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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.

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  • $\begingroup$ There is no consensus on how to calculate percentiles, including quartiles, in the literature. Usually, when this is used, a definition is given. $\endgroup$ – Parcly Taxel Jul 16 '18 at 8:08
  • $\begingroup$ @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.. $\endgroup$ – Arief Anbiya Jul 16 '18 at 8:11
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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 $78$

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