What is the lower quartile of the set of data?

Question

I came across this problem asking the lower quartile of the ungrouped data. My answer is 3, but other references say it should be 2.5. Here's the data:

1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 10, 11, 14, 15, 20, 22

What do you think?

Answer 1

From a purely mathematical point of view, the first quartile $Q_1$ is the $25$th percentile and represents a cutoff, that should have the following two properties:

• at most $25\%$ of the data is less than $Q_1$ and
• at most $75\%$ of the data is greater than $Q_1$

So, $\color{blue}{Q_1 =3}$.

Note, that $\color{red}{2.5}$ does not satisfy the above given conditions: $\frac{13}{17} \approx 76.5\%$ of the data items are larger than $2.5$. Hence, $2.5$ does not quality as a quartile.

Nevertheless, some sources will continue to report the result $2.5$, because they may use a method, where one determines the lower and upper quartile as the median of a so-called lower or upper "data half".

The result then depends on whether one "includes" the median of the whole data set in such a "data half" or not and whether the data items near the quartile postition are equal or not. (Just note that if the data set were $112\color{blue}{3}33...$ the result would be $3$)

So, in the case that the data items aren't equal around the quartile position, then

• If the median is included in the "data halves", then the median method is incorrect, if the data size is $4k+3$.
• If the median is excluded in the "data halves", then the median method is incorrect, if the data size is $4k+1$.