AQA GCSE Mathematics 8300: Statistics – Study Guide

This study guide teaches Topic 6, Statistics (specification references S1 to S6), of the AQA GCSE Mathematics (8300) specification, for teaching from September 2015 with exams from May/June 2017 (version 1.0). Statistics can be tested on any of the three papers at Foundation or Higher tier. Most of it is for both tiers. Histograms, cumulative frequency graphs (S3), box plots, quartiles and interquartile range (part of S4) are Higher tier only and are labelled below. Exams run every May/June and November.

Use it with the Statistics revision notes and the Statistics practice questions. The course hub is AQA GCSE Mathematics, the printable checklist lists every statement, and the free 10-minute diagnostics show where to start.

What this topic covers

Spec	What you must be able to do	Tier
S1	Infer properties of a population from a sample, knowing the limits of sampling	Both
S2	Read and draw frequency tables, bar charts, pie charts, pictograms, vertical line charts, and tables and line graphs for time series; choose a suitable diagram	Both
S3	Draw and interpret histograms (equal and unequal class widths) and cumulative frequency graphs	Higher tier only
S4	Compare distributions using graphs, averages (mean, median, mode, modal class) and range, including outliers	Both
S4	Box plots, quartiles and interquartile range	Higher tier only
S5	Apply statistics to describe a population	Both
S6	Scatter graphs, correlation, lines of best fit, interpolation and extrapolation; correlation is not causation	Both

Probability and statistics together make up about 15% of the marks at both tiers.

Types of data

You must know these four terms (S4 notes):

• Primary data: you collect it yourself, e.g. your own survey.
• Secondary data: someone else collected it, e.g. a website or a published table.
• Discrete data: can only take separate values, usually counts, e.g. number of siblings.
• Continuous data: measured, so it can take any value in a range, e.g. height, time, mass.

Populations and samples (S1, S5)

The population is the whole group you want to know about. A sample is the part you actually ask or measure. Sampling is quicker and cheaper, but a sample only gives an estimate.

A sample is biased if some members of the population are more likely to be chosen than others. Asking people at a gym about exercise habits will overstate how much the town exercises. A random sample gives every member an equal chance. A larger random sample gives a more reliable estimate.

Worked example 1. A factory makes 2000 phone cases a day. A random sample of 40 has 6 faulty cases.

Proportion faulty in sample = 6/40 = 0.15
Estimate for the population = 0.15 × 2000 = 300 faulty cases

The answer is an estimate. A different sample of 40 could give a different answer.

Tables, charts and diagrams (S2)

Diagram	Use it for
Bar chart	Categories (e.g. favourite sport); bars the same width with gaps
Pictogram	Categories; needs a key saying what one symbol stands for
Pie chart	Showing each category as a share of the whole
Vertical line chart	Ungrouped discrete data (e.g. shoe sizes)
Line graph	Time series: how a value changes over time

Worked example 2 (pie chart). 72 people were asked how they travel to work: walk 18, bus 26, car 20, cycle 8.

Angle for one person = 360° ÷ 72 = 5°
Walk 18 × 5 = 90°,  Bus 26 × 5 = 130°,  Car 20 × 5 = 100°,  Cycle 8 × 5 = 40°
Check: 90 + 130 + 100 + 40 = 360°

A pie chart shows proportions, not totals. Two pie charts with the same angle can stand for different numbers of people if the totals are different.

Time series. Plot time along the horizontal axis and join the points with straight lines. Describe the trend (the general rise or fall) and any repeating pattern, such as sales that peak every summer.

Averages and spread (S4)

Measure	How to find it
Mean	Total of values ÷ number of values
Median	Middle value when in order; position (n + 1)/2
Mode	Most common value (or modal class for grouped data)
Range	Largest − smallest; a measure of spread

Worked example 3 (frequency table). Goals scored in 20 matches:

Goals	0	1	2	3	4
Frequency	4	7	5	3	1

Total goals = 0×4 + 1×7 + 2×5 + 3×3 + 4×1 = 30
Mean = 30 ÷ 20 = 1.5 goals
Median: position (20 + 1)/2 = 10.5, so between the 10th and 11th values.
Running totals 4, 11, ... so the 10th and 11th are both 1. Median = 1
Mode = 1 (highest frequency).  Range = 4 − 0 = 4

Worked example 4 (grouped data). Time, t minutes, 40 people waited:

Time (minutes)	0 < t ≤ 10	10 < t ≤ 20	20 < t ≤ 30	30 < t ≤ 40
Frequency	4	15	12	9

Use the midpoint of each class as a stand-in for every value in it.

Midpoints: 5, 15, 25, 35
Σ(midpoint × frequency) = 20 + 225 + 300 + 315 = 860
Estimated mean = 860 ÷ 40 = 21.5 minutes
Modal class: 10 < t ≤ 20
Running totals 4, 19, 31, 40: the 20th and 21st values are in 20 < t ≤ 30

The mean is only an estimate because you do not know the exact values in each class.

Outliers

An outlier is a value far from the rest. For 12, 14, 15, 15, 16, 17, 18, 41 the mean is 18.5, but without 41 it is about 15.3. The median barely moves (15.5 to 15). The range drops from 29 to 6. When there is an outlier, the median is usually the better average, and the range is not a reliable measure of spread.

Comparing distributions

Always make two comparisons: one of an average and one of a spread, each in the context of the question.

Example: Class A has median 62 and range 30. Class B has median 55 and range 48. “On average Class A scored higher, because its median is higher. Class A’s scores were more consistent, because its range is smaller.”

Histograms and cumulative frequency (S3) – Higher tier only

Histograms

In a histogram the area of each bar stands for the frequency. The height is the frequency density:

frequency density = frequency ÷ class width

Worked example 5. Heights of 76 plants:

Height h (cm)	0 < h ≤ 10	10 < h ≤ 15	15 < h ≤ 20	20 < h ≤ 30	30 < h ≤ 50
Frequency	12	18	22	16	8
Frequency density	1.2	3.6	4.4	1.6	0.4

Estimate how many plants are between 12 cm and 25 cm:

10 < h ≤ 15: the part from 12 to 15 is 3/5 of the class → 3/5 × 18 = 10.8
15 < h ≤ 20: all 22
20 < h ≤ 30: the part from 20 to 25 is 5/10 of the class → 5/10 × 16 = 8
Total ≈ 10.8 + 22 + 8 = 40.8, about 41 plants

This assumes values are spread evenly within each class.

Cumulative frequency graphs

Add the frequencies as you go, then plot each running total at the upper class boundary. Join with a smooth curve or straight lines, starting from the lower boundary of the first class at a cumulative frequency of 0.

For 80 values in classes 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 8, 14, 26, 20, 12, plot (10, 8), (20, 22), (30, 48), (40, 68), (50, 80), starting at (0, 0).

• Median: read across at 80 ÷ 2 = 40. It is about 27.
• Lower quartile: read across at 80 ÷ 4 = 20. It is about 18.6.
• Upper quartile: read across at 3 × 80 ÷ 4 = 60. It is about 36.
• Interquartile range = upper quartile − lower quartile, about 17.4.

The values here come from straight lines between the points. A curve gives slightly different readings, which is why exam answers from graphs accept a range of values.

Box plots and interquartile range (S4) – Higher tier only

A box plot shows five values: minimum, lower quartile (LQ), median, upper quartile (UQ) and maximum. The interquartile range (IQR = UQ − LQ) is the spread of the middle half of the data. It is not affected by extreme values.

Worked example 6. Data in order: 3, 5, 6, 8, 9, 10, 12, 13, 15, 18, 22 (n = 11).

Median: (11 + 1)/2 = 6th value = 10
LQ: (11 + 1)/4 = 3rd value = 6
UQ: 3 × (11 + 1)/4 = 9th value = 15
IQR = 15 − 6 = 9;  range = 22 − 3 = 19

Draw the box from 6 to 15 with a line at 10, and whiskers out to 3 and 22, above a scale.

To compare two box plots, compare the medians (average) and the IQRs (consistency), in context.

Scatter graphs (S6)

A scatter graph plots pairs of values (bivariate data) to see if they are related.

• Positive correlation: as one goes up, the other tends to go up.
• Negative correlation: as one goes up, the other tends to go down.
• No correlation: no clear pattern.
• Strong correlation means the points lie close to a straight line; weak means they are more spread out.

A line of best fit is a straight line drawn by eye through the middle of the points, with roughly as many points above as below. Do not force it through the origin.

Worked example 7. Hours revised (x) and test score out of 100 (y) for 8 students: (2, 35), (3, 41), (4, 44), (5, 52), (6, 55), (7, 63), (8, 66), (9, 72). This shows strong positive correlation. The mean point is (5.5, 53.5), which is a good point for your line to pass through.

• Using the line to estimate within the range of the data (e.g. 6.5 hours) is interpolation and is fairly reliable.
• Using it outside the data (e.g. 20 hours) is extrapolation. The trend may not continue: a line through the mean point that fits these data predicts about 130 marks at 20 hours, which is impossible.

Correlation does not show causation. Ice-cream sales and sunburn cases both rise in summer. They are correlated because of a third factor, hot weather. Neither causes the other.

An outlier on a scatter graph is a point far from the pattern. Ignore it when drawing the line.

Common errors

• Drawing bar charts for continuous data with gaps, or histograms with frequency (not frequency density) on the vertical axis for unequal classes.
• Dividing Σ(midpoint × frequency) by the number of classes instead of the total frequency.
• Using the class width or upper boundary instead of the midpoint for an estimated mean.
• Plotting cumulative frequency at midpoints instead of upper class boundaries.
• Comparing only averages, or giving comparisons with no context.
• Saying one variable “causes” the other because the correlation is strong.

Next steps

Condense this into the revision notes, then test yourself with the practice questions. The Probability study guide covers the other half of this area of the course.

Official syllabus

AQA GCSE Mathematics (8300) specification, for teaching from September 2015, exams from May/June 2017, version 1.0, published by AQA – section 3.6 Statistics (S1 to S6).