Z-Score Calculator
Enter a value, mean, and standard deviation to get a z-score — a test score of 85 with mean 75 and sd 10 gives z = 1.0. Switch modes to convert any z-score into a percentile, or find the probability to the left, right, between, or outside two z-values, using the exact standard normal distribution rather than a printed table.
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Percentiles and probabilities use the exact standard normal CDF (Φ), not a printed z-table. How we calculate →
How to calculate a z-score
A z-score tells you how many standard deviations a value sits from the mean: z = (x − μ) ÷ σ, where x is your value, μ is the mean, and σ is the standard deviation. A test score of 85 with a class mean of 75 and a standard deviation of 10: z = (85 − 75) ÷ 10 = 1 — the score sits exactly one standard deviation above the mean.
A positive z-score means the value is above the mean; a negative z-score means it's below. A z-score of 0 means the value IS the mean, exactly.
Converting a z-score to a percentile
A z-score becomes a percentile using the cumulative standard normal distribution: percentile = Φ(z) × 100, where Φ(z) is the share of the normal distribution at or below z. A z-score of 1 converts to the 84.13th percentile — meaning roughly 84% of scores in a normal distribution fall at or below that value.
A z-score of 1.5 (test score 85, mean 70, sd 10) converts to the 93.32th percentile. This calculator computes Φ(z) directly with the Abramowitz & Stegun error-function approximation, not a lookup in a printed z-table, so it works for any z-value, not just the ones a table happens to list.
The 68-95-99.7 rule, checked against the exact math
The well-known "68-95-99.7 rule" says roughly 68% of values in a normal distribution fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. Running the exact calculation between z = −1 and z = 1 gives 68.27% — matching the commonly cited 68% (the exact figure is 68.27%, often rounded).
This is also why a 95% confidence interval uses a z-value close to 1.96, not exactly 2: the "between" probability from z = −1.96 to z = 1.96 is exactly 95%, and the "outside" probability (in either tail combined) is 5% — the standard 5% significance threshold used across statistics.
Left, right, between, and outside: the four probability questions a z-score answers
Once you have a z-score, four different questions come up depending on context: P(Z ≤ z) ("left-tail", e.g. what share scored at or below this?), P(Z > z) ("right-tail", what share scored higher?), P(z₁ < Z < z₂) ("between", what share falls in this range?), and P(Z < z₁ or Z > z₂) ("outside", what share falls in either extreme, used for two-tailed significance tests). All four modes are available above and share the same underlying Φ(z) calculation.
Frequently asked questions
How do you calculate a z-score?
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) ÷ σ. A value of 85 with mean 75 and standard deviation 10 gives z = (85 − 75) ÷ 10 = 1.
What percentile is a z-score of 1?
A z-score of exactly 1.0 corresponds to the 84.13th percentile — commonly rounded to "about the 84th percentile" (roughly 68% falls within ±1 SD of the mean, so 50% + 34% ≈ 84% falls at or below z = 1).
What does a negative z-score mean?
A negative z-score means the value is below the mean. A z-score of −1 corresponds to the 15.87th percentile — meaning roughly 16% of values fall at or below it.
How do I convert a percentile to a z-score?
This is the reverse operation of what the calculator does above (z to percentile): you'd need the inverse cumulative normal distribution rather than Φ(z) itself. Common reference points: the 50th percentile is z = 0, the 84th percentile is roughly z = 1, and the 97.5th percentile is roughly z = 1.96.
What z-score corresponds to a 95% confidence interval?
z ≈ 1.96 for a 95% two-tailed confidence interval — the probability between z = −1.96 and z = 1.96 is exactly 95%, leaving 2.5% in each tail (5% total in both tails combined).
What is the difference between a z-score and a percentile?
A z-score measures distance from the mean in standard deviations (can be negative, zero, or positive, with no fixed range). A percentile is the equivalent position expressed as "percent of values at or below this point" (always between 0 and 100). The two are directly convertible via the standard normal CDF, Φ(z).
Is the 68-95-99.7 rule exact?
It's a common rounding of the exact figures: 1 SD from the mean captures 68.27% (often rounded to 68%), 2 SD captures about 95.45%, and 3 SD captures about 99.73%. Use the "between" mode above for the exact percentage at any range, not just whole standard deviations.
Researched & verified by the Calcuris Data & Research Team. How we build and check our tools →