Question

Find the first, second, and third quartiles of the data. $$20,73,31,53,22,64,47$$

Short answer

The first quartile (Q1) is 22, the second quartile (Q2 or the median) is 47, and the third quartile (Q3) is 64.

Step-by-step solution

Arrange the data in ascending order

The first step is important because the quartiles are based on the order of the data. The arranged data would be \(20, 22, 31, 47, 53, 64, 73\).

Find the Median (Q2)

The median is the middle value of an arranged data set. If the set has an odd number of observations, the median is the middle number. If the set has an even number of observations, the median is calculated by averaging the two middle numbers. Since this set has 7 numbers, the median is the 4th number which is 47. So, Q2 or the median is 47.

Find Q1

Q1 is the median of the lower half of data (not including Q2 if the number of data points is odd). The lower half of our data is 20, 22, 31. So, Q1 is the second number in the lower half which is 22.

Find Q3

Q3 is the median of the upper half of data (not including Q2 if the number of data points is odd). The upper half of our data is 53, 64, 73. So, Q3 is the second number in the upper half which is 64.