Statistics & Probability: Median (Central Value)

The median is the exact middle value in a dataset when all numbers are ordered from least to greatest. It divides the data into two equal halves: at least 50% of the numbers are less than or equal to the median, and at least 50% are greater than or equal to it.

Notation (Subscripts): When a dataset of $n$ values is arranged in ascending order ($x_1 \le x_2 \le x_3 \le \dots \le x_n$), we use subscript notation $x_k$ to denote the number at position $k$ in the sorted list (e.g. $x_1$ is the first and smallest value, while $x_n$ is the last and largest).

To visually locate the median without manual index math, arrange the numbers in ascending order and pair off the smallest and largest remaining numbers from the outside inward until the center is reached.

When a dataset has an odd number of values ($n = 5, 7, 9, \dots$), there is exactly one number in the exact middle. Its position in the sorted list is given by the formula $\text{Position} = \frac{n + 1}{2}$.

When a dataset has an even number of values ($n = 6, 8, 10, \dots$), there is no single middle element. Instead, two numbers share the center at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$. The median is the arithmetic average (mean) of these two numbers.

Why do statisticians use both mean and median? The Mean is sensitive to extreme values (outliers), while the Median is resistant (robust). For example, if someone in a neighborhood earns $10,000,000, the mean income spikes drastically, while the median remains steady, reflecting a typical resident.