IB DP Mathematics: Analysis and Approaches – Sampling, data presentation, summary statistics and regression Study Guide

This study guide teaches the descriptive statistics unit of IB Diploma Programme Mathematics: Analysis and Approaches from scratch. It is aligned to the IB Mathematics: analysis and approaches guide, first assessment 2021, and covers syllabus sections 4.1–4.4 and 4.10. All of this content is common to SL and HL. It follows the IB guide for first assessment 2021, which remains the examined syllabus until the new course is first assessed in May 2029, so it applies to the May and November 2026, 2027 and 2028 SL and HL sessions.

When you have worked through it, use the statistics revision notes for final-weeks recall and the statistics practice questions to test yourself. For the whole course, see the IB DP Maths AA course hub and the printable syllabus checklist.

What this unit covers

Syllabus section	What you must be able to do	SL/HL
4.1	Use the terms population, sample, random sample, discrete and continuous data; judge reliability and bias; identify and interpret outliers; describe simple random, convenience, systematic, quota and stratified sampling	SL and HL
4.2	Read and draw frequency tables, frequency histograms (equal class intervals), cumulative frequency graphs and box and whisker diagrams; use them to find the median, quartiles, percentiles, range and IQR	SL and HL
4.3	Find the mean, median and mode; estimate the mean of grouped data; state the modal class; find the IQR, standard deviation and variance; know the effect of constant changes to the data	SL and HL
4.4	Draw and describe scatter diagrams; find and interpret Pearson’s r; find the regression line of y on x and use it to predict; interpret a and b in y = ax + b	SL and HL
4.10	Find the regression line of x on y and use it to predict x from y	SL and HL

Most calculations in this topic are done with technology, and the data set is treated as the population unless you are told otherwise. Paper 1 allows no technology, so be ready to do these by hand: sampling calculations, quartiles of short lists, outlier checks, mid-interval means, constant-change rules and using a given regression line.

4.1 Populations, samples and sampling

The population is every item of interest; a sample is the part you measure. In a random sample every item has an equal chance of selection.

• Discrete data can take only separate values, usually counts: number of siblings, goals scored.
• Continuous data can take any value in a range, usually measurements: time, mass, length.

Reliability and bias. Ask who collected the data and why. A sample is biased if some members of the population are more likely to be chosen than others. Also check for missing data and recording errors (a height typed as 1.7 instead of 170).

Sampling techniques

Method	How it works	Main weakness
Simple random	Number the population and choose with random numbers	Needs a full list; may miss small groups by chance
Convenience	Use whoever is easiest to reach	Very likely to be biased
Systematic	Choose every kth item from a list, random start	Biased if the list has a repeating pattern
Quota	Fill a fixed number from each group, not chosen at random	The interviewer chooses, so bias creeps in
Stratified	Split into groups (strata); sample each group in proportion to its size, at random	Needs the size of each group

Worked example 1. A school has 360 DP1 and 240 DP2 students. A stratified sample of 50 is needed.

Total = 360 + 240 = 600
DP1: (360/600) × 50 = 30
DP2: (240/600) × 50 = 20

For a systematic sample of 50 from the list of 600, take every 600/50 = 12th name, starting at a random position from 1 to 12.

Outliers

An outlier is a data item more than 1.5 × IQR from the nearest quartile.

lower fence = Q₁ − 1.5 × IQR
upper fence = Q₃ + 1.5 × IQR

In context, decide whether an outlier is a genuine value that belongs in the data or an error that should be checked or removed.

4.2 Presenting data

Class intervals are written as inequalities with no gaps, for example 10 ≤ t < 20. A frequency histogram uses equal class widths, so bar height is frequency. Frequency density histograms are not required.

A cumulative frequency is a running total. Plot each cumulative frequency against the upper class boundary and join the points. For n items, read across from:

• n/2 for the median
• n/4 for Q₁ and 3n/4 for Q₃
• p% of n for the pth percentile

Worked example 2. Waiting times, t minutes, for 60 patients.

t	0 ≤ t < 10	10 ≤ t < 20	20 ≤ t < 30	30 ≤ t < 40	40 ≤ t < 50
Frequency	6	14	22	12	6
Cumulative frequency	6	20	42	54	60

Plot (10, 6), (20, 20), (30, 42), (40, 54), (50, 60), starting from (0, 0). The median is at 60/2 = 30, which lies in the class 20 ≤ t < 30. Reading from a graph joined with straight lines (linear interpolation):

median ≈ 20 + (30 − 20)/22 × 10 ≈ 24.5 minutes
Q₁ (15th) ≈ 10 + (15 − 6)/14 × 10 ≈ 16.4
Q₃ (45th) ≈ 30 + (45 − 42)/12 × 10 = 32.5
IQR ≈ 32.5 − 16.4 ≈ 16.1 minutes
90th percentile (54th) = 40 minutes

Box and whisker diagrams

A box and whisker diagram shows the minimum, Q₁, median, Q₃ and maximum. Outliers are marked with a cross, and the whisker stops at the most extreme value that is not an outlier.

To compare two distributions, comment on a measure of centre (median) and a measure of spread (IQR or range), in context. If the median sits near the middle of the box and the whiskers are about the same length, the data are roughly symmetric and may be normally distributed.

4.3 Summary statistics

Mean x̄ = Σx / n, or Σfx / Σf from a frequency table. For grouped data, use mid-interval values to estimate the mean. The modal class is the class with the highest frequency (equal class intervals only).

Worked example 3. Estimate the mean waiting time in worked example 2.

mid-interval values: 5, 15, 25, 35, 45
Σfx = 5(6) + 15(14) + 25(22) + 35(12) + 45(6) = 1480
x̄ ≈ 1480 / 60 ≈ 24.7 minutes

The modal class is 20 ≤ t < 30.

Worked example 4. Twelve students recorded the number of hours of sport they did in a week:

3, 5, 6, 7, 7, 8, 9, 10, 11, 12, 14, 24

Σx = 116, so x̄ = 116/12 ≈ 9.67 hours
Mode = 7
Median = mean of 6th and 7th values = (8 + 9)/2 = 8.5
Lower half 3, 5, 6, 7, 7, 8  → Q₁ = (6 + 7)/2 = 6.5
Upper half 9, 10, 11, 12, 14, 24 → Q₃ = (11 + 12)/2 = 11.5
IQR = 11.5 − 6.5 = 5
Upper fence = 11.5 + 1.5 × 5 = 19, so 24 is an outlier
Lower fence = 6.5 − 7.5 = −1, so there are no low outliers

On a box plot, the whiskers reach 3 and 14 and 24 is a cross. Different quartile methods exist, so a GDC and a hand method can disagree on some data sets.

Standard deviation and variance. The guide expects these from technology only. Variance is the square of the standard deviation. On your GDC, treat the data as the population unless told otherwise and use σx. For the data above, σ ≈ 5.23 hours and variance ≈ 27.4 hours². Removing the outlier 24 would drop the mean to 8.36 but move the median only to 8: the median and IQR resist outliers.

Effect of constant changes

Change to every data item	Mean, median, quartiles	Standard deviation, IQR, range	Variance
Add or subtract c	Add or subtract c	Unchanged	Unchanged
Multiply by k (k > 0)	Multiply by k	Multiply by k	Multiply by k²

Worked example 5. Daily temperatures have mean 18.5 °C and standard deviation 4.2 °C. Convert to Fahrenheit using F = 1.8C + 32.

new mean = 1.8 × 18.5 + 32 = 65.3 °F
new SD = 1.8 × 4.2 = 7.56 °F   (adding 32 does not change the spread)

4.4 Correlation and the regression line of y on x

Bivariate data come in pairs (x, y). Plot them on a scatter diagram and describe the correlation as positive, negative or zero, and as strong or weak. A line of best fit by eye should pass through the mean point (x̄, ȳ).

Pearson’s product-moment correlation coefficient, r, measures the strength of a linear relationship, with −1 ≤ r ≤ 1. Values near 1 mean strong positive linear correlation, near −1 strong negative, and near 0 no linear correlation. Find r with technology. It is only meaningful for linear relationships: data on a clear curve can have r close to 0. Where a question gives a critical value of r, the correlation is significant if |r| is greater than it.

Correlation does not imply causation. Both variables may depend on a third.

The regression line of y on x, y = ax + b, is found with technology and is used to predict y from a given x. Here a is the gradient: the change in y for each increase of 1 in x. b is the value of y when x = 0, which may have no sensible meaning in context.

Worked example 6. Eight students recorded revision hours, x, and test score, y (out of 100).

x	2	4	5	7	8	10	11	13
y	38	52	45	50	64	58	70	66

Using a GDC (linear regression):

r ≈ 0.885  → strong positive linear correlation
y ≈ 2.62x + 35.7
mean point (7.5, 55.375) lies on this line

Each extra hour of revision is associated with about 2.62 more marks on average; 35.7 is the predicted score for no revision.

To predict the score for 9 hours: y ≈ 2.62 × 9 + 35.7 ≈ 59.3. Use unrounded GDC coefficients.

Predicting for 25 hours gives about 101, which is impossible for a test out of 100. This is extrapolation: x = 25 is far outside the data range of 2 to 13, so the prediction is unreliable.

4.10 The regression line of x on y

To predict x from a given y, use the regression line of x on y, found with technology by making y the independent variable. Do not rearrange the y on x line. The two lines are different unless r = ±1, and both pass through the mean point.

Worked example 6 continued. Estimate the revision hours for a score of 55.

x on y line: x ≈ 0.299y − 9.06
x ≈ 0.299 × 55 − 9.06 ≈ 7.39 hours

Rearranging the y on x line would give 7.36, but that is the wrong method, even though the numbers here are close. Likewise, do not use the x on y line to predict y.

Using your GDC

• One-variable statistics (with a frequency list if needed) gives x̄, σx, Q₁, median and Q₃.
• Two-variable linear regression gives a, b and r. Swap the lists to get the x on y line.
• Write the equation down before using it.

Common errors

• Plotting cumulative frequency at lower class boundaries or mid-points instead of upper boundaries.
• Measuring the outlier fence from the median instead of the nearest quartile.
• Adding a constant to the standard deviation, or multiplying the variance by k instead of k².
• Rearranging y = ax + b to predict x from y instead of using the x on y line.
• Treating a high r as proof of cause.
• Giving a prediction far outside the data range without a warning about extrapolation.

Where to go next

Next, use the revision notes and the practice questions. For other units, see the functions study guide and the calculus study guide. See also the AA syllabus guide and exam preparation guide.

Official syllabus

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