Relationship Among Median, Quartiles, Deciles and Percentiles

Because quartiles, deciles and percentiles are all built on the identical idea of locating a stated fraction of an ordered series, several of them must always coincide EXACTLY, for ANY dataset — this is not a coincidence of any one example, but follows directly from the underlying fractions themselves being equal:

since each of these locates precisely the halfway (50%) point of the series. In the same way,

because a decile's fraction ($i/10$) always equals the percentile fraction $10i/100$, and a quartile's fraction ($i/4$) always equals the percentile fraction $25i/100$. A genuine numerical check on any computed pair — for instance, verifying that a computed $P_{25}$ exactly equals the corresponding $Q_1$ worked out independently from the same data — is therefore a valid, INDEPENDENT check on the arithmetic, not merely the same calculation done twice (the worked examples use exactly this cross-check).

Quartile deviation and the middle 50%. Because $Q_1$ marks the 25% point and $Q_3$ marks the 75% point, exactly the MIDDLE 50% of all observations in the series lie between $Q_1$ and $Q_3$. The quartile deviation, $QD=(Q_3-Q_1)/2$, is therefore sometimes called the semi-inter-quartile range — half the range spanned by that middle half of the data. …