Probability · Distributions
Inverse Normal Distribution Calculator
Inverse normal — invNorm — runs the bell curve backwards: give it a probability and it returns the value (or z-score) that carves off exactly that area. It is how critical values and percentile cut-offs are found — the central 95% of a normal distribution lies between z = ±1.96; the 95th percentile sits at z = 1.645. This finds the cut-off for an area on the left, on the right, or split symmetrically in the middle, for any mean and standard deviation, with the curve shaded.
Central finds the symmetric ±cut-off enclosing that middle area (the two-tailed critical value). Left treats it as the area below the cut-off (a percentile); right as the area above. Defaults to the standard normal (μ = 0, σ = 1), where the answer is a pure z-score.
Result
In plain English
The ordinary normal calculation goes “value in, probability out” — how much of the curve lies below 115? The inverse goes the other way: “probability in, value out” — what value has 95% of the curve below it? Mathematically it is the quantile function, the inverse of the cumulative curve, and it is the tool behind every critical value, percentile and confidence-interval boundary in statistics.
- inverse normal / invNorm
- The function that turns an area into the value (or z-score) that bounds it: the inverse of the cumulative distribution, also called the quantile function or probit.
- z-score, then value
- For the standard normal the answer is a z-score; for a general N(μ, σ) the value is x = μ + σ·z. Set μ = 0 and σ = 1 to read z-scores directly.
- left vs right vs central
- The same area means different cut-offs depending on where it sits. Left: area below the value (a percentile). Right: area above (an upper-tail critical value). Central: the symmetric middle band, giving a ± pair.
- critical values
- The central mode is how two-tailed test and confidence-interval critical values come about: central 90% → ±1.645, central 95% → ±1.96, central 99% → ±2.576.
- the catch with central
- A central area of C leaves (1 − C)∕2 in each tail, so the ± cut-off is the (1 + C)∕2 percentile — 95% central uses the 97.5th percentile, not the 95th. Slipping between the two is a classic error.
- only as normal as the data
- These cut-offs assume the distribution really is normal. For skewed or heavy-tailed data the true percentile can sit well away from where the bell curve places it, especially far out in the tails.
Frequently asked
What does invNorm do?
invNorm is the inverse of the normal cumulative function: you give it a probability (an area under the curve) and it returns the value that has that much area to its left. On a calculator, invNorm(0.975) ≈ 1.96 means 97.5% of the standard normal lies below 1.96. With a mean and standard deviation supplied, it returns a value on that scale, x = μ + σ·invNorm(area); with μ = 0 and σ = 1 it returns a z-score. This page adds left, right and central (two-tailed) framings so you do not have to convert by hand.
How do I find the z-score for a 95% confidence interval?
Use the central mode with area 0.95: the answer is ±1.96. The reason it is 1.96 and not the 95th-percentile 1.645 is that a 95% central interval leaves 2.5% in each tail, so each bound is the 97.5th percentile — invNorm(0.975) = 1.96. For 90% confidence the critical value is ±1.645, and for 99% it is ±2.576. These are the multipliers you put on the standard error to build the interval.
What is the difference between the left-tail and central answers?
Left-tail invNorm(0.95) = 1.645 is the value with 95% below it — the 95th percentile, a one-sided cut-off. Central 0.95 gives ±1.96, the pair that encloses the middle 95% with 2.5% spilling into each tail. They answer different questions: a one-tailed test or a percentile uses the left (or right) value; a two-tailed test or a symmetric confidence interval uses the central pair. Picking the wrong one is among the most common mistakes in setting critical values.
How is this different from a z-score calculator?
A z-score calculator mostly goes value-to-score-to-percentile and back for a single point. This is built around the reverse, area-to-cut-off, direction and adds the two-tailed central mode that critical values need — give it a confidence level and it hands you the ± multiplier directly. For the everyday “what percentile is this score” question the z-score tool is more direct; for “what cut-off encloses this much probability” this one is.