IB Maths AI HLStatistics ToolkitPaper 1 & 2~7 min read
Interpreting Data
This is where the whole Statistics Toolkit comes together. You can calculate every measure and draw every diagram — now you have to choose wisely and explain in context. The two questions that run through every interpretation problem are: which average and spread should I use? and which diagram fits this data? The single most important idea is that outliers change the answer: when data has extreme values, the median and IQR are reliable while the mean and standard deviation get distorted. Master that, learn to compare two datasets in context, and you’ll pick up the “explain” and “comment” marks that students so often drop.
📘 What you need to know
Mean: uses every value, so it’s affected by outliers. Best for roughly symmetric data.
Median: the middle value, not affected by outliers. Best when there are extreme values.
Mode / modal class: most useful for qualitative data; quantitative data may have no unique mode.
Range: full spread, but affected by outliers (uses max and min).
IQR: spread of the middle 50%, not affected by outliers.
Standard deviation / variance: average spread about the mean, uses all values, so affected by outliers.
Comparing datasets: compare one measure of centre and one of spread, always in context.
Outliers present → use median & IQR. Roughly symmetric → use mean & standard deviation.
Choosing a diagram: box plot (compare quickly), cumulative frequency graph (grouped, percentiles), histogram (grouped, shape), scatter diagram (bivariate).
Whether a smaller or bigger average/spread is “better” depends entirely on the context.
Which measures do outliers affect?
This table is the heart of the topic. Once you know which measures are “robust” (unaffected) and which are “sensitive” (affected), every choice falls into place.
Measure
Type
Affected by outliers?
Mean
centre
Yes ✗
Median
centre
No ✓
Range
spread
Yes ✗
IQR
spread
No ✓
Standard deviation
spread
Yes ✗
Robust (safe with outliers)
median & IQR
Based on position in the middle, so extreme values don’t move them. Use when data is skewed or has outliers.
Sensitive (distorted by outliers)
mean, range & SD
Use every value (or the extremes), so a single outlier drags them. Use when data is roughly symmetric.
🧠 Memory aid — “outliers → middle measures”
If a question mentions outliers, skew, or extreme values, switch to the median and IQR — the “middle measures” that ignore the tails. If the data is described as symmetric or normal, use the mean and standard deviation, which squeeze more information out of every value. Match the tool to the shape.
Choosing which diagram to use
Each diagram in the toolkit suits a particular kind of data and a particular purpose. Pick by asking: is the data grouped? continuous? bivariate? do I need to compare or to find percentiles?
matching the diagram to the data
Box plots compare; cumulative frequency curves give percentiles for grouped data; histograms show shape; scatter diagrams reveal relationships between two variables.
Quick guide: ungrouped & comparing → box plot; grouped & need percentiles → cumulative frequency graph; grouped continuous & want shape → histogram; two variables → scatter diagram.
Comparing two or more datasets
This is the classic “compare in context” question. The IB mark scheme almost always wants two comparisons — one of centre, one of spread — each matched to the data’s shape and tied to the real situation.
🧭 Recipe — comparing datasets in context
Compare a measure of centre. Outliers present → use the median; roughly symmetric → use the mean.
Compare a measure of spread. Outliers present → use the IQR; roughly symmetric → use the standard deviation.
Decide what “better” means here. Smaller or bigger average/spread depends on the context.
Quote the actual values to back up each statement.
Relate it to the context and consider reasons (and the sampling/data-collection method).
🤔 Is a smaller average always “better”?
No — it depends entirely on what’s being measured. A smaller mean time to complete a puzzle is good (faster), but a bigger mean test score is good (higher marks). The same goes for spread: usually a smaller spread means more consistency, which is good for waiting times or manufacturing, but you’d never call a result “better” without saying what the variable represents. Always anchor the judgement to the context.
Worked examples
WE 1
Which average to use?
A small company’s salaries (£000s) are 28, 30, 31, 33, 150. Which measure of central tendency best represents a typical employee, and why?
spot the outlier150 is far above the rest (the director’s salary)mean vs medianmean = 272/5 = 54.4 (pulled up by 150)median = 31 (unaffected)use the median (£31k)with an outlier present, the median better reflects a typical value.
WE 2
Which spread to use?
For the same salary data (28, 30, 31, 33, 150), state which measure of dispersion you’d use to describe the spread, and justify it.
outlier present → robust measurethe range (150 − 28 = 122) and SD are inflated by the £150k outlier.the IQR uses only the middle 50%, so it’s not distorted.use the IQRpair the median with the IQR — both ignore the extremes.
WE 3
Choose the right diagram
State the most appropriate diagram for each. (a) The relationship between hours studied and exam score. (b) Comparing the waiting times at two clinics. (c) The shape of the distribution of 200 students’ heights grouped into classes.
(a) two variables → bivariate(a) scatter diagram(b) compare two ungrouped sets(b) box plots (side by side)(c) grouped continuous, want shape(c) histogrammatch the diagram to the data type and the purpose.
WE 4
Compare two box plots in context
Two surgeries’ waiting times (minutes) are shown as box plots. HealthHut: median 24, IQR 19. FitFirst: median 20, IQR 13. Compare the two distributions in context.
1. centre (median)FitFirst’s median (20) is lower than HealthHut’s (24), so on average patients are seen quicker at FitFirst.2. spread (IQR)FitFirst’s IQR (13) is smaller than HealthHut’s (19), so its waiting times are more consistent.FitFirst: quicker & more consistenttwo comparisons, both with values, both linked to context.
WE 5
Symmetric data — use mean & SD
Two classes sit the same test. Class A: mean 62, SD 5. Class B: mean 62, SD 11. Both sets are roughly symmetric with no outliers. Compare the classes’ performance.
centre (mean — data symmetric)both means = 62 → same average performancespread (SD — data symmetric)A: SD 5; B: SD 11Class A’s smaller SD means its marks are more tightly clustered (more consistent); Class B’s are more spread out.same average; Class A more consistentno outliers → mean & SD are the right tools here.
WE 6
Is “smaller” better?
A factory measures the time to assemble a product. Line 1 has mean 8.2 min, SD 0.4 min. Line 2 has mean 8.2 min, SD 1.5 min. Which line would you prefer, and why?
same mean → compare spreadLine 1 SD 0.4; Line 2 SD 1.5interpret in contexta smaller SD means assembly times are more predictable/consistent — better for planning production.prefer Line 1 (more consistent)here a smaller spread is “better” because consistency matters — always justify with context.
💡 Top tips
Outliers? Use median & IQR. Symmetric/no outliers? Use mean & standard deviation. This single rule drives most “which measure” marks.
Pair them correctly: median goes with IQR, mean goes with standard deviation.
Always make two comparisons — one of centre, one of spread — when comparing datasets.
Quote the values and say which is bigger/smaller, then explain what it means.
“Better” depends on context: faster time = smaller is better; higher score = bigger is better.
Match the diagram: box plot (compare), CF graph (percentiles, grouped), histogram (shape, grouped), scatter (two variables).
Consider the data collection — sampling method and possible bias can be worth a mark in “comment” questions.
⚠ Common mistakes
Using the mean when there are outliers. The mean is dragged by extremes — switch to the median.
Comparing only one thing. The IB wants both a centre and a spread comparison.
No context. “The median is bigger” earns little; say what it means for the real situation.
Calling a smaller average “better” automatically. It depends on what the variable is.
Mismatching measures: don’t compare one set’s mean with another’s median — use the same measure for both.
Wrong diagram for the data: e.g. a histogram for two-variable data (that needs a scatter diagram).
That completes the Statistics Toolkit! You’ve now covered sampling, central tendency, dispersion, frequency tables, linear transformations, outliers, box plots, cumulative frequency graphs, histograms and interpretation. The skills here — choosing the right measure and explaining in context — carry straight into the next units on correlation, regression and probability distributions.
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