AQA A-Level Mathematics: L: Data presentation and interpretation (7357) – Revision Notes

These notes condense Section L: Data presentation and interpretation (L1 to L4) of the AQA A-level Mathematics (7357) specification, version 1.3, for A-level exams from June 2018 onwards. The specification lists Section L under Paper 3. A calculator is required in every 7357 paper, and you must be able to use it to compute summary statistics. For full explanations and longer worked examples, use the Data presentation study guide.

Practise with the Data presentation practice questions, see every topic on the 7357 course hub, tick off outcomes on the printable checklist, and find your weak spots with the free 10-minute diagnostics.

Key definitions

• Frequency density: frequency ÷ class width. The height of a histogram bar.
• Relative frequency: frequency ÷ total frequency. Estimates a probability.
• Interquartile range (IQR): Q₃ − Q₁. The spread of the middle half of the data.
• Standard deviation: the square root of the variance; a typical distance of values from the mean.
• Correlation: how closely bivariate data follow a straight-line pattern (positive, negative or none; strong or weak).
• Regression line of y on x: a line of best fit used to predict y from a given x.
• Interpolation / extrapolation: predicting inside / outside the range of the observed x values.
• Outlier: a value far from the rest of the data, judged by a stated rule.
• Cleaning data: dealing with missing values, errors and outliers before analysis.

Formulas

Quantity	Formula
Mean (Appendix B, must recall)	x̄ = Σx / n = Σfx / Σf
Variance, divisor n	σ² = Σ(x − x̄)²/n = Σx²/n − x̄²
Variance, divisor n − 1	s² = (Σx² − n x̄²)/(n − 1)
Standard deviation	square root of the variance
Frequency density	frequency ÷ class width
Histogram frequency	bar area (× scale factor if needed)
IQR outlier limits	Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR
Standard deviation outlier limits	x̄ ± 2σ (or x̄ ± 3σ if the question says so)

The specification’s notation uses σ for a population standard deviation and s for a sample standard deviation. Calculators show both (often labelled σx and sx). Use the one the question asks for.

L1: Single-variable diagrams

Method: histogram questions

1. Find each class width from the class boundaries.
2. Frequency density = frequency ÷ width.
3. To find a frequency, find the bar’s area.
4. If the axis is not frequency density, use a known bar to find “people per cm²” (or per square) first.
5. For part of a class, take the matching fraction of that bar’s area (this assumes values are spread evenly within the class).

Reminder. The class 4 ≤ x < 6.5 has frequency 15, so its frequency density is 15 ÷ 2.5 = 6. An estimate of the frequency for 4 ≤ x < 5 is 1 × 6 = 6.

Other diagrams

• Box plot: read Q₁, median, Q₃, IQR and the range. A median nearer Q₁ with a long upper whisker shows positive skew.
• Cumulative frequency: plot at upper class boundaries. Median at n/2, quartiles at n/4 and 3n/4.
• Link to probability: relative frequency histograms have total area 1, so area over an interval estimates a probability. A symmetrical, bell-shaped histogram suggests a Normal model (Section N).

L2: Scatter diagrams and correlation

Describing a scatter diagram

Say the direction and the strength, in context: “strong negative correlation: cars with larger engines tend to travel fewer miles per gallon”.

Interpreting a regression line y = a + bx

• b: the estimated change in y when x increases by 1, in context, with units.
• a: the estimated y when x = 0. Often meaningless if x = 0 is far outside the data.
• Interpolation with strong correlation: reasonably reliable. Extrapolation: unreliable.
• Use a y on x line only to predict y from x.

Calculations involving regression lines are excluded from the specification. Focus on what the line means and when it is safe to use.

Must-know distinctions

Idea	What it means
Correlation	Two variables vary together
Causation	A change in one variable produces a change in the other
Distinct sections	Separate clusters in a scatter diagram, for example two regions

Correlation does not imply causation: a third variable may drive both. When a scatter diagram has distinct sections, describe each group separately. One line through all the points can suggest a correlation that does not hold within either group.

L3: Location and spread

Method: standard deviation from summary statistics

1. x̄ = Σx ÷ n.
2. σ² = Σx²/n − x̄². Check it is positive.
3. σ = √σ². Round to 3 s.f. at the end only.
4. If values are added or removed, update n, Σx and Σx² first, then recompute.

Reminder. n = 8, Σx = 96, Σx² = 1224. Then x̄ = 12, σ² = 153 − 144 = 9 and σ = 3.

Method: grouped data

Use class midpoints as x and the frequencies as f. Enter both into calculator statistics mode. The mean and standard deviation are estimates.

Choosing a measure

Data	Location	Spread
Roughly symmetrical, no outliers	Mean	Standard deviation
Skewed or with outliers	Median	IQR

When comparing two data sets, give one location and one spread comparison, each in context.

L4: Outliers, cleaning and choosing diagrams

Method: outlier check

1. Use the rule given in the question.
2. Work out the limits exactly; show them.
3. Compare the suspect value with the limit and state a conclusion.

Reminder. Q₁ = 18 and Q₃ = 26, so IQR = 8. The limits are 18 − 12 = 6 and 26 + 12 = 38. A value of 4 is an outlier; a value of 37 is not.

Cleaning data

• Missing values: leave out, say so, reduce n. Never record them as 0.
• Errors: impossible or mistyped values. Correct if known, otherwise remove.
• Outliers: investigate. Remove only if they are errors or not from the population.

Choosing a diagram

• Grouped continuous data: histogram or cumulative frequency diagram.
• Comparing distributions: box plots on a common scale.
• Bivariate data: scatter diagram.
• Categorical data: bar chart or pie chart.

Quick self-test

1. A class 15 ≤ x < 25 has frequency 32. Find its frequency density.
2. On a histogram, a bar of area 4 cm² represents 10 people. How many people does a bar of area 9.2 cm² represent?
3. n = 10, Σx = 85, Σx² = 785. Find the mean and standard deviation (divisor n).
4. Q₁ = 30 and Q₃ = 42. Is 61 an outlier under the 1.5 × IQR rule?
5. A data set has mean 50 and standard deviation 6. Is 63 an outlier under the rule “more than 2 standard deviations from the mean”?
6. A data set is strongly positively skewed. Which measures of location and spread should you use?
7. Towns with more swimming pools have more doctors. Does building pools increase the number of doctors? Explain.
8. Describe the correlation when points lie close to a line sloping down from left to right.
9. A sleep survey records one value as −4 hours. What should you do with it?
10. Estimate the mean of: 0 ≤ x < 10, f = 5; 10 ≤ x < 20, f = 10; 20 ≤ x < 40, f = 5.
11. A regression line was found from data with x between 20 and 60. Which prediction is more reliable: x = 45 or x = 90? Why?
12. For the data in question 3, an 11th value of 13.5 is added. Find the new mean.

Answers

1. 32 ÷ 10 = 3.2
2. 10 ÷ 4 = 2.5 people per cm², so 9.2 × 2.5 = 23
3. x̄ = 8.5; σ² = 78.5 − 72.25 = 6.25, so σ = 2.5
4. IQR = 12; upper limit 42 + 18 = 60. 61 > 60, so yes.
5. Limits 50 ± 12, that is 38 and 62. 63 > 62, so yes.
6. Median and IQR: they are not distorted by the long tail.
7. No. Correlation does not imply causation. Larger towns have more of both.
8. Strong negative correlation.
9. It is impossible, so it is an error: correct it if the true value is known, otherwise remove it and reduce n.
10. Midpoints 5, 15, 30: Σfx = 25 + 150 + 150 = 325, Σf = 20, so x̄ ≈ 16.25
11. x = 45: it lies inside the data range (interpolation); 90 is extrapolation.
12. (85 + 13.5) ÷ 11 = 8.95 (3 s.f.)

Where marks are usually lost

• Bar heights read as frequencies on a histogram with unequal class widths.
• The area scale factor not found when the axis is not labelled frequency density.
• Cumulative frequency plotted at class midpoints rather than upper class boundaries.
• Σx² confused with (Σx)², or x̄² not subtracted in the variance.
• Standard deviation updated from the old σ instead of from new n, Σx and Σx².
• Outlier limits not shown, so the conclusion has no support.
• An outlier removed with no reason, when it is a genuine value.
• Comparisons made with numbers but no context, or with two measures of location and none of spread.
• “Causes” written when the data only show correlation.
• Predictions from a regression line trusted outside the data range.

Related notes

Data you present usually come from a sample, so revise the statistical sampling revision notes alongside these.

Official syllabus

AQA A-level Mathematics (7357) specification, version 1.3 (31 January 2018), for A-level exams June 2018 onwards, published by AQA. Section 3.13, L: Data presentation and interpretation.