qnormp

qnormp is the quantile function for a standard normal distribution. Given a probability value p, it returns the z‑score such that the probability of the cumulative distribution function (CDF) not exceeding that z‑score is equal to p. Mathematically, qnormp(p) = Φ⁻¹(p), where Φ denotes the standard normal CDF.

In computer implementations the function is typically named to emphasize its inverse‑CDF nature. For instance, in Microsoft Excel the analogous function is NORM.S.INV, in the R programming language it is qnorm, and in Python’s scipy library it is scipy.stats.norm.ppf. The qnormp name is sometimes used in specialized statistical software packages or in educational materials to highlight the relationship between probability and quantile.

Parameters:

- p: A numeric value or array of values. The function requires 0 < p < 1. Values outside this interval trigger errors or return ±∞ in vectorized contexts.

Behavior:

- For p = 0.5, qnormp returns 0, the median of the standard normal.

- For p close to 0, qnormp yields large negative values; for p close to 1, it yields large positive values.

- The function is monotonic increasing.

Applications:

- Computing confidence interval bounds for normal‑distributed estimates.

- Simulating random variates via inverse transform sampling when combined with a uniform random generator.

- Determining critical values in hypothesis testing where a normal approximation is justified.

Related functions:

- qnorm (inverse standard normal CDF)

- pnorm (standard normal CDF)

- rnorm (random sampling from a normal distribution)

History:

The concept of the inverse CDF dates back to early 20th‑century work on the normal distribution. The term qnorm, and by extension qnormp, gained common usage with the advent of statistical software in the 1980s. The choice of the “p” suffix sometimes signals compatibility with probability‑based inputs in domain‑specific libraries.

For further reading, consult textbook references on probability theory or the documentation of statistical libraries that implement the inverse normal function.