Question

in the data set below, what is the upper quartile? 18 19 38 69 73 75 77 85

Explanation:

Step1: Calculate position

The formula for the position of the upper - quartile ($Q_3$) for a data set of size $n$ is $i=\frac{3(n + 1)}{4}$. Here, $n = 8$, so $i=\frac{3\times(8 + 1)}{4}=\frac{3\times9}{4}=6.75$.

Step2: Interpolate

Since the position is not an integer, we interpolate between the 6th and 7th ordered data - values. The 6th value is 75 and the 7th value is 77. Using the interpolation formula $Q_3=x_{j}+(i - j)(x_{j + 1}-x_{j})$, where $j = 6$, $x_{j}=75$, $x_{j + 1}=77$, and $i = 6.75$. So $Q_3=75+(6.75 - 6)\times(77 - 75)=75+0.75\times2=75 + 1.5=76$.

Answer:

76