How this Q-Q plot generator works

This tool sorts your sample into order statistics \(x_{(1)}, \dots, x_{(n)}\) and pairs each one with a corresponding theoretical quantile from the chosen distribution. For the normal distribution, we first estimate the sample mean \(\hat{\mu}\) and standard deviation \(\hat{\sigma}\), then transform standard normal quantiles to match.

Formula for the hyperbolic plotting positions

We use the popular rule:

Then the theoretical quantile is \(q_i = F^{-1}(p_i)\), where \(F^{-1}\) is the inverse CDF of the selected distribution:

• Normal: \(q_i = \hat{\mu} + \hat{\sigma} \Phi^{-1}(p_i)\)
• Uniform(0,1): \(q_i = p_i\)
• Exponential(1): \(q_i = -\ln(1 - p_i)\)

Interpretation tips

• Points ~ straight line → good fit.
• Concave up/down → skewness.
• Ends deviating → heavy or light tails than expected.