Percentile to Z-Score Calculator
Convert a percentile into the standard-normal z-score that sits at it — the inverse of the normal CDF.
How it works
z = Φ⁻¹(p ÷ 100) · Φ⁻¹ = inverse standard normal CDF (quantile function)
This is the inverse of the z-score-to-percentile conversion: given a cumulative proportion, it returns the z-value with exactly that much area to its left. It answers questions like "what z marks the top 10%?" and produces the critical values used to build confidence intervals and set hypothesis-test cutoffs. The computation uses Acklam’s rational approximation to the normal quantile function, accurate to roughly one part in 10⁹ across the usual range.
Worked example
For the 90th percentile: z = Φ⁻¹(0.90) = 1.2816, meaning 90% of a standard normal distribution lies below 1.28 standard deviations above the mean. The two-sided 90% critical value is Φ⁻¹(0.95) = 1.645.
Sources: Normal distribution — quantile function — Wikipedia
Frequently asked questions
What z-score corresponds to the 95th percentile?
z = 1.645. This one-sided value is the cutoff for the top 5% and underlies one-tailed 5% tests. For a two-sided 95% interval you instead want the 97.5th percentile, z = 1.96.
Why does the calculator also show a two-sided critical value?
Confidence intervals split the excluded probability across both tails. A 90% interval leaves 5% in each tail, so its critical value is the 95th-percentile z (1.645), not the 90th (1.2816). Showing both avoids a common mix-up.
Can the percentile be exactly 0 or 100?
No — the normal distribution extends infinitely in both directions, so the 0th and 100th percentiles correspond to ∓∞. This calculator requires a percentile strictly between 0 and 100.
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This calculator is for educational purposes. Double-check important results before acting on them.