Measures of Spread (AI HL)
The mean tells you where a data set is centred, but not how tightly the values cluster around it - that's the job of a measure of spread. This page focuses on the interquartile range and the 1.5 x IQR outlier rule, the piece of this syllabus point examiners return to most often, with worked examples and the mistakes that lose marks. It's part of the broader Statistics & Sampling topic.
12 questions on this sub-topic.
IQR and outlier boundaries
Covered under IB syllabus reference SL4.3: measures of dispersion (interquartile range, standard deviation, variance), calculated using technology, plus the interpretation of outliers via the \(1.5\times\)IQR rule from SL4.1.
Interquartile range
\(\text{IQR} = Q_3 - Q_1\)
\(Q_1\) and \(Q_3\) come straight from your GDC's one-variable statistics screen - you're not expected to find them by hand once the data set is large.
Outlier boundaries
\(Q_1 - 1.5\times\text{IQR}\ \ \text{and}\ \ Q_3 + 1.5\times\text{IQR}\)
A value is an outlier if it falls outside this range. This is standard technique that IB defines explicitly in the syllabus, rather than a formula-booklet entry.
Need the full syllabus wording and formula-booklet reference table? See Statistics & Sampling.
Worked examples
A shop's monthly sales figures ($1000s) have \(Q_1=70\) and \(Q_3=100.\) One particular month had sales of $20,000. Using the 1.5\(\times\)IQR rule, determine whether this month should be treated as an outlier.
Worked solution
\(\text{IQR}=100-70=30.\) M1
Lower boundary \(=Q_1-1.5\times\text{IQR}=70-1.5(30)=70-45.\) A1
\(20<25,\) so the value lies below the lower boundary. R1
The month's sales of $20,000 are an outlier. A1
The scores, out of 50, of 12 students on a test are:
5, 22, 25, 28, 30, 31, 33, 35, 36, 38, 40, 49
(a) Use your GDC to find the interquartile range.
(b) Find the lower and upper outlier boundaries.
(c) State which score(s), if any, are outliers.
Worked solution
(a) \(Q_1=26.5,\ Q_3=37.\) M1
\(\text{IQR}=37-26.5=10.5.\) A1
(b) Lower \(=26.5-1.5(10.5)=26.5-15.75,\) Upper \(=37+1.5(10.5)=37+15.75.\) M1
\(=10.75\) and \(52.75.\) A1
(c) Only \(5\) lies below \(10.75;\) no score exceeds \(52.75.\) So \(5\) is the only outlier. A1
Common mistakes
- Comparing the raw value to \(Q_1\) or \(Q_3\) instead of the boundary. A value below \(Q_1\) is not automatically an outlier - it has to fall below \(Q_1 - 1.5\times\text{IQR}\), which is often a good deal lower again.
- Using the wrong quartile for the wrong boundary. The lower boundary subtracts \(1.5\times\text{IQR}\) from \(Q_1\); the upper boundary adds it to \(Q_3\). Mixing these up gives boundaries the wrong way round.
- Reading \(Q_1\) and \(Q_3\) off the GDC incorrectly. Different calculator models label these slightly differently in the one-variable statistics output - always check you're reading \(Q_1\) and \(Q_3\), not the min/max or the median, before subtracting.
Ready to practise properly?
11 measures-of-spread questions, marked instantly like the real exam.
Quick answers
How do you find the interquartile range?
\(\text{IQR} = Q_3 - Q_1\). On the IB course you find \(Q_1\) and \(Q_3\) using the one-variable statistics function on your GDC rather than by hand.
How do you test whether a value is an outlier?
A value is an outlier if it lies below \(Q_1 - 1.5\times\text{IQR}\) or above \(Q_3 + 1.5\times\text{IQR}\).