Problem :
If I have a table of numerical data
$$ \begin{matrix} Value & Freq \\
40-49 & 7 \\
50-59 & 6 \\
60-69 & 10 \\
70-79 & 8 \\
80-89 & 9 \\
Total & 40
\end{matrix} $$
How do I calculate the upper quartile of this grouped data?
I have seen a resource that says that the $Q_{3}$ is calculated as below.
$0.75 \cdot 40 = 30$
so the upper quartile should be in class $70-79$.
$$ Q_{3} = (70-0.5) + (79-70+1) \cdot \left( \frac{8-10}{(8-10) + (8-9)} \right) \approx 75…$$
But the multiple choices differ. What is the correct method? thanks.
Best Answer
There are several different methods, but I suspect the one you are quoting from starts with
\begin{matrix} Value & Freq & Cum.Freq\\ 40-49 & 7 & 7\\ 50-59 & 6 & 13\\ 60-69 & 10 & 23\\ 70-79 & 8 & 31\\ 80-89 & 9 & 40\\ Total & 40 \end{matrix}
and might suggest
$$Q_{3} = (70-0.5) + (79+0.5-(70-0.5)) \cdot \frac{ 30 - 23}{31 - 23} \approx 78.25$$
Similar methods might be slightly different in detail, e.g. ignoring the $\pm 0.5$ or using $30.5$ instead of $30$, but they should all give an answer near $78$