# An explanation of the choice of QQ-line in a normal QQ-plot

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 ??