IB Maths AA HL Topic 4 — Statistics & Probability Paper 1 & 2 ~7 min read

Interpreting Data

All the tools assembled — averages, dispersion, box plots, histograms, cumulative graphs — boil down to one practical skill: which one to use, and what to say about the result. The answer depends on whether outliers are present, whether the data is symmetric, and what real-world claim you’re trying to support.

📘 What you need to know

Outliers present → use median + IQR (both resistant).
Roughly symmetric, no outliers → use mean + SD (both use every value).
Qualitative data → use the mode; a mean or median doesn’t apply.
Bigger / smaller is “better” depends on the context: smaller times for races, bigger marks for tests.
Smaller spread = more consistent; bigger spread = more variable.
Choosing a graph: bar chart (qualitative), histogram (continuous grouped), CF graph (positions and percentiles), box plot (compare two distributions), scatter (bivariate).
Always relate findings to context — “B has a smaller IQR” is a half-answer; “B’s commute times are more predictable” is the full one.
Consider the sampling method when commenting on reliability or generalisability.

Choosing the right statistical measure

Situation	Average to use	Spread to use
Symmetric, no outliers	mean	standard deviation
Skewed or has outliers	median	interquartile range
Qualitative / categorical data	mode	range (or comparison of frequencies)

Pair them correctly: mean ↔ standard deviation; median ↔ IQR. Mixing them (like “median + SD”) is technically valid but unusual.

Choosing the right graph

Graph type	Best for…
Bar chart	qualitative or discrete data — gaps between bars
Histogram	continuous grouped data — shape of distribution
Cumulative frequency graph	continuous grouped data — medians, quartiles, percentiles, “how many ≤”
Box plot	quick centre-and-spread comparison; ideal for two or more distributions
Scatter plot	bivariate data — relationship between two variables

Framework for comparing two distributions

Step 1: Centre

compare averages

“X has a higher/lower mean (or median) than Y”

Step 2: Spread

compare dispersion

“X is more/less consistent than Y”

Then add step 3: relate to context. Translate “smaller IQR” into “more predictable journey times”, “consistent quality”, “tighter exam scores”, or whatever the scenario calls for.

🧭 Recipe — interpret and compare

Check for outliers in the data — if present, prefer median and IQR.
Pick paired measures: mean+SD or median+IQR.
Compare the centres: which group has the higher/lower average?
Compare the spreads: which group is more consistent?
State the conclusion in context — translate numbers into a real-world claim.
Comment on reliability: was the sample large enough, randomly chosen, free of bias?

Worked examples

WE 1

Choose the most representative average

The annual salaries at Company X have a mean of $58,000 and a median of $44,000. State which measure is more representative of a typical employee’s earnings, and justify your answer.

Step 1: Compare the two values Mean ($58k) > median ($44k) Step 2: Diagnose A larger mean than median indicates the distribution is right-skewed → a few high earners (e.g., directors) are pulling the mean up Step 3: Decision Median is resistant to outliers; mean is not Median ($44k) is more representative of a typical employee salary data almost always uses median for this reason

WE 2

Compare two athletes using mean and SD

Two long-jumpers have the following recent results (in metres):
Athlete A: mean = 6.85, SD = 0.20 Athlete B: mean = 7.05, SD = 0.40.
Compare the two athletes in context.

Step 1: Compare means (centres) B’s mean (7.05 m) > A’s mean (6.85 m) → B jumps further on average Step 2: Compare SDs (spreads) A’s SD (0.20 m) < B’s SD (0.40 m) → A is more consistent Step 3: Context B better for peak performance (higher average) A more reliable (less variable from jump to jump) B jumps further on average; A is more consistent “better” is context-dependent — depends on whether the team values peaks or reliability

WE 3

Compare two groups using median and IQR

The commute times of two groups of workers are summarised below.
Group A: median = 28 min, IQR = 12 min Group B: median = 24 min, IQR = 25 min.
Compare the two groups in context.

Step 1: Compare medians B’s median (24 min) < A’s median (28 min) → B has shorter typical commute Step 2: Compare IQRs A’s IQR (12 min) < B’s IQR (25 min) → A’s commute times are more consistent Step 3: Context B is faster on average, but A’s commutes are more predictable B has a shorter typical commute; A is more predictable median+IQR pairing suggests the data has outliers or skew — quite possible for commute times

WE 4

Compare two box plots

Two factories assemble the same product. Box plots of assembly times (in minutes) are summarised:
Factory P: Min=10, Q₁=15, Med=20, Q₃=25, Max=35.
Factory Q: Min=8, Q₁=14, Med=22, Q₃=32, Max=45.
Compare the two factories’ performance.

Step 1: Compare medians P: 20 min; Q: 22 min → P faster on average Step 2: Compare IQRs P: 25 − 15 = 10; Q: 32 − 14 = 18 → P more consistent (middle 50% spans only 10 min vs 18 min) Step 3: Compare ranges P: 35 − 10 = 25; Q: 45 − 8 = 37 → Q has wider total spread Step 4: Context Factory P assembles products faster on average and more consistently all three measures (median, IQR, range) point the same way — strong conclusion

WE 5

Match the graph type to the data

For each scenario, state the most appropriate graph type. (a) Showing how 100 students answered “what’s your favourite ice cream flavour?”. (b) Heights of 50 students grouped into 5 cm intervals. (c) Investigating whether hours studied predicts exam scores. (d) Quickly comparing the player heights of two basketball teams.

(a) Qualitative data → bar chart (with gaps) (b) Continuous grouped data → histogram (or CF graph if quartiles needed) (c) Bivariate data → scatter plot (d) Quick centre-and-spread comparison → side-by-side box plots (a) bar chart; (b) histogram; (c) scatter plot; (d) box plots “box plots” are unbeatable for at-a-glance comparison of two or more distributions

WE 6

Quality-control interpretation

A factory uses two processes to make a part with target weight 100 g. Sampling gives:
Process A: mean = 100.1 g, SD = 0.4 g Process B: mean = 99.9 g, SD = 1.5 g.
(a) Comment on which process is closer to the target on average. (b) Comment on consistency. (c) Recommend which process to use for quality-controlled production and justify.

(a) Distance from target A: |100.1 − 100| = 0.1 g B: |99.9 − 100| = 0.1 g → both equally close to the target on average (b) SDs A’s SD (0.4) ≪ B’s SD (1.5) → A’s outputs cluster more tightly around their mean (c) Recommendation For quality control, consistency matters most A’s smaller SD means fewer parts deviate from target Use Process A — both processes hit the target on average, but A is much more consistent when both means are equal, SD becomes the deciding factor

💡 Top tips

Pair the measures correctly: mean+SD or median+IQR — never mean+IQR or median+SD without a reason.
For comparison questions, always cover BOTH centre and spread — half-answers lose marks.
End every comparison with a contextual sentence that translates the numbers into a real-world claim.
“Consistent” maps to small spread; “varies a lot” maps to large spread.
If asked which is “better”, decide what “better” means in the context (e.g., faster races, more accurate measurements).

⚠ Common mistakes

Stating numerical differences without context — e.g., “A’s median is 5 less” rather than “Process A is 5 minutes faster on average”.
Using mean + SD when outliers are present — both get distorted; switch to median + IQR.
Concluding from centre alone when the spreads tell a different story.
Picking a graph without considering data type — qualitative data needs a bar chart, not a histogram.
Forgetting that “better” is context-dependent — bigger SD isn’t bad if reaching extremes is desirable (e.g., a creative class might value variety).

That closes the Statistics Toolkit. From sampling to interpreting, you now have the full vocabulary for describing data: where it sits, how it spreads, how to visualise it, and how to compare two distributions in plain language. The probability sub-section starts next — building from sample spaces and Venn diagrams up to conditional probability, independence, and Bayes’ theorem.

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