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The standard precedure for making a QQ-line in a normal quantile-plot is to draw a line through the first and third sample quantile.

My question is why?

If data follows a normal distribution, then sample quantiles should lie approximately on a straight line when plottet against the quantiles of $N(0,1)$.

In fact, if $X\sim N(\mu,\sigma^2)$ and $q_p$ is the $p'th$ quantile of $N(0,1)$, then we have that $$ p=P(\frac{X-\mu}{\sigma} \leq q_p)=P(X\leq \mu + \sigma q_p), $$ which means that $\mu + \sigma q_p $ is the $p'th$ quantile of $X$.

When considering this, wouldn't it make more sense to use the line $f(t)=\bar{x}+\hat{\sigma} t$ as the qqline in a normal-qqplot, rather than the line through first and third sample quantile ??

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I think the idea is that the data may contain outliers that can have a lot of influence on the standard deviation. You want to show the "robust" fit based on the interquartile range. I.e. you want the line to reflect the normal distribution that fits the bulk of the data, not the maximum likelihood estimate.

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