Q.Find the three quartiles $Q_1$, $Q_2$ and $Q_3$ for the data: $12, 18, 25, 30, 35, 40, 48, 52, 60$.

The data are already in ascending order with $n = 9$, so use the raw-data quartile positions $\frac{i(n+1)}{4}$ with $n+1 = 10$.

$Q_1 = 21.5$, $Q_2 = 35$, $Q_3 = 50$.

The nine values are already ascending, so $n = 9$ and $n + 1 = 10$.

$Q_1$: position $= \dfrac{1(10)}{4} = 2.5$ → between the 2nd ($18$) and 3rd ($25$) observations.

$Q_2$: position $= \dfrac{2(10)}{4} = 5$ → the 5th observation exactly, which is $35$. So $Q_2 = 35$.

$Q_3$: position $= \dfrac{3(10)}{4} = 7.5$ → between the 7th ($48$) and 8th ($52$).

Independent check. $Q_2$ should equal the median: with $n = 9$ (odd) the median is the $\tfrac{9+1}{2} = 5$th value $= 35$ ✓. Also $Q_1 \le Q_2 \le Q_3$ ($21.5 \le 35 \le 50$) holds, as it must.

$Q_3$ can be cross-checked from the top: it should have three-quarters of the data below it. Six of the nine values ($12$ to $40$) lie below $50$ and one ($48$) is just under it, consistent with $Q_3 = 50$ sitting near the upper quarter mark.

Forgetting the '$+1$' and using $\frac{i\,n}{4}$ instead of $\frac{i(n+1)}{4}$, or not interpolating when the position is fractional (e.g. reporting the 2nd value $18$ for $Q_1$ instead of interpolating to $21.5$). Also, never compute partition values before ordering the data.