Normal Distribution Percent Point (Quantile) Calculator

Find the percent point x and z-score of a normal distribution N(mu, sigma squared) for a probability p, using lower, upper, or two-sided tails.

Input

For a normal distribution N(mu, sigma squared), find the percent point (quantile) x and z-score for a given probability. Choose lower-tail, upper-tail, or two-sided.

Probability mode

Finds the point x whose cumulative probability, the area from the left, equals p.

Probability p

A value greater than 0 and less than 1, such as 0.975

Mean mu

Center of the distribution

Standard deviation sigma

Enter a positive value

Result

Percent point x

1.95996398

z-score

1.95996398

Input probability p

0.975

Cumulative probability at the point

0.975

Mean mu

Standard deviation sigma

Probability density and selected area

The curve is the standard normal probability density. The shaded region shows the probability area for the selected mode.

How it works

The standard normal cumulative distribution function Phi(z) is computed from the error function erf, evaluated by a Maclaurin series for small z and a continued fraction for large z.
The inverse function Phi inverse of p uses Acklam rational approximation for an initial value, refined by one Newton step.
Lower-tail mode finds the point whose cumulative probability equals p. Upper-tail mode finds the point whose upper probability equals p, so its cumulative probability is 1 minus p.
Two-sided mode finds the symmetric interval that contains probability p in the center. Each tail holds probability (1 minus p) divided by 2.
A point of a general normal distribution is x equals mu plus sigma times z, converting from the standard normal z.
Enter a probability p strictly between 0 and 1, and a positive standard deviation sigma.

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