IB DP Mathematics: Analysis and Approaches – Sampling, data presentation, summary statistics and regression Revision Notes

These are condensed revision notes. For full explanations and longer worked examples, read the statistics study guide first.

The notes cover the descriptive statistics unit of IB Diploma Programme Mathematics: Analysis and Approaches. They are aligned to the IB Mathematics: analysis and approaches guide, first assessment 2021, syllabus sections 4.1–4.4 and 4.10, and all of the content is common to SL and HL. They follow the IB guide for first assessment 2021, which remains the examined syllabus until the new course is first assessed in May 2029, so they apply to the May and November 2026, 2027 and 2028 SL and HL sessions.

When you can recall everything here, test yourself with the statistics practice questions. The IB DP Maths AA course hub and the printable syllabus checklist show where this unit sits in the course.

Definitions

• Population: every item of interest. Sample: the part you measure. Random sample: every item has an equal chance of selection.
• Discrete data: separate values, usually counts. Continuous data: any value in a range, usually measurements.
• Bias: some members of the population are more likely to be chosen than others, so the sample does not represent it.
• Reliability: consider who collected the data, missing data and recording errors.
• Outlier: a data item more than 1.5 × IQR from the nearest quartile.
• Modal class: the class with the highest frequency (equal class intervals only).
• Variance: the square of the standard deviation.
• Mean point: (x̄, ȳ). Lines of best fit by eye and both regression lines pass through it.
• Extrapolation: predicting outside the range of the data. It is unreliable.

At SL the data set is treated as the population unless you are told otherwise.

Sampling methods

Method	Key feature	Watch out for
Simple random	Random numbers from a full list	Needs a complete list
Convenience	Whoever is easy to reach	Usually biased
Systematic	Every kth item, random start, k = N/n	Patterns in the list
Quota	Fixed number per group, not random	Interviewer chooses
Stratified	Random sample from each group, in proportion	Needs group sizes

Method in steps: stratified sample

1. Find the total population N.
2. For each group, multiply (group size / N) by the sample size n.
3. Round sensibly and check the numbers add up to n.

Formulas and results

Quantity	Result	How you find it
Mean	x̄ = Σx / n or Σfx / Σf	By hand or GDC
Grouped mean (estimate)	Σfx / Σf with x = mid-interval value	By hand or GDC
Median	Value at position (n + 1)/2 of ordered data	By hand or GDC
IQR	Q₃ − Q₁	Quartiles by GDC; by hand for short lists
Outlier fences	Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR	By hand
Standard deviation	σ (use σx on the GDC)	Technology only
Variance	σ²	Square the standard deviation
Pearson’s r	−1 ≤ r ≤ 1	Technology
y on x line	y = ax + b	Technology
x on y line	x = cy + d	Technology (swap the lists)

Constant changes to every data item

Change	Mean, median, quartiles	SD, IQR, range	Variance
+ c	+ c	unchanged	unchanged
× k (k > 0)	× k	× k	× k²

Worked reminder: mean 50, SD 8. Every item becomes 3x − 10. New mean = 3(50) − 10 = 140. New SD = 3 × 8 = 24. New variance = 576.

Data displays

Method in steps: cumulative frequency graph

1. Add a cumulative frequency column (running total).
2. Plot each total against the upper class boundary; start at zero at the lowest boundary.
3. Join the points with a smooth curve or straight lines.
4. Read across at n/2 (median), n/4 (Q₁), 3n/4 (Q₃) or p% of n (pth percentile).

• Class intervals are inequalities with no gaps, such as 20 ≤ t < 30.
• Frequency histograms use equal class widths; bar height is frequency. Frequency density is not required.
• Box and whisker diagram: minimum, Q₁, median, Q₃, maximum. Outliers are crosses, and the whisker ends at the most extreme value that is not an outlier.
• Comparing two box plots: one comment on the median, one on the IQR or range, both in context.
• Possibly normal: median near the centre of the box and whiskers of similar length.

Worked reminder: data 2, 3, 5, 6, 8, 9, 11, 20. Median = (6 + 8)/2 = 7. Q₁ = (3 + 5)/2 = 4, Q₃ = (9 + 11)/2 = 10, IQR = 6. Upper fence = 10 + 9 = 19, so 20 is an outlier.

Correlation and regression

Method in steps: bivariate data

1. Plot a scatter diagram and look at the shape. Is it linear?
2. Find r with the GDC. Describe the direction (positive or negative) and the strength (strong or weak).
3. If a critical value is given, the correlation is significant when |r| is greater than it.
4. Find the regression line with the GDC and write the equation down.
5. Substitute to predict, using unrounded coefficients. Check the value is inside the data range.

• a in y = ax + b: the change in y for each increase of 1 in x.
• b: the value of y when x = 0. It may have no sensible meaning in context.
• r is only meaningful for linear relationships. A clear curve can give r near 0.
• Correlation does not imply causation. A third variable may drive both.

Worked reminder: y = 4.2x + 11 and x̄ = 5. Then ȳ = 4.2(5) + 11 = 32, because the line passes through (x̄, ȳ).

Must-know distinctions

• y on x vs x on y: predict y from x with the y on x line; predict x from y with the x on y line. Never rearrange one to get the other. The lines are different unless r = ±1.
• Interpolation vs extrapolation: inside the data range is reasonable when correlation is strong; outside it is unreliable.
• Correlation vs causation: r measures association, not cause.
• Median and IQR vs mean and SD: the median and IQR resist outliers; the mean and SD do not.
• Discrete vs continuous: counts vs measurements.
• Quota vs stratified: both use groups, but only stratified chooses at random within each group.
• σx vs sx on the GDC: use σx unless the question says otherwise.
• Hand quartiles vs GDC quartiles: different methods exist, so answers may differ on some data sets.

Quick self-test

1. State whether each is discrete or continuous: (i) the number of emails received in a day, (ii) the time taken to run 100 m.
2. A gym has 1200 members, of whom 450 are under 30. How many under-30s should be in a stratified sample of 80?
3. A data set has Q₁ = 20 and Q₃ = 32. Is 51 an outlier?
4. A data set has mean 12 and standard deviation 3. Every value is doubled and then 5 is subtracted. Find the new mean, standard deviation and variance.
5. Estimate the mean of this grouped data and state the modal class: 0 ≤ x < 4 (5), 4 ≤ x < 8 (9), 8 ≤ x < 12 (4), 12 ≤ x < 16 (2).
6. Find the median and IQR of 3, 4, 4, 6, 7, 9, 10, 13.
7. The regression line of y on x is y = −1.5x + 40. Interpret −1.5 and predict y when x = 10.
8. The regression line of y on x is y = 2.5x + 3 and x̄ = 6. Find ȳ.
9. Describe the correlation when r = −0.92.
10. You are given y and need to estimate x. Which regression line should you use?
11. A survey asks every 20th customer who enters a shop. Name the sampling method.
12. A scatter diagram shows points on a clear U-shaped curve and r = 0.1. What can you conclude?

Answers

1. (i) Discrete. (ii) Continuous.
2. (450/1200) × 80 = 30.
3. IQR = 12, upper fence = 32 + 18 = 50. 51 > 50, so yes.
4. Mean = 2(12) − 5 = 19; SD = 2 × 3 = 6; variance = 36.
5. Mid-intervals 2, 6, 10, 14: Σfx = 10 + 54 + 40 + 28 = 132, Σf = 20, mean ≈ 6.6. Modal class 4 ≤ x < 8.
6. Median = (6 + 7)/2 = 6.5. Q₁ = 4, Q₃ = 9.5, IQR = 5.5.
7. y decreases by 1.5 on average for each increase of 1 in x. y = −15 + 40 = 25.
8. ȳ = 2.5(6) + 3 = 18.
9. Strong negative linear correlation.
10. The regression line of x on y.
11. Systematic sampling.
12. There is no linear correlation, but there is a clear non-linear relationship, so r is not a suitable measure here.

Where marks are usually lost

• Rearranging the y on x line to predict x, instead of using the x on y line.
• Measuring the 1.5 × IQR fence from the median instead of the nearest quartile.
• Using class boundaries instead of mid-interval values for a grouped mean.
• Plotting cumulative frequency at mid-points instead of upper class boundaries.
• Adding the constant to the standard deviation, or multiplying variance by k instead of k².
• Using sx from the GDC when the data are the population.
• Rounding a and b to 3 s.f. before predicting, which can change the third significant figure of the answer.
• Describing correlation as “strong” without saying positive or negative, or without saying “linear”.
• Comparing box plots without context, or giving two comments on centre and none on spread.
• Making a prediction far outside the data range without saying it is unreliable.

Official syllabus

International Baccalaureate Organization, Diploma Programme, Mathematics: analysis and approaches guide, first assessment 2021 (published February 2019, updated November 2020).