# Formal definition of the qqline used in a Q-Q plot

I'm doing some distribution fitting work and I'm looking at Q-Q plots and how they can be used visually to interpret goodness of fit.

My data is heavy-tailed so I am looking at Weibull, log-normal, Pareto and log-logistic distributions initially.

For a Weibull distribution, I understand how the points on the Q-Q plot are constructed (using the quantiles of observed data vs. the quantiles of an estimated Weibull distribution). The piece I am not clear on is how the line used in Q-Q plots is calculated/constructed.

The R documentation for the qqplot() function provides the following description:

qqnorm is a generic function the default method of which produces a normal QQ plot of the values in y. qqline adds a line to a “theoretical”, by default normal, quantile-quantile plot which passes through the probs quantiles, by default the first and third quartiles.

Another post on Cross Validated seems to indicate that the line is essentially a line constructed from the parameters of the theoretical (estimated) distribution. Is this a true statement and correct interpretation?

If a link to a formal definition could be provided I'd very much appreciate it.

• This question rather hinges on what R does. Other implementations don't necessarily make the same (default) choice. More generally, the reference line is just psychological. People would sometimes be better served by untilting the graph, and plotting residual versus fitted, as Wilk and Gnanadesikan did flag in their paper (and was an idea much used earlier by Tukey and others). May 13 '20 at 15:34

Sort of "both" - the line depends both on the observed quantiles (which define the y-axis of the QQ plot) and the expected/theoretical/reference quantiles (which the define the x-axis). The documentation (which you quote) should always be taken as the canonical reference:

‘qqline’ adds a line to a “theoretical”, by default normal, quantile-quantile plot which passes through the ‘probs’ quantiles, by default the first and third quartiles.

If in doubt, USTL ("Use the Source, Luke") , which can be found here: here's a slightly abridged and commented version

 ## quantiles (.25 and 0.75 by default) of data
y <- quantile(y, probs, names=FALSE, type=qtype, na.rm = TRUE)
## quantiles of reference/theoretical distribution
x <- distribution(probs)
## ...
slope <- diff(y)/diff(x)  ## observed slope between quantiles
int <- y[1L]-slope*x[1L]  ## intercept
abline(int, slope, ...)   ## draw the line


For what it's worth, I believe that this approach (line connecting central quantiles) is used because it fulfills the following criteria for exploratory/diagnostic approaches:

• quick (e.g. no need to run a linear regression, just find the quantiles and draw a straight line)
• robust (it only depends on the behavior of the central part of the distribution, won't be thrown off by weird tails)

I think it simply adds a line segment between the points (x1, y1) & (x2, y2) for given probabilities (p1, p2)

(x1, x2) are the quantiles of the theoretical distribution; (y1, y2) for the data comparison. Function qline has simple code under the hood. This is a simple e.g. in R

# sample data
set.seed(2)
y <- rt(100, df = 5)

# get the values
probs <- c(0.25, 0.75)
x1 <- qnorm(probs[1])
x2 <- qnorm(probs[2])
y1 <- quantile(y, probs[1])
y2 <- quantile(y, probs[2])

# plot
qqnorm(y)
segments(x1, y1, x2, y2, col = "red", lwd = 2)
qqline(y, lty = 2)
# theoretical match is straight line. If you add more samples, qqline should
# converge to this
abline(0,1)