Question

Find the first, second, and third quartiles of the data. $$4,20,14,6,8,18,2,10,12,16$$

Short answer

The first quartile (Q1) is 6, the second quartile (Q2 or median) is 11, and the third quartile (Q3) is 16.

Step-by-step solution

Sort the data in ascending order

First, arrange the given data set in ascending order. This makes it easier to find the quartiles. So, the sorted data set is: \(2, 4, 6, 8, 10, 12, 14, 16, 18, 20\)

Find the second quartile (Q2)

The second quartile (Q2) is the median of the entire data set. Since there are an even number of observations, the median is the average of the two middle numbers. Here, those numbers are 10 and 12. So, \(Q2 = \frac{10 + 12}{2} = 11\)

Find the first quartile (Q1)

The first quartile (Q1) is the median of the lower half of the data not including Q2. Since there are an even number of observations in the lower half (2, 4, 6, 8, 10), the median is the middle number. So, \(Q1 = 6\)

Find the third quartile (Q3)

The third quartile (Q3) is the median of the upper half of the data not including Q2. Since there are an even number of observations in the upper half (12, 14, 16, 18, 20), the median is the middle number. So, \(Q3 = 16\)