The median (or \(50\)th percentile), usually denoted by \(m\), is the value of \(X\) such that the line \(x = m\) splits the area under the curve \(y = \mathrm{f}(x)\) into two halves.
\[\text{i.e.}\hspace{0.5em} \int_{a}^{m} \mathrm{f}(x) \, \mathrm{d}x = 0.5 \text{ or}\hspace{0.5em} F(m) = 0.5\]
The lower quartile (or \(25\)th percentile), \(r\), is the value of \(X\) such that&nbsp;\(\int_{a}^{r} \mathrm{f}(x) \, \mathrm{d}x = 0.25\) or \(F(r) = 0.25\).
The upper quartile (or \(75\)th percentile), \(s\), is the value of \(X\) such that&nbsp;\(\int_{a}^{s} \mathrm{f}(x) \, \mathrm{d}x = 0.75\) or \(F(s) = 0.75\).
In general, the \(k^{\mathrm{th}}\) percentile, \(t\), is the value of \(X\) such that&nbsp;\(\int_{a}^{t} \mathrm{f}(x) \, \mathrm{d}x = \frac{k}{100}\) or \(F(t) = \frac{k}{100}\).
The interquartile range = upper quartile – lower quartile.

The median (or \(50\)th percentile), usually denoted by \(m\), is the value of \(X\) such that the line \(x = m\) splits the area under the curve \(y = \mathrm{f}(x)\) into two halves.

\[\text{i.e. } \int_{a}^{m} \mathrm{f}(x) \, \mathrm{d}x = 0.5 \text{ or } F(m) = 0.5\]

The lower quartile (or \(25\)th percentile), \(r\), is the value of \(X\) such that \(\int_{a}^{r} \mathrm{f}(x) \, \mathrm{d}x = 0.25\) or \(F(r) = 0.25\).

The upper quartile (or \(75\)th percentile), \(s\), is the value of \(X\) such that \(\int_{a}^{s} \mathrm{f}(x) \, \mathrm{d}x = 0.75\) or \(F(s) = 0.75\).

In general, the \(k^{\mathrm{th}}\) **percentile**, \(t\), is the value of \(X\) such that \(\int_{a}^{t} \mathrm{f}(x) \, \mathrm{d}x = \frac{k}{100}\) or \(F(t) = \frac{k}{100}\).

Median and IQR of a CRV

Median and IQR of a CRV