median

The median is a measure of central tendency used in statistics and probability. In probability theory, the median of a random variable is any value m such that the probability of X being at most m is at least 1/2 and the probability of X being at least m is also at least 1/2. In a finite data set, the median is the middle value when the data are arranged in ascending order. It divides the data into two equal halves.

For a data set with n observations, ordered as x1 ≤ x2 ≤ ... ≤ xn:

- If n is odd, the median is x_{(n+1)/2}.

- If n is even, the median is the average of the two middle values, (x_{n/2} + x_{n/2+1})/2.

This approach does not rely on any assumption about the distribution and is robust to extreme values.

For grouped or continuous data, the exact median may not be identifiable from the data alone; it is often estimated. A common estimate uses the median class, with the formula: L + ((n/2 − cf) / f) × w, where L is the lower boundary of the median class, cf is the cumulative frequency before that class, f is the frequency of the median class, and w is the class width.

Properties and relationships: the median is the 50th percentile of a distribution. It is less sensitive to outliers than the mean and may differ from the mean in skewed distributions. When a distribution is symmetric, the median equals the mean.

Applications: the median is widely used in descriptive statistics, box plots, and nonparametric analyses, especially when data are skewed or contain outliers.