Qnorm

qnorm is the quantile function of the normal distribution. For a given probability p between 0 and 1, qnorm(p, mu, sigma) returns the p-th percentile (the value x such that P(X ≤ x) = p) for a normal distribution with mean mu and standard deviation sigma. Equivalently, qnorm(p, mu, sigma) = mu + sigma · z_p, where z_p is the p-th quantile of the standard normal distribution.

In the standard normal case where mu equals 0 and sigma equals 1, qnorm(p, 0, 1) reduces to z_p, the inverse of the standard normal CDF Φ. Thus qnorm is the inverse function of the CDF; it undoes the operation of pnorm, the CDF of the normal distribution.

Properties and common values are straightforward. The function is monotone increasing in p, with qnorm(0.5, mu, sigma) = mu. Values near p = 0.975 or p = 0.025 correspond to mu ± 1.96·sigma, commonly used for 95% confidence intervals. Extreme tails produce larger magnitudes and may require careful numerical handling.

Uses include constructing confidence intervals, determining critical values for z-tests, and transforming standard normal quantiles to a general normal distribution in simulations or probabilistic computations. In many programming environments, qnorm is implemented with parameters for mu and sigma; if p is outside (0,1), the result may be NaN or an error, and qnorm(0) and qnorm(1) typically yield −∞ and ∞ for standardized cases.

Overall, qnorm provides a direct way to map probabilities to values in a normal distribution, serving as a fundamental tool in statistical analysis and modeling.