Key takeaway: The five-number summary describes a quantitative data set with its minimum, first quartile, median, third quartile and maximum, which together mark the spread of the lowest quarter, middle half and highest quarter.
Grade 12 – AP StatisticsAP Statistics Unit 1Statistics and Data
Definition of Five-Number Summary
The five-number summary condenses a list of numbers into five landmarks. Sort the data, then locate the smallest value, the median, the largest value, and the medians of the lower and upper halves. Each stretch between neighbors holds about a quarter of the data.
Because it relies on positions in the ordered list, the summary is resistant to extreme values. It is the exact information drawn in a box plot and the starting point for the interquartile range and the outlier fences.
Formula and notation
In order from least to greatest:
\[ \text{Min},\quad Q_1,\quad \text{Median},\quad Q_3,\quad \text{Max}. \]
The interquartile range is \(IQR=Q_3-Q_1\). With an odd number of values, the AP convention leaves the overall median out when finding \(Q_1\) and \(Q_3\).
Key facts
- The lowest quarter of values lies between the minimum and Q1, and the highest quarter between Q3 and the maximum.
- The middle half of the data lies between Q1 and Q3.
- Unlike the mean and standard deviation, these five values are barely moved by one extreme outlier.
- Values beyond 1.5 times the IQR from the quartiles are flagged as potential outliers.
Five-Number Summary example
A basketball player scored 12, 15, 17, 21, 24, 26, 29, 33, 38, 41 and 52 points in 11 games. Give the five-number summary and the IQR.
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The list is already sorted. Minimum \(=12\), maximum \(=52\), and the 6th value is the median, \(26\).
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Lower half (without the median): 12, 15, 17, 21, 24. Its middle value gives \(Q_1=17\).
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Upper half: 29, 33, 38, 41, 52. Its middle value gives \(Q_3=38\).
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Spread of the middle half: \(IQR=38-17=21\).
The five-number summary is 12, 17, 26, 38, 52, with an IQR of 21 points.
Common mistake
Watch out
For the same scores, a student keeps 26 in the lower half, uses 12, 15, 17, 21, 24, 26, and reports \(Q_1=\frac{17+21}{2}=19\).
Why it is wrong
With an odd count, the median belongs to neither half. Excluding it leaves five values below and five above, so the quartiles are single data values: \(Q_1=17\) and \(Q_3=38\). Always decide the halves before taking their medians.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data