Question

Find the first, second, and third quartiles of the data. $$1,12,6,5,4,7,5,10,3,4$$

Short answer

The first quartile Q1 is 4, the second quartile Q2 is 5, and the third quartile Q3 is 7.

Step-by-step solution

Arrange the data

Firstly, the data needs to be sorted in increasing order. The given data set \(1,12,6,5,4,7,5,10,3,4\) becomes \(1,3,4,4,5,5,6,7,10,12\).

Determine the second quartile Q2

The second quartile Q2 is simply the median of the given data set. As the data set has even number of observations, we find Q2 by calculating the average of the two middle numbers. Thus, \(Q2 = \frac{5+5}{2} = 5\).

Determine the first quartile Q1

The first quartile Q1 is the median of the lower half of the data; this is the data to the left of the Q2, not including Q2. This subset is \(1,3,4,4,5\). Since it has an odd number of observations, Q1 is simply the middle number, so \(Q1 = 4\).

Determine the third quartile Q3

The third quartile Q3 is the median of the upper half of the data; this is the data to the right of the Q2, not including Q2. This subset is \(5,6,7,10,12\). As this set has an odd number of observations, Q3 is simply the middle number, so \(Q3 = 7\).