spread, which loses the second comparison mark.
• Picking the middle of the list before putting the values in order.
• Reading a back-to-back stem-and-leaf diagram as if both sides shared one ordered list, instead of treating each side as its own data set.
• Reading quartiles straight from an unordered list, so the box is built from the wrong values.
• Drawing the whiskers to the quartiles and the box edges to the extremes, which turns the plot inside out.
• Subtracting the wrong way round and giving a negative inter-quartile range.
• Saying one group did better without quoting either the median or the inter-quartile range.
• Quoting a mean to several decimal places when the data are whole counts, which claims more precision than the data carry.
• Giving an average on its own, with nothing about how spread out the values are.
• Using the mean on categorical data, where only the mode has any meaning.
• Dropping the units, so the figure describes nothing in particular.
• Joining the points up dot to dot, which turns a scatter graph into a line graph.
• Writing "correlation" with no direction, so the answer never says whether the pattern rises or falls.
• Calling a clear downward pattern no correlation because the values are getting smaller.
• Judging the strength of the correlation with an obvious outlier included in the pattern.
• Drawing the line through the first and last points instead of through the whole pattern.
• Predicting well beyond the plotted values and still calling the answer reliable.
• Claiming that one quantity makes the other happen because the pattern is strong.
• Letting an outlier drag the line of best fit away from the bulk of the points.
• Students often think the mean is always the best average to use. In fact, the median is better when there are extreme values (outliers) because it is not affected by them.
• Many students confuse the interquartile range with the range. The range is the difference between the highest and lowest values, while the interquartile range is the spread of the middle 50% of data.
• When drawing histograms, students frequently use frequency on the vertical axis instead of frequency density. For unequal class widths, you must calculate frequency density = frequency ÷ class width.

1. 1Week 1: Revise the definitions and calculations for mean, median, mode and range. Practice with simple data sets and frequency tables.
2. 2Week 1: Learn to construct and interpret bar charts, pie charts and scatter graphs. Focus on labelling axes and understanding correlation.
3. 3Week 2: Master cumulative frequency graphs and box plots. Practice finding median, quartiles and interquartile range.
4. 4Week 2: Study histograms with unequal class widths. Learn to calculate frequency density and draw histograms accurately.
5. 5Week 2: Complete past paper questions on statistics, timing yourself to build exam confidence. Review mistakes carefully.

• 📋Calculation questions: Find the mean, median, mode or range from a list or frequency table. Always show your method.
• 📋Graph interpretation: Read values from a cumulative frequency graph or box plot to find median, quartiles and interquartile range. Use a ruler and state units.
• 📋Comparison questions: Compare two data sets using averages and measures of spread. Comment on consistency and typical values in context.
• 📋Construction questions: Draw a bar chart, pie chart, histogram or box plot from given data. Ensure correct scales and labels.