Continuous random upper quartile

Remember that the lower quartile of any distribution is the value where 75% of the data is below it and 25% of the data is above it.

When talking about probability and distributions, the upper quartile is where the cumulative probability is 0.75.

Finding a formula

Knowing our formula for finding the probability of a continuous random variable, P(a \leq X \leq b) = \int_a^b f(x)\,dx, as well as that the upper quartile is where the probability is 0.75, we can find that:

\int_0^af(x)dx=0.75

We can use definite integration and find the value of a - that’s the upper quartile!