Statistical Diagrams (AA SL)
Histograms, cumulative frequency graphs and box plots each display the same underlying data in a different way, and IB questions usually expect you to read values off the diagram rather than recompute them from scratch. This page covers how to extract a median from a cumulative frequency curve and how to judge skewness from a box plot, with worked examples and the mistakes markers see most often. It's part of the broader Descriptive Statistics & Correlation topic.
11 questions on this sub-topic.
Reading the diagrams
Covered under IB syllabus reference SL4.2: presentation of data (discrete and continuous), including frequency distributions, histograms and cumulative frequency graphs, plus production and understanding of box and whisker diagrams.
Box and whisker diagrams
A box plot displays the five-number summary: minimum, \(Q_1\), median, \(Q_3\), maximum. The box spans the IQR, and the whiskers extend to the most extreme values that aren't outliers.
Outliers are plotted as separate crosses or dots, not included in the whisker.
Cumulative frequency graph
Plots running totals against the upper class boundary; the curve rises from \(0\) to \(n\).
The median is read at cumulative frequency \(\tfrac{n}{2}\); \(Q_1\) at \(\tfrac{n}{4}\) and \(Q_3\) at \(\tfrac{3n}{4}\).
Need the full syllabus wording and formula-booklet reference table? See Descriptive Statistics & Correlation.
Worked examples
A cumulative frequency curve for 80 students has the 40th value at \(x=62.\) State the estimated median and explain.
Worked solution
For \(n = 80\), the median is the value at cumulative frequency \(40.\) A1
Reading across, the median \(\approx 62.\) M1 A1
A box plot has \(Q_1=20,\) median \(=30,\) \(Q_3=55.\) Describe the skewness.
Worked solution
From \(Q_1\) to the median is \(30 - 20 = 10\); from the median to \(Q_3\) is \(55 - 30\) M1
\(= 25.\) A1
The upper half is more spread, M1
so the distribution is positively (right) skewed. A1
Common mistakes
- Reading the median at the wrong cumulative frequency. The median sits at \(\tfrac{n}{2}\) on the cumulative frequency axis, not at the middle of the \(x\)-axis - always work from the vertical axis first, then read across and down.
- Judging skewness from the whiskers instead of the box. Skewness is about which side of the median the box (the middle 50%) is more stretched, not which whisker looks longer - a long whisker can exist on either side regardless of the overall skew.
- Including outliers in the whisker length. A box plot's whiskers should stop at the most extreme non-outlier value; genuine outliers are marked separately, so don't stretch a whisker out to catch them.
Ready to practise properly?
11 statistical-diagram questions, marked instantly like the real exam.
Quick answers
How do you find the median from a cumulative frequency graph?
Find \(\tfrac{n}{2}\) on the cumulative frequency axis, draw a horizontal line across to the curve, then a vertical line down to the horizontal axis - that value is the estimated median.
How do you tell if a box plot is skewed?
Compare the two halves of the box either side of the median. If the gap from \(Q_3\) to the median is bigger than the gap from the median to \(Q_1\), the distribution is positively (right) skewed; if the lower gap is bigger, it's negatively (left) skewed. See the parent topic's GDC guidance for building diagrams from raw data.