= Normal Q-Q plot
= Normal probability plot
{synonym}
A <normal Q-Q plot> compares <order statistics> with <standard normal quantiles> $z_i=\Phi^{-1}(p_i)$. For a <normal distribution> $N(\mu,\sigma^2)$ with $\sigma>0$, the population <quantile function> is $\mu+\sigma\Phi^{-1}(p)$, so the diagnostic sample relation is
$$
x_{(i)}\approx\mu+\sigma z_i.
$$
The appropriate reference is a line with an intercept and positive slope, not necessarily the identity line. A line can be fitted through two central <sample quantiles>; R's `qqline` uses the <quantile> levels $p=0.25$ and $p=0.75$ by default. Neither that line nor its fitting rule is part of the definition of a <normal Q-Q plot>. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/qqnorm.html[R stats documentation, qqnorm and qqline]{ref}
To isolate curvature, retain the same 19 <plotting positions> and construct two more ordered sequences:
$$
b_i=2p_i-1,\qquad c_i=\sinh(0.7z_i).
$$
The $b_i$ are <continuous uniform distribution> <quantiles> on $[-1,1]$, while the $c_i$ apply the <hyperbolic sine> to the reference <quantiles>. Both examples are symmetric about zero. At ranks $1,10,19$, $b_i$ is approximately $-0.947368,0,0.947368$ and $c_i$ is approximately $-1.812618,0,1.812618$. Again, these are constructed values rather than random draws or real data.
\Image[/qq-plot-tail-shapes.png]
{title=Normal Q-Q curvature distinguishes compressed and stretched outer quantiles}
{description=Left: the constructed <uniform distribution> <quantiles> $b_i=2p_i-1$ bend inward relative to the central tangent $b=\sqrt{2/\pi}\,z$. Right: the constructed values $c_i=\sinh(0.7z_i)$ bend outward relative to the central tangent $c=0.7z$. Both panels use 19 <standard normal quantiles> at $p_i=(i-1/2)/19$. Dashed lines are analytic central tangents, not fitted lines or uncertainty bands; only the chosen finite range is illustrated.}
Read these shapes relative to their central slope. The <uniform distribution> example has $b(z)=2\Phi(z)-1$, with <derivative> $b'(0)=\sqrt{2/\pi}\approx0.797885$; at the upper endpoint the tangent predicts $1.546246$, while $b_{19}=0.947368$. Its lower end lies above the tangent and its upper end below it: the outer <quantiles> are compressed. The <hyperbolic sine> example has $c'(0)=0.7$; its tangent predicts $1.356552$ at the upper endpoint, while $c_{19}=1.812618$. Its lower end lies below the tangent and its upper end above it: the outer <quantiles> are stretched. Unequal departures at the two ends can suggest <skewness>. The direction of these descriptions assumes that reference <quantiles> are horizontal and observed values vertical. The graphs illustrate finite-range shape differences; a sample pattern cannot establish an asymptotic <tail probability> law or membership in a <heavy-tailed distribution> class.
A nearly straight <normal Q-Q plot> is a diagnostic, not a proof of a <normal distribution> or a formal <hypothesis test>. Random <statistical samples> fluctuate, particularly at extreme <order statistics>, and apparent curvature depends on <sample size>, <plotting positions> and the reference line. A calibrated <hypothesis test> or uncertainty band requires its own assumptions and must account for parameters estimated from the same observations. Sorting also discards the original observation order: rearranging a time series leaves its <Q-Q plot> unchanged, so the plot cannot establish <independence>.
For a <normal linear model>, R's diagnostic <normal Q-Q plot> uses <standardized regression residuals>, also called <internally studentized residuals>. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/plot.lm.html[Maindonald and Maechler, R plot.lm documentation, Details]{ref} Even with independent errors, the <regression residuals> have <covariance matrix> $\sigma^2(I-H)$, where $H$ is the <hat matrix>; fitting parameters generally creates dependencies. Dividing by a shared estimated <standard deviation> creates further dependence, and the <distribution of an internally studentized Gaussian residual> is not exactly the <standard normal distribution>. The <normal Q-Q plot> therefore helps assess error shape alongside other diagnostics rather than certifying independent <normal random variables>.
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