Normal CDF & z from percentile

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Standard normal tools: approximate $\Phi(z) = P(Z \le z)$ for $Z \sim N(0,1)$, or recover the $z$ whose left-tail probability matches a percentile.

Method. CDF uses the error-function approximation in Abramowitz & Stegun (formula 7.1.26) via $\Phi(z) = \tfrac12\bigl(1+\mathrm{erf}(z/\sqrt{{2}})\bigr)$. The inverse uses a rational approximation in the style of Acklam / Beasley–Springer–Moro. These are educational approximations, not certified library math.

Mode z-score Percentile (e.g. 97.5) or probability in (0,1)

For inverse mode, values above 1 are treated as percentages (97.5 → 0.975).

What it computes. Left-tail probabilities and quantile $z$-scores for the standard normal.

Limitations. Tail accuracy degrades far from the mean; this is not a substitute for a statistical package. See the disclaimer.