Track 5 · Probability Distributions · Lesson 9 of 10

Quantiles and inverse probability

Find quantiles of continuous and discrete distributions, and use them as critical values for one- and two-tailed procedures.

30 minIntermediate8 graded exercisesMembers

A warehouse wants to stock enough units to meet demand on 95% of days; a quality engineer needs the cut-offs that leave 2.5% of a distribution in each tail; an analyst building a confidence interval needs “the 1.96”. All of these are the same question asked backwards: not what is the probability of this value? but which value has this probability below it?

A cumulative distribution function (cdf) turns a value into a probability: $F(x) = P(X \le x)$ . The quantile function runs the other way: it turns a probability $p$ into the value $Q(p)$ that has (at least) a proportion $p$ of the distribution at or below it. The median is $Q(0.5)$ , the quartiles are $Q(0.25)$ and $Q(0.75)$ , and the 90th percentile is $Q(0.90)$ .

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Find a quantile as the inverse of the cdf, for continuous and discrete distributions
Explain why a discrete quantile usually covers more than the target probability
Choose α/2 or α for two-sided and one-sided critical values
Find t critical values with n − 1 degrees of freedom and asymmetric chi-square cut-offs
Compare a sample percentile with the percentile of a fitted model

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Formula sheet

The quantile function (the inverse of the cdf)
$Q(p) = \min\{\, x : F(x) \ge p \,\}$
Standard normal critical values
$z_{\alpha/2} = Q_Z\!\left(1 - \tfrac{\alpha}{2}\right) \quad\text{(two-sided)}, \qquad z_{\alpha} = Q_Z(1 - \alpha) \quad\text{(one-sided)}$

Reference

Quantiles, percentiles and quartiles
The p-th quantile is a value with about a proportion p of the data below it; software differs in how it interpolates, and Statistics Quest uses Hyndman–Fan type 7 by default.
Random variables: pmf, pdf, cdf, sf and quantiles
A random variable assigns a number to each outcome; its distribution is described by a pmf or pdf, a cdf, a survival function and a quantile function.
Normal distribution
The normal distribution is the symmetric bell-shaped curve fully described by its mean μ and standard deviation σ.
Student's t distribution
The t distribution is the symmetric, heavier-tailed sampling distribution of a standardised mean when σ is estimated by the sample SD.

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.