Statistics
KS3MA-KS3-D009
Collecting, representing, interpreting and comparing data using appropriate graphical representations and measures
National Curriculum context
Statistics at KS3 develops pupils' ability to work with data — collecting, organising, representing, analysing and interpreting it to answer questions and make inferences. Pupils extend their primary experience of simple charts and averages to a full range of graphical representations including scatter graphs, histograms and cumulative frequency, and to formal measures of average (mean, median, mode) and spread (range, quartiles). The statutory curriculum requires pupils to describe, interpret and compare distributions, and to begin reasoning about the relationship between two variables using correlation. Statistical reasoning is foundational to science, geography, economics and social sciences, and the curriculum prepares pupils for quantitative work across all these disciplines.
6
Concepts
3
Clusters
3
Prerequisites
6
With difficulty levels
Lesson Clusters
Calculate and interpret measures of central tendency and spread
introduction CuratedMean, mode, median and range (with outliers) are tightly co-taught (C084 lists C083). These foundational summary statistics must be established before data representation work.
Represent and compare data distributions using appropriate charts
practice CuratedData distribution, data representation (bar charts, pie charts, pictograms) and grouped data are co-taught (C085 lists C082, C083, C086; C084 lists C082, C083, C086). This is the data representation and analysis cluster.
Describe relationships between two variables using scatter graphs
practice CuratedBivariate data and scatter graphs represent a distinct statistical idea (correlation between two variables) that is meaningfully different from univariate distribution work.
Prerequisites
Concepts from other domains that pupils should know before this domain.
Concepts (6)
Data distribution
skill AI DirectMA-KS3-C082
Describing and comparing distributions using appropriate graphs and measures
Teaching guidance
Teach pupils to select appropriate representations for different types of data: bar charts for categorical data, histograms for continuous data, line graphs for time series. Compare distributions using back-to-back stem-and-leaf diagrams or comparative bar charts. Focus on interpretation: what does the distribution tell us about the context? Practise describing distributions using measures of central tendency and spread. Include real-world datasets where pupils must choose how to represent and describe the data. Emphasise that different representations can tell different stories about the same data.
Common misconceptions
Pupils often use bar charts for continuous data when a histogram would be more appropriate. The distinction between discrete and continuous data affects the choice of representation, but pupils commonly ignore this. When comparing distributions, pupils may only compare averages without considering spread. Some pupils think taller bars always mean 'better' without considering what the data represents.
Difficulty levels
Can read information from a simple chart or table and describe a data set in general terms (e.g. 'most people chose blue').
Example task
This bar chart shows favourite colours in a class. Which colour was most popular? How many chose red?
Model response: Blue was most popular (12 students). 7 students chose red.
Compares two data sets using basic measures (mean, range) and appropriate graphical representations.
Example task
Class A's test scores have mean 65 and range 40. Class B has mean 62 and range 15. Compare the two classes.
Model response: Class A has a slightly higher average (65 vs 62) but much greater spread (range 40 vs 15). Class B is more consistent — most students scored near the average — while Class A has a wider spread of abilities.
Selects and constructs appropriate graphical representations for different data types and uses them to compare distributions.
Example task
You want to compare the heights of Year 7 boys and girls. Which type of graph would you use and why?
Model response: Dual bar charts or back-to-back stem and leaf diagrams would allow direct comparison. Box plots would be best for comparing the medians, quartiles and ranges of both distributions side by side. Histograms would show the shape of each distribution but are harder to compare directly.
Describes, interprets and compares complex distributions using shape (skewness), outliers and multiple summary statistics.
Example task
Dataset A has median 45, mean 52, IQR 12. Dataset B has median 50, mean 50, IQR 20. Compare and interpret.
Model response: Dataset A: mean > median suggests positive skew (a few high values pull the mean up). The IQR of 12 indicates relatively consistent data. Dataset B: mean ≈ median suggests a roughly symmetric distribution. The IQR of 20 shows greater variability. Despite B having a higher median, A has less spread. The skew in A suggests a few exceptional high values — the median (45) is more representative of the typical value than the mean (52).
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.
Measures of central tendency
skill AI DirectMA-KS3-C083
Understanding and calculating mean, mode and median
Teaching guidance
Teach all three averages (mean, median, mode) and when each is most appropriate. Use datasets where the three averages differ significantly to motivate the discussion. Calculate the mean from raw data and from frequency tables. Find the median by ordering data and locating the middle value (or mean of the two middle values for even datasets). Identify the mode as the most frequent value. Discuss which average best represents a dataset: the mean is affected by outliers, the median is resistant to outliers, the mode is useful for categorical data.
Common misconceptions
Pupils commonly forget to order data before finding the median. When finding the mean from a frequency table, pupils may divide by the number of categories rather than the total frequency. Some pupils think there must always be exactly one mode, not recognising bimodal distributions or datasets with no mode. The belief that the mean is always the 'best' average persists despite counterexamples with skewed data.
Difficulty levels
Can calculate the mean of a small data set by adding values and dividing by the number of values.
Example task
Find the mean of: 4, 7, 3, 8, 3.
Model response: Sum = 4 + 7 + 3 + 8 + 3 = 25. Number of values = 5. Mean = 25/5 = 5.
Calculates mean, median and mode from ungrouped data and knows when each is appropriate.
Example task
Find the mean, median and mode of: 3, 5, 7, 7, 8, 10, 15.
Model response: Mean = (3+5+7+7+8+10+15)/7 = 55/7 = 7.86 (2 d.p.). Median = 7 (middle value of 7 ordered values). Mode = 7 (appears twice). The median and mode both suggest 7 is typical, while the mean is pulled up by the outlier 15.
Calculates the mean from a frequency table and understands the effect of outliers on different averages.
Example task
Calculate the mean from this frequency table: Score 1 (freq 3), Score 2 (freq 5), Score 3 (freq 8), Score 4 (freq 4).
Model response: Total frequency = 3+5+8+4 = 20. Σfx = 1(3)+2(5)+3(8)+4(4) = 3+10+24+16 = 53. Mean = 53/20 = 2.65.
Uses the mean to solve problems algebraically, including finding missing values and understanding the mean as a balance point.
Example task
The mean of five numbers is 8. Four of the numbers are 5, 7, 9, 11. Find the fifth number.
Model response: Sum of all five = 5 × 8 = 40. Sum of known four = 5+7+9+11 = 32. Fifth number = 40 - 32 = 8. The mean acts as a balance point — the total deviation above the mean equals the total deviation below.
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.
Measures of spread
knowledge AI DirectMA-KS3-C084
Understanding range and consideration of outliers
Teaching guidance
Introduce range as the simplest measure of spread: range = highest value − lowest value. Discuss its limitations: it depends on only two values and is heavily affected by outliers. Identify outliers visually and using rules of thumb (e.g., values more than 1.5 × IQR beyond the quartiles, though this may be simplified for KS3). Use comparative datasets where two groups have the same mean but different spreads to show why a measure of spread is needed alongside an average. Introduce the concept of consistency — lower spread means more consistent data.
Common misconceptions
Pupils often confuse range with the highest value rather than the difference between highest and lowest. Some pupils include outliers in range calculations without recognising their disproportionate effect. The concept of an outlier is often applied inconsistently — pupils may exclude values that are merely unusual rather than genuinely anomalous. Some think a larger range always means 'worse' data.
Difficulty levels
Can calculate the range of a data set (largest minus smallest) and understands it measures how spread out the data is.
Example task
Find the range of: 3, 7, 2, 11, 5.
Model response: Range = 11 - 2 = 9.
Understands that range is sensitive to outliers and can compare data sets using both an average and the range.
Example task
Data set A: 5, 6, 6, 7, 7, 8. Data set B: 2, 6, 6, 7, 7, 50. Compare using mean and range.
Model response: A: mean = 6.5, range = 3. B: mean = 13, range = 48. The outlier (50) in B dramatically increases both the mean and range. The median would be a better comparison: both have median 6.5, showing the typical values are similar.
Calculates and interprets the interquartile range (IQR) as a measure of spread that is resistant to outliers.
Example task
Find the IQR of: 2, 3, 5, 7, 8, 9, 11, 13, 15.
Model response: Q1 = 4 (median of lower half: 2,3,5,7). Q3 = 12 (median of upper half: 9,11,13,15). IQR = Q3 - Q1 = 12 - 4 = 8. The IQR tells us the range of the middle 50% of the data.
Uses IQR to identify outliers (values more than 1.5 × IQR from the quartiles) and evaluates which measure of spread is most appropriate for a given data set.
Example task
Data: 2, 5, 7, 8, 9, 10, 11, 12, 35. Identify any outliers using the IQR method.
Model response: Q1 = 6, Q3 = 11.5. IQR = 5.5. Lower fence: Q1 - 1.5(IQR) = 6 - 8.25 = -2.25. Upper fence: Q3 + 1.5(IQR) = 11.5 + 8.25 = 19.75. Value 35 > 19.75, so 35 is an outlier. Range = 33 (heavily influenced by the outlier). IQR = 5.5 (not affected). For this data, the IQR is a more representative measure of spread.
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.
Data representation
skill AI DirectMA-KS3-C085
Constructing and interpreting frequency tables, bar charts, pie charts, pictograms
Teaching guidance
Build on primary experience with bar charts, pictograms and line graphs. Teach pupils to select appropriate representations for different data types: pictograms for small datasets, bar charts for categorical data, vertical line charts for discrete data, pie charts for proportional comparison. Emphasise the mechanics of construction: equal bar widths, appropriate scales, clear labels and titles. Include frequency table construction as a data-organising step before graphical representation. Critique real-world charts for misleading features (truncated axes, 3D effects).
Common misconceptions
Pupils frequently use unequal bar widths or leave gaps between bars for continuous data. Pie chart construction errors include not measuring sectors accurately or not making sectors proportional to the data. Scale selection causes problems — choosing scales that either compress or stretch the data. Some pupils think pie charts show absolute values rather than proportions.
Difficulty levels
Can read information from bar charts, pictograms and simple tables.
Example task
How many pupils chose chocolate in this pictogram? (Each symbol represents 4 pupils, chocolate has 3.5 symbols.)
Model response: 3.5 × 4 = 14 pupils chose chocolate.
Constructs bar charts, pie charts and frequency tables from raw data, choosing appropriate scales and labels.
Example task
Draw a bar chart for: Red 8, Blue 12, Green 5, Yellow 3.
Model response: I draw a bar chart with the y-axis from 0 to 14 (suitable scale), bars of equal width, gaps between bars, and labels on both axes. The bars have heights 8, 12, 5, 3.
Constructs and interprets pie charts (calculating sector angles), frequency polygons and dual bar charts.
Example task
In a survey, 90 people were asked their favourite sport: Football 35, Tennis 20, Swimming 25, Other 10. Draw a pie chart.
Model response: Total = 90. Football: (35/90) × 360° = 140°. Tennis: (20/90) × 360° = 80°. Swimming: (25/90) × 360° = 100°. Other: (10/90) × 360° = 40°. Check: 140+80+100+40 = 360° ✓.
Selects the most appropriate representation for a given data type and purpose, and critically evaluates misleading graphs.
Example task
A newspaper shows a bar chart where the y-axis starts at 90 instead of 0, making a small difference between two political parties look dramatic. Explain the problem and draw a corrected version.
Model response: Starting the y-axis at 90 exaggerates the visual difference. If Party A has 95% and Party B has 92%, the truncated chart makes it look like A has roughly double B's support. The corrected chart with y-axis from 0 to 100 shows the bars are nearly the same height. This is a common way to mislead with data — the visual impression contradicts the numerical reality. Always check the axis scales before interpreting a graph.
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.
Grouped data
skill AI DirectMA-KS3-C086
Working with discrete, continuous and grouped data
Teaching guidance
Introduce the distinction between discrete data (counted, specific values) and continuous data (measured, any value within a range). Show that continuous data must be grouped because individual values are unlikely to repeat. Teach frequency table construction with class intervals, emphasising that intervals must not overlap and must cover all values. Discuss the effect of different class widths on the appearance of the data. Calculate estimated means from grouped frequency tables using midpoints. Introduce histograms where frequency density is used for unequal class widths.
Common misconceptions
Pupils commonly create overlapping class intervals (10-20, 20-30) rather than non-overlapping ones (10 ≤ x < 20, 20 ≤ x < 30). When calculating the estimated mean, pupils may use the lower boundary instead of the midpoint. The difference between a bar chart and a histogram (bar chart has gaps, histogram does not; histogram uses frequency density for unequal widths) is frequently confused.
Difficulty levels
Understands the difference between discrete data (counted) and continuous data (measured) and can organise data into simple groups.
Example task
Classify each as discrete or continuous: (a) number of siblings (b) height in cm (c) shoe size (d) time to run 100m.
Model response: (a) Discrete (counted, whole numbers). (b) Continuous (measured, any value). (c) Discrete (set sizes: 3, 3.5, 4, ...). (d) Continuous (measured, any positive value).
Constructs grouped frequency tables with appropriate class intervals, ensuring no gaps or overlaps.
Example task
Group these test scores into a frequency table: 45, 52, 67, 71, 38, 59, 82, 63, 55, 74. Use groups 30-39, 40-49, etc.
Model response: 30-39: 1, 40-49: 1, 50-59: 3, 60-69: 2, 70-79: 2, 80-89: 1. Total: 10 ✓.
Estimates the mean from grouped frequency tables using midpoints and understands why it is an estimate.
Example task
Estimate the mean from: 0-10 (freq 4), 10-20 (freq 7), 20-30 (freq 5), 30-40 (freq 2).
Model response: Midpoints: 5, 15, 25, 35. Σfm = 4(5)+7(15)+5(25)+2(35) = 20+105+125+70 = 320. Σf = 18. Estimated mean = 320/18 = 17.8 (1 d.p.). This is an estimate because we assume all values in each group equal the midpoint, which may not be true.
Works with grouped continuous data to estimate median, draw and interpret cumulative frequency curves and histograms.
Example task
From a cumulative frequency graph with total 80, estimate the median, Q1 and Q3.
Model response: Median at 80/2 = 40th value: read across from 40 on the y-axis to the curve, then down to the x-axis. Q1 at 80/4 = 20th value. Q3 at 3(80)/4 = 60th value. The positions are read from the smooth cumulative frequency curve, giving estimates not exact values.
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.
Bivariate data
skill AI DirectMA-KS3-C087
Describing relationships between two variables using scatter graphs
Teaching guidance
Introduce scatter graphs using real data: height versus arm span, temperature versus ice cream sales, hours of study versus test score. Plot ordered pairs with the independent variable on the x-axis and the dependent variable on the y-axis. Discuss correlation: positive (both increase together), negative (as one increases the other decreases), and no correlation. Draw lines of best fit by eye, balancing points above and below the line. Use the line of best fit to make predictions and discuss the reliability of interpolation versus extrapolation.
Common misconceptions
Pupils often confuse correlation with causation — just because two variables correlate does not mean one causes the other. When drawing lines of best fit, pupils may try to connect all the points rather than drawing a straight line that represents the overall trend. Some pupils think negative correlation means there is no relationship. Others place the line of best fit through the origin regardless of the data.
Difficulty levels
Can plot points on a scatter graph when given paired data and describe the overall trend informally.
Example task
Plot these pairs on a scatter graph: (2,3), (4,5), (6,8), (8,9), (10,11). What do you notice?
Model response: The points roughly form an upward line. As x increases, y tends to increase too.
Describes correlation (positive, negative, none) and draws a line of best fit by eye.
Example task
A scatter graph shows that as temperature increases, ice cream sales increase. Describe the correlation.
Model response: There is a positive correlation — as temperature increases, ice cream sales tend to increase.
Uses a line of best fit to estimate values (interpolation) and understands the limitations of extrapolation.
Example task
Using a scatter graph and line of best fit, estimate the ice cream sales when the temperature is 25°C (within the data range) and 45°C (outside the data range). Comment on reliability.
Model response: At 25°C: reading from the line gives approximately 180 sales (interpolation — reliable, within data range). At 45°C: the line suggests about 320 sales, but this is extrapolation (outside the data range) and unreliable — there may be a ceiling effect or other factors that change the relationship at extreme temperatures.
Critically evaluates bivariate data analysis, distinguishes correlation from causation, and identifies lurking variables.
Example task
A study finds a strong positive correlation between shoe size and reading ability in children aged 5-16. Does having big feet help you read better?
Model response: No — this is a classic spurious correlation. The lurking variable is age: as children get older, both their feet grow (shoe size increases) and their reading improves (more years of practice). Age causes both variables to increase, creating a correlation between them, but there is no causal link between shoe size and reading ability. To test for a genuine relationship, you would need to control for age (compare children of the same age).
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.