Quartiles — Meaning and Computation for Individual, Discrete and Continuous Series

Quartiles split an ordered series into four equal parts. $Q_1$ (the lower quartile) marks the point below which the smallest 25% of observations lie; $Q_2$ (the middle quartile) is identical to the median, marking the 50% point; $Q_3$ (the upper quartile) marks the point below which the largest 25% of observations lie.

Individual series (data listed item by item, arranged in ascending order). Locate the position $k = iN/4$ for $Q_i$ ($i=1,2,3$):

• If $k$ is not a whole number, round it UP to the next whole number; the quartile is the value of the item at that position.
• If $k$ is a whole number, the quartile is the AVERAGE of the item at that position and the item immediately after it — exactly the convention already used for the median of an even-sized individual series.

Discrete series (distinct values X, each with frequency f). Build the cumulative frequency (cf) column exactly as for the median. Locate $k=iN/4$; the quartile is the value of X at the first class whose cf is equal to or just greater than $k$.

Continuous (grouped) series (class intervals, each with frequency f). First locate the quartile class — the class whose cumulative frequency is the first to equal or exceed $k=iN/4$ — then interpolate within it:

where $L$ is the lower boundary of the quartile class, $cf$ is the cumulative frequency of the class immediately BEFORE it, $f$ is the frequency of the quartile class itself, and $h$ is its width. Setting $i=2$ in this same formula reproduces exactly the median formula from the Measures of Central Tendency chapter — a first confirmation that $Q_2$ and the median are the same value (Section 5 develops this fully). …