13  Mode, Median and Percentiles

13.1 Mode

The mode is the value \(x\) which maximises the pmf (discrete case) or the pdf (continuous case). That is \(x = \arg\max_{x_i} P(X = x_i)\) or \(x = \arg \max_{x} f(x)\). It is also a way of expressing, in a (usually) single number, important information about a distribution.

Example: Let X be the discrete random variable with pmf given by \(p(x) = \frac{1}{2^x}\) , \(x = 1,2,···\) and \(0\) otherwise. Find the mode of \(X\).

Solution: The value of \(x\) that maximizes \(p(x)\) is \(x = 1\). Thus, the mode of \(X\) is \(1\).

Example: Let \(X\) be the continuous random variable with pdf given by \(f(x) = 0.75(1− x^2)\) for \(−1\le x \le 1\) and \(0\) otherwise. Find the mode of \(X\).

Solution: The pdf is maximized at \(x=0\). Thus, the mode of \(X\) is \(0\).

The mode might not be unique; for example, the uniform distribution. Sometimes, the mode for a continuous variable is the local maxima of the pdf; and hence raising the notion of a multimodal distribution.

13.2 Percentiles and Median

The (100p)-th percentile is the value of \(x\) that satisfies: \[P(X <x)\le p\le P(X \le x)\] This is equivalent to \[x = \inf_x \{x : F(x) \ge p\}\] The median of \(X\) is defined to be the 50th percentile of \(X\).

Remark: Sometimes, the (100p)-th percentile for a discrete random variable is linearly interpolated from \(x_i\) and \(x_{i+1}\) in the support of \(X\) where \(P(X < x_1) \le p \le P(X \le x_2)\).