Statistical Diagrams (AA HL)
Box plots and cumulative frequency graphs both summarise a data set visually, but they're read in different ways and lose marks for different reasons. This page covers how to pull \(Q_1\), the median and \(Q_3\) out of raw data, how to test for outliers, and how to read values off a cumulative frequency curve accurately. It's part of the broader Descriptive Statistics topic.
11 questions on this sub-topic.
Key working
Covered under IB syllabus reference SL4.2: presentation of data in frequency tables and histograms, cumulative frequency graphs for the median, quartiles and IQR, and box-and-whisker diagrams. Neither of these is a formula-booklet entry - they're definitions you apply directly to the ordered data.
Interquartile range
\(\text{IQR} = Q_3 - Q_1\)
\(Q_1\) is the median of the lower half of the ordered data, \(Q_3\) the median of the upper half. On the GDC, 1-Var Stats returns both directly once the data is entered.
Outlier boundaries
\(Q_1 - 1.5 \times \text{IQR}\) and \(Q_3 + 1.5 \times \text{IQR}\)
Any value beyond these two boundaries is plotted separately on a box plot rather than as the end of a whisker.
Worked examples
A data set in order is \(3, 5, 6, 8, 9, 11, 14, 18\).
(a) Find the range.
(b) Find the interquartile range.
Worked solution
(a) Range. \(\text{Range}=18-3=15.\) A1
(b) \(Q_1\).
With \(n=8,\) lower half \(3,5,6,8\Rightarrow Q_1=\dfrac{5+6}{2}=5.5;\)
\(Q_3\)
upper half \(9,11,14,18\Rightarrow Q_3=\dfrac{11+14}{2}=12.5.\) M1 A1
So \(\text{IQR}=12.5-5.5=7.\) A1
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 \(40.\) M1
Reading across, the median \(\approx 62.\) M1 A1
Common mistakes
- Reading the median off the wrong axis. On a cumulative frequency graph, you go in at \(\frac{n}{2}\) on the vertical axis, across to the curve, then down - not the other way round.
- Eyeballing outliers instead of testing them. A point looking far from the box isn't automatically an outlier - it only counts if it's beyond \(Q_1 - 1.5\times\text{IQR}\) or \(Q_3 + 1.5\times\text{IQR}\), so calculate the boundaries first.
- Mixing up which half gives \(Q_1\) and \(Q_3\). \(Q_1\) always comes from the lower half of the ordered data and \(Q_3\) from the upper half - swapping them flips the whole box plot.
Ready to practise properly?
11 statistical-diagrams questions, marked instantly like the real exam. Need a calculator refresher first? See the parent topic's GDC guidance.
Quick answers
How do you find the interquartile range from a data set?
Split the ordered data into a lower half and upper half, find \(Q_1\) as the median of the lower half and \(Q_3\) as the median of the upper half, then \(\text{IQR}=Q_3-Q_1\).
How do you decide whether a value is an outlier on a box plot?
A value is an outlier if it lies below \(Q_1 - 1.5\times\text{IQR}\) or above \(Q_3 + 1.5\times\text{IQR}\). Values inside these boundaries set the ends of the whiskers; anything beyond is plotted separately.