Five-Number Summary
In short: Five-number summary boils a data set down to five ordered landmarks, lowest value, lower quartile, middle value, upper quartile and highest value, which a boxplot then draws.
AP StatisticsStatisticsAP Statistics Unit 1
What is Five-Number Summary?
The five-number summary squeezes a whole data set into five landmarks: the smallest value, the value one quarter of the way up, the middle value, the value three quarters of the way up, and the largest value. In order, they are the minimum, \(Q_1\), the median, \(Q_3\) and the maximum.
Consecutive landmarks fence off roughly a quarter of the ordered data. The gap \(Q_3 – Q_1\) is the interquartile range (IQR). These numbers depend on rank rather than on sums, so extreme values hardly move them; statisticians call such a description resistant.
How to spot it
- A question asks for the median and quartiles of a list, or for the data behind a boxplot.
- You must compare two groups using a resistant description because a few values look extreme.
- The word “outlier” appears and a 1.5 times IQR rule is expected.
- The distribution is skewed, so mean and standard deviation would mislead.
The method
- Sort the values from least to greatest and count them.
- Locate the median, which cuts the list into two halves (when \(n\) is odd, set the median itself aside).
- \(Q_1\) is the middle value of the bottom half, and \(Q_3\) is the middle value of the top half.
- Write min, \(Q_1\), median, \(Q_3\), max in that order, then test for outliers with the fences \(Q_1 – 1.5\,\text{IQR}\) and \(Q_3 + 1.5\,\text{IQR}\).
Five-Number Summary: solved example
Problem
Fourteen employees at a Denver design studio reported commute times in minutes: 8, 11, 12, 15, 17, 18, 20, 22, 24, 25, 28, 31, 36, 52. Build the summary and test for outliers.
Solution
The list is already sorted and \(n = 14\). Average the 7th and 8th entries to get the median: \(\frac{20 + 22}{2} = 21\).
Lower half (8 to 20): median is the 4th value, \(Q_1 = 15\). Upper half (22 to 52): median is the 4th value, \(Q_3 = 28\).
The summary is 8, 15, 21, 28, 52. Then \(\text{IQR} = 13\) and the upper fence is \(28 + 1.5(13) = 47.5\), so 52 is an outlier. The lower fence is \(15 – 19.5 = -4.5\); nothing falls below it.
Try it
Question 1
Give the five-number summary for the list 3, 5, 7, 9, 11.
Show the answer
Median 7. Lower half 3, 5 gives \(Q_1 = 4\); upper half 9, 11 gives \(Q_3 = 10\). Summary: 3, 4, 7, 10, 11.
Question 2
A data set has \(Q_1 = 62\) and \(Q_3 = 80\). What values would be flagged as outliers?
Show the answer
\(\text{IQR} = 18\) and \(1.5 \times 18 = 27\). Values below \(62 – 27 = 35\) or above \(80 + 27 = 107\) are outliers.
Five-Number Summary vs. Interquartile Range
The summary is the full set of five landmarks. The IQR is a single number built from two of them, \(Q_3 – Q_1\), and it captures how wide the central half of the data is; the summary also shows where the data begins, ends and centers.
Keep going
Lessons
Worksheets and quizzes
- Five-Number Summary, Boxplots and Comparing Distributions – Worksheet
- Five-Number Summary, Boxplots and Comparing Distributions – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics