Mode And Median

1. Core Concept Overview & Intuition

The mode and the median are two measures of central tendency associated with a probability distribution. They complement the mean by describing where the "centre" of a distribution lies, but each does so through a different mechanism.

The mode is the value of the random variable at which the probability distribution attains its maximum. Intuitively, it is the most probable value for a discrete random variable, and the point of highest density for a continuous random variable. A distribution may have one mode (unimodal), several modes (multimodal), or, in the discrete case, a plateau of equally probable values.

The median is the value that splits probability mass into two equal halves: half of the probability lies at or below it, and half lies at or above it. For a discrete random variable the median is defined through cumulative probabilities, while for a continuous random variable it is the point at which the cumulative distribution function equals $0.5$.

Both quantities are derived directly from the probability distribution function $f(x)$ and its cumulative distribution function $F(x)$, so the tools required are those of random variables and distribution functions: summation for discrete distributions, integration for continuous distributions, and differentiation for locating maxima of continuous densities.

2. Mathematical Formalization & Core Definitions

2.1 The Mode of a Distribution

> Definition (Mode). Let $X$ be a random variable with probability distribution $f(x)$ and corresponding cumulative distribution function $F(x)$. A mode of $f(x)$ is that value of the random variable $X$ which maximizes $f(x)$. That is, a value $\mu_0$ of $X$ is called a mode of $f(x)$ if it satisfies

$$
f(\mu_0)=\max f(x)
$$

> where $f(x)$ is maximized over all possible values of $X$.

The definition applies uniformly to discrete and continuous random variables. The essential distinction lies in the method of locating the maximum:

• For a discrete random variable, the mode is found by comparing the probabilities $f(x)$ at the admissible values of $X$; if $f(x)$ is monotone decreasing over the support, the mode is the smallest admissible value.
• For a continuous random variable, the mode is located by the calculus of maxima: solve $f'(x)=0$ over the support and confirm a maximum by the second-derivative test.

2.2 The Median of a Distribution

> Definition (Median, discrete case). The median of $f(x)$ is a value $\lambda$ of the random variable $X$ which satisfies

$$
P(X<\lambda)\le 0.5 \quad \text{and} \quad P(X\le\lambda)\ge 0.5
$$

> when $X$ is a discrete random variable.

For a discrete random variable with admissible values $x=0,1,2,\dots$, these two conditions are evaluated through partial sums of the probability function:

$$
\sum_{x=0}^{\lambda-1} f(x)\le 0.5 \quad \text{and} \quad \sum_{x=0}^{\lambda} f(x)\ge 0.5
$$

The first sum is the probability that $X$ is strictly below $\lambda$; the second is the probability that $X$ is at most $\lambda$. The median is the smallest admissible value at which the cumulative probability first reaches or exceeds $0.5$.

> Definition (Median, continuous case). If $X$ is a continuous random variable, then the median $\lambda$ satisfies

$$
F(\lambda)=0.5
$$

> equivalently,

$$
\int_{\text{lower limit}}^{\lambda} f(x)\,dx=0.5 \quad \text{or} \quad \int_{\lambda}^{\text{upper limit}} f(x)\,dx=0.5
$$

For a continuous random variable the median is therefore obtained by integrating the density from the lower limit of the support up to $\lambda$, setting the result equal to $0.5$, and solving for $\lambda$. Equivalently, one may integrate from $\lambda$ to the upper limit of the support and set that equal to $0.5$; both routes yield the same value because the total probability is $1$.

2.3 Summary of the Two Measures