= Q-Q plot
{c}
{wiki=Q–Q_plot}
= Quantile-quantile plot
{synonym}
A <Q-Q plot> is a <scatter plot> pairing <quantiles> at the same <probability> levels. For a <statistical sample> of size $n$, sort the observations into <order statistics> $x_{(1)}\leq\cdots\leq x_{(n)}$. Choose <plotting positions> $p_i$ and a reference <quantile function> $Q_0$, then plot
$$
(Q_0(p_i),x_{(i)}),\qquad p_i=\frac{i-1/2}{n},\qquad i=1,\ldots,n.
$$
Here the horizontal coordinate is a reference value, and the vertical coordinate is an observed value; neither axis is the <probability> $p_i$. The displayed <plotting positions> are our chosen convention, placing every level strictly between zero and one. They are not a claim that the $i$th <order statistic> equals that population <quantile>. For two <statistical samples>, use <sample quantiles> at matching levels, with a stated interpolation convention when their sizes differ. https://www.itl.nist.gov/div898/handbook/eda/section3/qqplot.htm[NIST/SEMATECH, Quantile-Quantile Plot, definition]{ref}
As a computed example, take $n=19$ and the <standard normal quantiles> $z_i=\Phi^{-1}(p_i)$, where $\Phi$ is the <standard normal cumulative distribution function>. Construct the ordered values $a_i=10+2z_i$. At ranks $1,10,19$, the <plotting positions> are $1/38,1/2,37/38$, the reference <quantiles> are approximately $-1.937932,0,1.937932$, and the constructed values are approximately $6.124137,10,13.875863$. Every point lies on $a=10+2z$ by construction. These 19 values are deliberately chosen, not random draws or real measurements.
\Image[/qq-plot-location-and-scale.png]
{title=An exact constructed Q-Q line encodes location and scale}
{description=Left: all 19 pairs $(z_i,a_i)$ lie on $a=10+2z$, with <plotting positions> $p_i=(i-1/2)/19$. Right: subtracting the specified location 10 and dividing by the specified scale 2 puts the same points on the identity line. These parameters are construction constants, not <sample mean> and sample <standard deviation> estimates. The 19 values are deterministic examples, not random draws.}
The usefulness of a <Q-Q plot> is that it turns a comparison of <probability distributions> into geometry: if a reference <random variable> $Z$ is transformed into $X=\mu+\sigma Z$ with $\sigma>0$, its <quantile function> satisfies $Q_X(p)=\mu+\sigma Q_Z(p)$. This follows because a positive scaling preserves ordering. Thus the <location-scale family> produces a straight population <Q-Q plot>, and an approximate sample line suggests a compatible shape over the observed range. Its intercept describes location and its slope describes scale. The identity line requires the same location, scale and units on both axes; even then its displayed angle is 45 degrees only with equal axis scales. Curvature is often more informative than a <histogram> because it shows which <quantiles> depart from the reference. https://www.itl.nist.gov/div898/handbook/eda/section3/qqplot.htm[NIST/SEMATECH, Quantile-Quantile Plot, location and scale comparisons]{ref}
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