OxfordAQA International GCSE Mathematics: Interpreting graphs, comparing distributions and correlation (9260)

This study guide is for OxfordAQA International GCSE Mathematics (9260), written to the specification Version 5.1 (for exams May/June 2018 onwards). It covers references S6, S7 and S8 in section 3.4.2, Interpretation. All three are Core content, so they can also be assessed on the Extension papers. There is no Extension-only content in this unit.

Use it with the revision notes and the practice set. See the OxfordAQA International GCSE Mathematics course hub, the printable checklist lets you tick off skills, and the free diagnostics show where to focus.

What you must be able to do

Ref	Skill	Tier
S6	Read and interpret a wide range of graphs and diagrams and draw conclusions	Core
S7	Compare distributions and make inferences	Core
S8	Recognise correlation; draw and/or use lines of best fit by eye, understanding what they represent	Core

The specification notes that you should know the terms positive correlation, negative correlation, no correlation, weak correlation and strong correlation. Drawing the charts themselves, and finding the averages you will use here, sit in section 3.4.1.

1. Reading and interpreting graphs and diagrams (S6)

Every question of this type rewards the same habit: read before you calculate.

1. Read the title, the axis labels, the units and the key.
2. Work out the scale. Check what one small square is worth on each axis.
3. Read the exact values you need, using ruler lines if you are allowed to draw on the diagram.
4. Describe the pattern in the context (rising, falling, peak, steady, seasonal).
5. Write a conclusion that the data can support, and no more.

What to look for in each diagram

Diagram	What it shows	Watch for
Line graph (time series)	How a quantity changes over time	Steepest part = fastest change; flat part = no change
Dual bar chart	Two groups side by side	Compare the same category in each group
Composite bar chart	Total split into parts	Only the bottom part starts at zero
Pie chart	Share of a whole	Angle ÷ 360 gives the fraction; it shows no totals unless you are told one
Frequency polygon, histogram (equal widths)	Shape of a distribution	Peak, symmetry, tails
Stem-and-leaf	Every value, in order	Read the key; count leaves for the total
Scatter graph	Link between two variables	Direction, tightness, outliers

Worked example 1: pie chart

A survey of 240 students asked how they travel to school. The pie chart angles are walk 90°, bus 135°, cycle 60°, car 75°.

Check the angles add to 360°: 90 + 135 + 60 + 75 = 360 ✓

Each student is worth 360° ÷ 240 = 1.5°. So:

walk  = 90 ÷ 1.5  = 60 students
bus   = 135 ÷ 1.5 = 90 students
cycle = 60 ÷ 1.5  = 40 students
car   = 75 ÷ 1.5  = 50 students

Bus users make up 135/360 = 3/8 of the students. Cycling is 60/360 = 1/6 of them. The bus group is 30 students bigger than the walking group.

Worked example 2: line graph

The line graph shows visitors to a museum (in thousands) at the end of each month from January to June: 12, 16, 22, 30, 27, 18.

Change each month:  +4, +6, +8, -3, -9

• The greatest rise was from March to April (+8 thousand). That is the steepest section of the line.
• Visitors peaked in April at 30 thousand.
• From April to June visitors fell from 30 to 18. That is 12 ÷ 30 = 40% down.
• A fair conclusion is “visitors fell after April”. It is not safe to say “visitors will keep falling in July”, because the graph gives no information beyond June.

Misleading graphs

A graph can mislead without any wrong numbers. Look for a vertical axis that does not start at zero, bars of different widths, a scale that is uneven, or a 3-D effect that exaggerates a slice. If a bar chart has its axis starting at 50, a bar of height 56 looks three times as tall as a bar of height 52, because the visible parts are 6 and 2. The real difference is only 56 ÷ 52 ≈ 1.08 times.

2. Comparing distributions and making inferences (S7)

To compare two sets of data you need two things, each used for both sets:

• an average (mean, median or mode), which tells you what is typical
• a measure of spread (usually the range), which tells you how consistent the data is

A comparison is only complete when it uses numbers from both sets and says what they mean in the context.

Choose your average with care. The mean includes every value, so a single extreme figure drags it up or down. The median is unaffected by extremes. The mode or modal class is the only sensible average for category data, or when you want the most common value. The range is easy to find but depends only on the largest and smallest values, so one unusual value inflates it.

Worked example 3: raw data

Two couriers record how many minutes late each of seven deliveries was.

Courier X: 4, 6, 7, 8, 9, 11, 25
Courier Y: 7, 8, 9, 10, 10, 12, 14

	Mean	Median	Range
X	70 ÷ 7 = 10	8	25 - 4 = 21
Y	70 ÷ 7 = 10	10	14 - 7 = 7

Comparison: “The mean lateness is the same for both couriers (10 minutes), but Y is more consistent because Y’s range is 7 minutes against 21 for X. X’s median is lower (8 against 10), so most of X’s deliveries are less late than Y’s. X’s mean is raised by one very late delivery (25 minutes). Without it X’s mean would be 7.5 minutes.”

Notice how the answer uses a measure of average and a measure of spread, names both couriers, and gives a reason for the difference.

Worked example 4: back-to-back stem-and-leaf diagram

Two classes timed themselves solving a puzzle (seconds). Each class has 11 pupils.

      Class A            |  Stem  |      Class B
               8  5  3   |    2   |
         7  4  4  1      |    3   |  1  3  6  8
               5  2      |    4   |  0  1  3  4  6  7  9
                  3  1   |    5   |

Key: for Class A, 4 | 3 means 34 seconds (leaf 4 on the left of stem 3). For Class B, 4 | 1 means 41 seconds (stem 4, leaf 1 on the right).

With 11 values the median is the 6th value.

	Class A	Class B
Median	34	41
Range	53 - 23 = 30	49 - 31 = 18
Mean	403 ÷ 11 ≈ 36.6	448 ÷ 11 ≈ 40.7

Comparison: “Class A solved the puzzle faster on average (median 34 s against 41 s), and for a puzzle time a lower value is better. Class B’s times were more consistent (range 18 s against 30 s). Class A has both the quickest pupil and some of the slowest.”

Making inferences

An inference is a conclusion about a wider group or a cause, based on the data. Good inferences are cautious.

• With few values, say the sample is small.
• If the two groups were chosen differently, the comparison may not be fair.
• A difference in averages that is small compared with the spread may not mean much.
• You can talk about the data you have. Be careful about predicting beyond it.

3. Correlation and lines of best fit (S8)

A scatter graph plots pairs of values. Correlation describes the relationship between the two variables.

Type	Pattern	Example
Positive	As x rises, y tends to rise	Height and arm length
Negative	As x rises, y tends to fall	Age of a car and its value
None	No pattern	Shoe size and test mark

Strength says how close the points lie to a straight line. Points tightly along a line show strong correlation. Points that form a loose band show weak correlation. A good description has two parts, for example “strong negative correlation”.

Correlation is not cause

Two variables can move together without one causing the other. Sales of sun cream and ice lollies both rise in hot weather. Neither causes the other: a third factor, the temperature, affects both. Write “there is a relationship” or “is linked to”, and look for a third factor before you say “causes”.

Drawing a line of best fit by eye

A line of best fit is a straight line that follows the trend of the points.

1. Work out the mean of each variable if the question gives you the data. A sensible line passes through, or very close to, the mean point.
2. Place a ruler so the line follows the direction of the points, with roughly equal numbers of points above and below it.
3. Draw it as one straight line. It does not need to pass through the origin or any particular point.
4. Ignore outliers when positioning the line.

Worked example 5

Ten pupils recorded hours of revision (x) and test score out of 100 (y).

x: 1  2  2  3  4  5  5  6  7  8
y: 38 45 41 52 55 60 66 68 75 79

Sum of x = 43, so the mean x = 4.3. Sum of y = 579, so the mean y = 57.9. The mean point is (4.3, 57.9).

The points rise from bottom left to top right and lie close to a line, so there is strong positive correlation.

Suppose your line passes through (1, 38) and (8, 80), which keeps it close to the mean point.

gradient = (80 - 38) ÷ (8 - 1) = 42 ÷ 7 = 6
y = 6x + c, so 38 = 6(1) + c, giving c = 32
line: y = 6x + 32

Check: at x = 4.3 the line gives 6(4.3) + 32 = 57.8, almost exactly the mean y.

• Estimate for 6.5 hours: y = 6(6.5) + 32 = 71 (interpolation, inside the data range 1 to 8).
• The gradient 6 means each extra hour of revision is linked to about 6 more marks.
• The intercept 32 means about 32 marks with no revision. This is only a little outside the data, so treat it as rough.
• For 12 hours the line gives 104, which is impossible for a score out of 100. This is extrapolation, outside the data range, and is unreliable.

Because the line is drawn by eye, other sensible lines are accepted. Your estimates should be close to these but need not match exactly.

Common errors

• Reading a pie chart as if the angle were the frequency. Convert using the total.
• Comparing only averages and never mentioning spread, or comparing a number of one set with nothing from the other.
• Writing “Class A is higher” without saying higher in what (time, score) and whether higher is better.
• Calling a weak pattern “no correlation”, or a clear pattern “strong” when the points are scattered.
• Using a line of best fit that is joined point to point, or that is forced through the origin.
• Estimating far outside the data and treating the answer as reliable.
• Saying correlation proves one variable causes the other.

Where to go next

Work through the practice set, then use the revision notes for the final check. For the earlier skills of charts and averages, use the course hub.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), specification Version 5.1 (for exams May/June 2018 onwards), published by OxfordAQA, section 3.4.2 Interpretation, references S6, S7 and S8.