Standard Normal Percent Point (Quantile) Calculator
Find the z value from a probability p and tail type (lower, upper, or two sided). This standard normal quantile tool computes critical values such as 1.96 with high precision.
Input
Enter a probability p and a tail type to find the standard normal percent point (quantile) z value, including critical values used for confidence intervals.
Tail type
Returns the z whose probability of being at or below it equals p.
Enter a value greater than 0 and less than 1. Example: 0.975
Result
Percent point z
1.95996398
Lower probability
97.5 percent
Upper probability
2.5 percent
Density phi of z
0.05844507
Standard normal distribution
Density PDF
Cumulative CDF
Common percent points (lower probability)
Lower probability
z value
90 percent
1.28155157
95 percent
1.64485363
97.5 percent
1.95996398
99 percent
2.32634787
99.5 percent
2.5758293
99.9 percent
3.09023231
How it works
- The standard normal percent point is the value z at which the cumulative distribution function Phi of z equals a given probability. It is also called the inverse CDF or quantile function.
- Choosing lower probability returns z such that Phi of z equals p, meaning the probability of being at or below z is p.
- Choosing upper probability returns z such that the probability of being at or above z is p, that is Phi of z equals 1 minus p.
- Choosing two sided returns the positive critical value whose two tails together hold probability p, computed from Phi of z equals 1 minus p over 2.
- The value is found as z equals square root of 2 times erfinv of 2p minus 1, using the inverse error function with a Winitzki approximation refined by Newton iteration.
- Common results include z near 1.959964 for a lower probability of 0.975 and z near 2.575829 for 0.995, matching the familiar 1.96 and 2.576 used for confidence intervals.
- Enter a probability p strictly between 0 and 1. Values of 0, 1, or outside this range cannot be computed.
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Standard Normal Percent Point (Quantile) Calculator