quantile() — Quantiles

quantile(x)

Key idea

A quantile is a cut-off value: a chosen share of the sorted data falls below it.
The 0.5 quantile is the median, with half the data below it. (e.g. 0.5 quantile = median)
The 0.25 quantile is Q1, meaning 25% of values fall below it. (e.g. 0.25 quantile = Q1)
The 0.75 quantile is Q3, with 75% of values below it. (e.g. 0.75 quantile = Q3)
Quantiles let you describe any position, such as the 90th percentile (the top-10% cut-off). (e.g. 0.90 quantile = top 10% cutoff)

Definition

$Q (p) = the value x with P (X \leq x) = p$

$p$: a fraction between 0 and 1
$Q (p)$: the value at that quantile
$P (X \leq x)$: share of data at or below x

Where you'd use it

You have everyone’s exam scores and want the mark for the top 10%. The 90% quantile might land at 88, so a student needs about 88 to be in the best tenth of the class.
For a set of finish times, the 25% quantile is the cut-off for the fastest quarter of runners; if it comes out at 22 minutes, anyone faster than 22 minutes made the top 25%.
Given a list of incomes, the 25% quantile tells you the level below which a quarter of people fall — for example $28k — a position-based figure that outliers cannot distort.

Frequently asked questions

What is a quantile?: The value at a given proportion: the 0.25 quantile is Q1, 0.5 is the median, 0.75 is Q3.

Statistics Playground

An interactive playground for probability and statistics. Pick a function, edit its parameters, and see the distribution curve, the computed value, and a plain-English interpretation — all computed in your browser (no server, no install). The functions mirror familiar R syntax (for example pnorm, dbinom, t.test) and the numbers come from numpy running under Pyodide.

The normal distribution

pnorm(q, mean = 0, sd = 1) — Cumulative probability for the normal distribution.
dnorm(x, mean = 0, sd = 1) — Height of the bell curve at a point (the density).
qnorm(p, mean = 0, sd = 1) — The inverse of pnorm; turns a probability back into a value.
rnorm(n, mean = 0, sd = 1) — Draws random values from a normal distribution.
scale(x) — Re-expresses values as z-scores (mean 0, sd 1).

Binomial & Poisson distributions

dbinom(x, size, prob) — Probability of exactly x successes in n trials.
pbinom(q, size, prob) — Probability of q or fewer successes.
dpois(x, lambda) — Probability of exactly x events at rate λ.
ppois(q, lambda) — Probability of q or fewer events at rate λ.

The chi-square distribution & hypothesis testing

pchisq(q, df) — Cumulative probability for the χ² distribution.
t.test(x, mu = 0) — One-sample t-test against a hypothesised mean.

Correlation & regression

cor(x, y) — Strength of a linear relationship, −1 to +1.
lm(y ~ x) — Fits a least-squares straight line through the points.

Descriptive statistics

mean(x) — The arithmetic average.
median(x) — The middle value of the data.
sd(x) — Standard deviation; typical distance of a value from the mean.
var(x) — Variance; the mean squared deviation from the mean.
IQR(x) — Span of the middle 50% of the data.
range(x) — The minimum and maximum values.
quantile(x) — Sample quantiles that split the data into equal slices.
summary(x) — The five-number summary plus the mean.
weighted.mean(x, w) — An average where each value carries a weight.

Combinatorics, sampling & visualisation

factorial(x) — x!, the ways to arrange x distinct items in order.
choose(n, k) — Ways to choose k items from n, order ignored.
sample(x, size, replace = FALSE) — Draw a random sample.
hist(x, breaks) — A frequency histogram of the data.
boxplot(x) — A box-and-whisker summary of the spread.