Statistical Diagrams (AI SL)
A statistical diagram turns a list of raw numbers into something you can read at a glance - a frequency table groups values, a cumulative frequency graph lets you locate the median and quartiles without sorting the whole data set by hand, and a box plot summarises the spread in one compact picture. This page covers how to build and read each of them, with worked examples and the mistakes that cost marks. It's part of the broader Descriptive Statistics topic.
11 questions on this sub-topic.
Reading the diagrams
Covered under IB syllabus reference SL4.2: presentation of data in frequency tables, histograms and cumulative frequency graphs, and using cumulative frequency to find the median, quartiles, percentiles, range and IQR, including box and whisker diagrams.
Median and quartiles from a cumulative frequency curve
Median at position \(\dfrac{n}{2}\); \(Q_1\) at \(\dfrac{n}{4}\); \(Q_3\) at \(\dfrac{3n}{4}\).
Not in the formula booklet - it's a reading skill, not a formula. Trace across from the vertical axis to the curve, then down to the horizontal axis.
Box plot from the five-number summary
Minimum, \(Q_1\), median, \(Q_3\), maximum.
The box spans \(Q_1\) to \(Q_3\) with a line at the median; the whiskers run to the minimum and maximum, not to a 1.5 x IQR fence.
Need the full syllabus wording for descriptive statistics? See Descriptive Statistics.
Worked examples
For the data \(3, 5, 5, 6, 8, 8, 8, 10\):
(a) Find the mode.
(b) Find the median.
Worked solution
(a) Mode. The mode is the most frequent value; \(8\) appears three times (more than any other), so the mode is \(8.\) A1
(b) Median. With \(n=8\) (even) the median is the mean of the 4th and 5th values: \(\dfrac{6+8}{2}\) M1
\(=7.\) A1
A data set is grouped into four classes with frequencies 4, 9, 12 and 5. State the total number of data values.
Worked solution
\(4+9+12+5\) M1
\(=30.\) A1
Common mistakes
- Reading cumulative frequency instead of frequency, or vice versa. A cumulative frequency graph always rises (or stays flat) - if your curve dips, you've plotted plain frequency by mistake.
- Using the class midpoint where the boundary is needed. Histograms and cumulative frequency graphs plot against class boundaries, not midpoints - midpoints only belong in mean calculations.
- Assuming IB box plots trim outliers. Unless told otherwise, the whiskers go all the way to the minimum and maximum value - there's no automatic 1.5 x IQR outlier fence to apply here.
Ready to practise properly?
11 statistical-diagram questions, marked instantly like the real exam.
Quick answers
How do I read the median from a cumulative frequency graph?
Find \(\dfrac{n}{2}\) on the cumulative frequency axis, draw across to the curve, then drop down to the horizontal axis to read the median value.
Do IB box plots show outliers separately?
No. Unless a question tells you otherwise, IB box plots run from the minimum to the maximum value - there is no automatic \(1.5 \times \text{IQR}\) outlier rule to apply. See using your GDC on the full topic page for how to generate one on a calculator.