STAT1008 Chap.4 Quartiles, Boxplots and Outliers
Quartiles, Boxplots and Outliers
Quartiles, Boxplots and Outliers covers the course’s hand rule for percentiles, where the location index is P/100 times n, averaged when whole and rounded up when fractional. It builds the five-number summary, draws boxplots with fences at 1.5 IQR beyond the quartiles, explains why whiskers stop at the last value inside the fence, and compares the boxplot rule with the z-score rule for outliers.
It also shows how boxplots reveal skew and compare groups without calculation. Assessment items in this area are short but convention-sensitive. They ask for a percentile position, a five-number summary, a boxplot with its fences, or a judgement about which summary to report for skewed data, and they reward saying which outlier rule was applied.
What this chapter covers
- 01
Percentiles and the location index
- 02
Quartiles and the interquartile range
- 03
The five-number summary
- 04
Boxplot construction and fences
- 05
Two outlier rules that can disagree
- 06
Comparing groups with boxplots
Worked example · free
Quartiles and outliers for delivery times at depot A
- 1Q1: i = 0.25 × 12 = 3, a whole number, so Q1 = (18 + 20)/2 = 19 minutes.
- 1Median: i = 6, so (22 + 24)/2 = 23 minutes.
- 2Q3: i = 9, so (27 + 29)/2 = 28 minutes, and IQR = 28 − 19 = 9 minutes.
- 1Fences: 19 − 1.5(9) = 5.5 and 28 + 1.5(9) = 41.5 minutes.
- 1Only 52 lies outside the fences, so it is an outlier and the upper whisker ends at 33.
Key terms
- Interquartile range
- The interquartile range is Q3 minus Q1, the spread of the middle half of the ordered data.
- Five-number summary
- The five-number summary lists the minimum, first quartile, median, third quartile and maximum.
- Percentile
- The P-th percentile is the value with P per cent of the data below it and the remainder above.
- Outlier fence
- An outlier fence is a boundary 1.5 interquartile ranges below Q1 or above Q3, beyond which values are plotted as outliers.
Quartiles, Boxplots and Outliers FAQ
Why do my quartiles differ from Excel’s?
Excel’s PERCENTILE.INC and PERCENTILE.EXC use their own location formulas and interpolate between values, while the course’s hand rule averages or rounds up. On small datasets the results differ slightly. The lecture accepts either Excel function unless one is specified.
Is the end of the whisker the maximum?
Only when there are no outliers. Whiskers end at the furthest observations that remain within the fences at 1.5 IQR beyond the quartiles; any value beyond is plotted separately and is the true minimum or maximum.
Which outlier rule should I use in an answer?
Use the rule the question names. If none is named, state your choice, for example the 1.5 IQR boxplot rule or the plus or minus 3 z-score rule, because the two can classify the same observation differently.
What does the IQR tell me?
The interquartile range measures the spread of the middle half of the data, from the first to the third quartile. Because it ignores the extreme quarter at each end, it is a resistant measure and pairs naturally with the median for skewed data.
How do I read skew from a boxplot?
Compare the two halves. A longer upper whisker, outliers above it and a median sitting closer to the first quartile indicate right skew; the mirror image indicates left skew. Roughly equal whiskers and a central median suggest symmetry.
Exam move
Practise the percentile rule on datasets of 10, 12, 15 and 20 values so that both the whole-number and round-up cases become automatic. For each dataset draw the boxplot by hand with fences, then describe its shape in one sentence. Finally, compute the z-score of the most extreme value and compare the two outlier verdicts.
When you draw a boxplot, label the five numbers and the fences on the sketch itself, then write a one-sentence comparison if two groups are shown: medians first, box widths second, whiskers and outliers last. Practising that order makes comparison questions quick and keeps the answer structured.
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