# Calculate Upper Quartile of Grouped Data

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.

• There is no consensus on how to calculate percentiles, including quartiles, in the literature. Usually, when this is used, a definition is given. – Parcly Taxel Jul 16 '18 at 8:08
• @ParclyTaxel This problem is from highschool test sample. Is there an elementary formula that is used regularly in school? because as I remembered there must be one.. – Arief Anbiya Jul 16 '18 at 8:11

$$Q_{3} = (70-0.5) + (79+0.5-(70-0.5)) \cdot \frac{ 30 - 23}{31 - 23} \approx 78.25$$
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 $78$