Continuous random lower quartile

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

When talking about probability and distributions, the lower quartile is where the cumulative probability is 0.25.

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 lower quartile is where the probability is 0.25, we can find that:

\int_0^af(x)dx=0.25

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

Question	Answer
lower quartile	The value where 25% of data is below it and 75% is above it; for a probability distribution, the cumulative probability is 0.25.
How do you find the lower quartile of a continuous random variable?	Solve \int_0^a f(x)\,dx = 0.25 for a, using definite integration.
What equation defines the lower quartile a for a continuous random variable with PDF f(x)?	\int_0^a f(x)\,dx = 0.25