When data are arranged from smallest to largest, the values that divide the whole data set into four equal parts are called quartiles.
First, Second, and Third Quartiles
- Second quartile (Q2): Median (the middle of the entire data set)
- First quartile (Q1): Median of the lower half of the data
- Third quartile (Q3): Median of the upper half of the data
Example ①
For the data: 2, 4, 5, 7, 8, 10, 12, 14, find Q1, Q2, and Q3.
Solution: Q2 = (7+8)/2 = 7.5. Lower half: 2,4,5,7 → Q1 = (4+5)/2 = 4.5. Upper half: 8,10,12,14 → Q3 = (10+12)/2 = 11.
For the data: 2, 4, 5, 7, 8, 10, 12, 14, find Q1, Q2, and Q3.
Solution: Q2 = (7+8)/2 = 7.5. Lower half: 2,4,5,7 → Q1 = (4+5)/2 = 4.5. Upper half: 8,10,12,14 → Q3 = (10+12)/2 = 11.
Key Points
- First arrange the data in ascending order
- For an even number of data values, Q2 is the average of the two middle values
- Q3−Q1 is called the interquartile range (IQR)
Practice Problems
- For the data: 3, 5, 7, 9, 11, 13, 15, find Q1, Q2, and Q3.
Answers and Explanations
- Answer: Q2=9 (median). Lower half: 3,5,7 → Q1=5. Upper half: 11,13,15 → Q3=13.