Suppose I have a set of quantitative data $x_i$, with $1 \le i \le N$. This set has a median $M = x_m$, where the median $M=x_m$ where $m=\frac{N+1}{2}$ if $N$ is odd, or $M=\frac{x_\mu + x_{\mu+1}}{2}$, where $\mu \equiv \frac{N}{2}$ if $N$ is even.
From a conceptual perspective, the first quartile ($Q_1$) is the median of the lower half of the data and the third quartile ($Q_3$) is the median of the upper half of the data.
Another way of thinking of the quartiles is in terms of percentiles. In this picture $Q_1$ and $Q_3$ equal the 25th percentile ($P_{25}$) and 75th percentile ($P_{75}$), respectively.
My intuition tells me that these pictures are equivalent for some values of $N$ (e.g. N=7), but because of the way $M$ is defined for even $N$, they might not be the same in all circumstances.
Under what circumstances are these pictures identical or not identical?
