How Do You Find the Mode and Median of a Continuous Random Variable?

The mode is the value of the continuous random variable where its probability density function is greatest. The median is the value that divides the total area under the probability density function into two equal halves.

This is IB Mathematics: Analysis and Approaches HL content from AHL 4.14: Properties of discrete and continuous random variables.

Before calculating either measure, first verify that $f(x)$ is a valid density: it must be non-negative on its domain and have total area $1$. If the density contains an unknown constant, use the normalization condition $\int_{-\infty}^{\infty} f(x)\,dx=1$ to find that constant before continuing.

For example, suppose $X$ has probability density function

$$f(x)=2x, \quad 0\leq x\leq 1.$$

Since $f(x)=2x$ increases throughout the interval, its maximum occurs at $x=1$. Therefore,

$$\operatorname{mode}(X)=1.$$

To find the median, set the accumulated probability equal to $\frac{1}{2}$:

$$\int_{0}^{m}2x\,dx=\frac{1}{2}.$$

Therefore,

$$\left[x^2\right]_{0}^{m}=\frac{1}{2} \quad\Rightarrow\quad m^2=\frac{1}{2} \quad\Rightarrow\quad m=\frac{1}{\sqrt{2}}.$$

Only the positive solution is valid because $0\leq m\leq1$. Hence the median is $\frac{1}{\sqrt{2}}$.

These values need not be equal because the mode depends on the height of the density while the median depends on accumulated area. For a skewed distribution, the density's highest point and halfway-area point usually occur at different $x$-values.

A common misconception is that the mode equals the maximum value of $f(x)$. This is incorrect: the mode is the corresponding value of $x$, not the height of the density curve.

In an IB exam response, write the integral equation before solving for the median. For the mode, justify the maximum using the derivative, the behaviour of the function, or a comparison of stationary points and endpoints. This AHL topic may appear analytically on Paper 1 or with technology on Papers 2 and 3. Always show the equation used on a GDC.