Quartiles
In short: Quartiles are three cut points, Q1, the median and Q3, that slice a sorted list into four slices holding roughly equal numbers of entries.
6th GradeStatistics6.SP.B.5c
What is Quartiles?
If the median splits a sorted data set into halves, quartiles go one step further and split it into quarters. They show where the lowest fourth ends, where the middle sits and where the top fourth begins.
The median itself is the second one, \(Q_2\). The first, \(Q_1\), is the middle of the bottom half; the third, \(Q_3\), is the middle of the top half. With an odd count the overall median joins neither half.
How to spot it
- A question asks for \(Q_1\), \(Q_3\) or a “lower” and “upper” quartile.
- A box plot needs its box edges calculated from raw data.
- A report says a score is in the top quarter or bottom quarter of a group.
- You are asked to describe the middle half of a data set.
The method
- Put the entries in increasing order and mark the center.
- Separate a bottom group from a top group; with an odd count, set the center aside.
- The center of the bottom group is \(Q_1\).
- The center of the top group is \(Q_3\).
Quartiles: solved example
Problem
A track team in Eugene records these 100-meter times in seconds: 12.4, 12.9, 13.1, 13.4, 13.8, 14.0, 14.5, 15.2, 15.6. Find \(Q_1\), the median and \(Q_3\).
Solution
Nine values, so the median is the 5th: 13.8. Lower half 12.4, 12.9, 13.1, 13.4 has median \(\frac{12.9 + 13.1}{2} = 13.0\). Upper half 14.0, 14.5, 15.2, 15.6 has median \(\frac{14.5 + 15.2}{2} = 14.85\).
So \(Q_1 = 13.0\), median \(= 13.8\), \(Q_3 = 14.85\).
Try it
Question 1
Locate the lower and upper cut points for 3, 5, 9, 11, 16, 20.
Show the answer
Bottom group 3, 5, 9 gives \(Q_1 = 5\); top group 11, 16, 20 gives \(Q_3 = 16\).
Question 2
If \(Q_3 = 38\), roughly what percent of the data is above 38?
Show the answer
About 25%, the top quarter.
Quartiles vs. Interquartile Range
The quartiles are three locations in the data. The interquartile range is one number made from two of them, \(Q_3 – Q_1\), that measures the width of the middle half.
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Whole dictionary: A–Z · 6th Grade · Statistics