IB Maths AA SL Topic 4 — Statistics Toolkit Paper 1 & 2 ~10 min read

Interpreting Data

Knowing how to find the mean, median, and SD is one thing — knowing which one to use and how to write a comparison answer is what wins exam marks. This note pulls the whole Statistics Toolkit together into a clear decision-making guide.

📘 What you need to know

Different statistics tell different stories — pick the one that suits the data and the question.
If the data has outliers: use median and IQR.
If the data is symmetric / outlier-free: use mean and standard deviation.
Different graphs suit different data — box plots, cumulative frequency graphs, histograms, scatter diagrams.
To compare two data sets: quote a measure of centre AND a measure of spread, then link to context.
Always relate your answer back to what the data actually represents (waiting times, ages, scores, etc.) — that’s where marks live.

Which statistic should I use?

The toolkit has three “centres” (mean, median, mode) and three “spreads” (range, IQR, standard deviation). Picking between them is mostly about whether your data is clean (symmetric) or messy (outliers, skewed).

Measure of central tendency — which one?

Use this…	When…	Why
Mean	Data is roughly symmetric, no outliers	Uses every value — most “complete” answer
Median	Data has outliers or is skewed	Outliers don’t drag it off course
Mode	Data is qualitative (words/categories)	It’s the only “average” that works for words

Measure of dispersion — which one?

Use this…	When…	Why
Range	Quick rough check	Easy, but ruined by a single outlier
IQR	Data has outliers or is skewed	Ignores the top and bottom 25%
Standard deviation	Data is roughly symmetric, no outliers	Uses every value — pairs naturally with the mean

🧠

Memory trick: “Mean & SD travel together. Median & IQR travel together.”

Both pairs use a measure of the centre + a measure of the spread around that centre. Pick one pair based on whether the data is clean (mean & SD) or messy (median & IQR). Don’t mix and match — quote the matching pair.

If a question doesn’t tell you whether the data has outliers, look for clues — the word “skewed”, an outlier mentioned in a previous part, or a box plot with a long whisker. When in doubt, the median & IQR is the safer choice.

Is smaller or bigger better?

Once you’ve picked the right measure, ask: in the context of the question, do you want a bigger or smaller value?

What you’re measuring	Smaller is better	Bigger is better
Time to finish a task	✓
Test scores		✓
Sales / revenue		✓
Errors / faults	✓
Wait time	✓
Customer satisfaction		✓

And what about the spread?

For spread, smaller is usually better — it means the data is more consistent, more reliable, more predictable. A small SD or IQR shows the values cluster tightly around the centre. A large spread shows wild variation.

📍

The two questions to ask

(1) Is a bigger or smaller centre preferred here? (2) Is a bigger or smaller spread preferred? Answer both — and your comparison will hit the marks every time.

Which graph should I use?

Different graph types suit different data — pick the one that shows what the question is actually asking about.

📊 Box plot

Use for: ungrouped data when comparing two data sets quickly, or showing the 5-number summary.

Best at: showing range, IQR, median, and outliers in one image.

📈 Cumulative frequency graph

Use for: grouped continuous data when you need to find the median, quartiles, or percentiles.

Best at: showing running totals, finding “how many below x” type answers.

📉 Histogram

Use for: grouped continuous data with equal class widths, when you want to see the shape of the distribution.

Best at: showing modal class, symmetry, skewness.

🔵 Scatter diagram

Use for: ungrouped data with two variables (bivariate data), when looking for a relationship.

Best at: showing correlation between two variables.

How to write a comparison answer

“Compare the two data sets” is one of the most common Paper 2 commands. The pattern for full marks is always the same.

The 3-step comparison structure

Compare centre

Quote a measure of central tendency for both groups. Say which one is bigger and by how much. Then link it to context: “On average, group A’s wait time is shorter than group B’s…”

Compare spread

Quote a measure of dispersion for both groups. Say which has more variation. Link to context: “Group A’s IQR is smaller, so wait times at A are more consistent.”

Connect to context

Always end with what this means in real-world terms. Don’t just say “the median is bigger” — say “shoppers at store A spent more on average than at store B”.

What full-mark vs no-mark answers look like

✗ No marks

“The median of A is bigger than B. The IQR of B is smaller.”

✓ Full marks

“A’s median wait time (24 min) is greater than B’s (20 min), so on average people waited longer at A. B’s IQR (13 min) is smaller than A’s (15 min), so wait times at B were more consistent.”

The bad answer just lists numbers. The good answer (1) names the actual statistic with values, (2) compares them, (3) links to context (wait times, people, on average). Same maths — totally different mark.

The comparison sentence pattern

“[Group A’s] [statistic] ([value]) is [bigger/smaller] than [Group B’s] ([value]),

so [context-linked conclusion].”

Beyond the maths — context matters

Sometimes a “compare the data sets” question wants a bit more than just numbers. Look out for opportunities to mention:

Sampling method — was the sample random? Is bias likely?
Sample size — bigger samples are more reliable. A small sample of 5 may not be representative.
Time of measurement — were the two groups measured at different times of day, week, year?
Possible outliers — does one data set have an extreme value that skews the comparison?

📍

Don’t over-claim

If a data set is small, has outliers, or comes from a biased sample, your conclusion should reflect that. Saying “this proves shop A is better” from a sample of 10 customers is too strong — say “based on this sample, shop A appeared to perform better, but a larger sample would be needed to confirm”.

Worked examples

WE 1

Choose the right measure of centre and spread

For each scenario, state which measure of central tendency and which measure of dispersion would be most appropriate.

(a) The salaries at a small company, where the owner earns much more than everyone else.

(b) The heights of 50 randomly chosen 16-year-olds (no outliers).

(c) The favourite ice-cream flavour of 20 students.

Identify whether each set has outliers, is symmetric, or qualitative.part (a) Owner’s salary is an outlier — drags the mean. Median & IQRpart (b) Random sample, no outliers — symmetric. Mean & standard deviationpart (c) Flavours = words = qualitative data. Mode (no spread measure for qualitative data) match the centre and spread to the data type — outliers? skewed? words?

WE 2

Compare two distributions in context

The box plots below show waiting times in minutes at HealthHut and FitFirst surgeries. Compare the two distributions in context.

Apply the 3-step comparison structure: centre → spread → context.step 1 — centre Read medians off box plots: HealthHut median = 20 min, FitFirst median = 24 min HealthHut’s median (20) is smaller than FitFirst’s (24). Patients are seen quicker at HealthHut on averagestep 2 — spread IQR for each: HealthHut IQR = 30 − 15 = 15 min FitFirst IQR = 31 − 18 = 13 min FitFirst’s IQR is smaller. Wait times are more consistent at FitFirststep 3 — context Wrap it together: “On average HealthHut sees patients faster (lower median), but FitFirst is more predictable (smaller IQR).” HealthHut quicker, FitFirst more consistent always use BOTH centre AND spread, BOTH linked to context

WE 3

Choose the most appropriate graph

For each scenario, state the most appropriate graph type.

(a) Showing whether height and weight of 50 students are related.

(b) Showing the shape of the distribution of 200 reaction times (grouped).

(c) Comparing test scores in two different classes (raw data).

(d) Estimating the median height from grouped data.

Match the question to the chart that’s best at showing it.part (a) Two variables (height & weight) — looking for a relationship. Scatter diagrampart (b) Grouped continuous data — wanting to see distribution shape. Histogrampart (c) Two raw data sets — comparing centre and spread. Box plots (drawn on the same axis)part (d) Need to estimate median from grouped data. Cumulative frequency graph match the chart to the question — there’s usually one obviously right answer

WE 4

Justify your choice when outliers exist

The number of books read by 8 students last summer: 3, 4, 4, 5, 6, 7, 8, 50.

(a) Calculate the mean and median. (b) Which is more representative of a typical student? Justify your answer.

50 is clearly an outlier. We need to see how each measure responds.part (a) Mean: (3+4+4+5+6+7+8+50) ÷ 8 = 87/8 ≈ 10.9 Sort: 3, 4, 4, 5, 6, 7, 8, 50 → middle two: 5, 6 Median: (5+6)/2 = 5.5 Mean ≈ 10.9, Median = 5.5part (b) Most students read 3–8 books — the value 50 is extreme. Mean of 10.9 is misleading: nobody read 10.9 books, and 7 of 8 read fewer. Median of 5.5 sits in the middle of the typical range. Median is more representative — outlier doesn’t drag it when an outlier exists, the mean lies — the median tells the truth

WE 5

Write a full comparison with sample-size caveat

Two restaurants are reviewed. Restaurant A’s last 5 ratings (out of 10): 9, 9, 10, 9, 10. Restaurant B’s last 50 ratings: mean = 7.8, SD = 1.2.

Compare the two and comment on the reliability of the comparison.

Both samples roughly symmetric — use mean & SD. But sample sizes differ wildly!step 1 — centre A mean: (9+9+10+9+10) ÷ 5 = 47/5 = 9.4 B mean: 7.8 A’s mean (9.4) is bigger than B’s (7.8) — A scored higher on average.step 2 — spread A’s SD (using GDC): σ ≈ 0.49 B’s SD: 1.2 A’s SD is smaller → A’s ratings are more consistent.step 3 — context & reliability In context: A averages higher and is more consistent than B. BUT — A’s sample is only 5 reviews, B has 50. A’s small sample may not be representative. A appears better, but A’s sample size is too small for a reliable comparison always check sample size before drawing big conclusions — top mark answer!

💡 Top tips

Mean & SD travel together. Median & IQR travel together. Pick one pair based on whether the data is symmetric (mean/SD) or has outliers (median/IQR).
For qualitative data, only the mode is meaningful — there’s no useful spread measure.
Smaller spread = more consistent (a good thing in most contexts).
Always link your answer to the context. Say “shorter wait times” not just “smaller median”.
The 3-step comparison: compare centre → compare spread → connect to context. Hit all three for full marks.
Mention sample size or outliers when relevant — examiners reward awareness of these caveats.
Match the graph to the question: scatter diagrams for relationships, histograms for shape, box plots for comparison, cum freq for medians/quartiles from grouped data.
Don’t over-claim. A small or biased sample doesn’t “prove” anything — it only suggests.

⚠ Common mistakes

Listing numbers without comparing them. “Median A = 20, Median B = 24” — that’s just data, not a comparison. You need to say which is bigger and link to context.
Mixing the wrong pairs. Don’t quote a mean for one group and a median for the other — use the same measure for both.
Forgetting the context. “The median is bigger” → Why does that matter? Connect it to wait times, scores, sales, etc.
Using mean & SD when there’s a clear outlier. One huge value distorts both — use median & IQR instead.
Picking the wrong graph. Don’t use a histogram to compare two specific data sets — that’s what side-by-side box plots are for.
Over-claiming from a tiny sample. 5 ratings can’t “prove” one place is better — always note the size limitation.
Forgetting units in your comparison sentences (e.g. “20 minutes” not just “20”).
Calling the mode the “average” of qualitative data without explaining. Be specific: “the modal favourite flavour is chocolate”.

🎉 You’ve finished the Statistics Toolkit! You can now collect data, summarise it with the right measures, choose the right graph, and write a marker-friendly comparison. Next up in Topic 4 is regression and correlation — analysing the relationship between two variables.

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