Question

Find the first, second, and third quartiles of the data. $$12,5,3,8,10,7,6,5$$

Short answer

The first quartile is 5, the second quartile is 6.5, and the third quartile is 9.

Step-by-step solution

Organize the Data in Ascending Order

Arrange the provided set of numbers from lowest to highest: \(3, 5, 5, 6, 7, 8, 10, 12\).

Find the Second Quartile (Median)

Find the median of the dataset. As the number of observations (8) is even, the median or second quartile \(Q2\) is the average of the two middle numbers, which are 6 and 7. So, \(Q2 = \frac{6+7}{2}= 6.5\).

Determine the First Quartile

The first quartile \(Q1\) is the median of the lower half of the data, not including the overall median. This would be the dataset {3, 5, 5, 6}. With an even number of observations, the first quartile is the average of the two middle numbers, which are 5 and 5. So, \(Q1 = \frac{5+5}{2}= 5\).

Determine the Third Quartile

The third quartile \(Q3\) is the median of the upper half of the data, again, not including the overall median. This would be the dataset {7, 8, 10, 12}. The third quartile is the average of the two middle numbers, which are 8 and 10. So, \(Q3 = \frac{8+10}{2}= 9\).