Enter a value, the mean, and the standard deviation to get the z-score and the matching percentile — how much of a normal distribution your value beats. Or flip it around: enter a z-score and get the raw value back. The formula substitution is shown at every step.
A value of 85 is higher than about 93.3% of values in a normal distribution with mean 70 and standard deviation 10.
How it was calculated
z = (x − μ) / σ
z = (85 − 70) / 10 = 15 / 10 = 1.5
Percentile = Φ(1.5) × 100 ≈ 93.32 (standard normal CDF)
📌 Percentiles assume the data follows a normal (bell-curve) distribution. The CDF is computed with the Abramowitz–Stegun approximation of the error function (accurate to about 7 decimal places).
Share:
The z-score formula, worked out
A z-score standardizes any value: z = (x − μ) / σ, where μ is the mean and σ the standard deviation. Say SAT-style test scores average 70 with a standard deviation of 10, and you scored 85. Then z = (85 − 70) / 10 = 1.5 — your score sits one and a half standard deviations above the mean. On the normal curve that is about the 93.3rd percentile, so you scored higher than roughly 93% of test-takers.
The reverse direction uses x = μ + zσ. If a program admits everyone above z = 2.0, the cutoff score is 70 + 2 × 10 = 90. The percentile comes from the standard normal cumulative distribution function Φ(z), which this calculator evaluates with the classic Abramowitz–Stegun error-function approximation.
Quick z-table
The table shows the percentile — the share of a normal distribution below the given z-score.
Z-score
Percentile
Higher than
−3.0
0.13
0.13% of values
−2.5
0.62
0.62% of values
−2.0
2.28
2.28% of values
−1.5
6.68
6.68% of values
−1.0
15.87
15.87% of values
−0.5
30.85
30.85% of values
0.0
50.00
50.00% of values
+0.5
69.15
69.15% of values
+1.0
84.13
84.13% of values
+1.5
93.32
93.32% of values
+2.0
97.72
97.72% of values
+2.5
99.38
99.38% of values
+3.0
99.87
99.87% of values
Frequently asked questions
A z-score of 2 means the value sits exactly 2 standard deviations above the mean. In a normal distribution that puts it at roughly the 97.7th percentile — higher than about 97.7% of all values. For example, if test scores average 70 with a standard deviation of 10, a score of 90 has z = 2 and beats nearly 98% of the class.
It depends entirely on what is being measured. For things where higher is better (test scores, income), a positive z-score is above average and z above 1 is clearly strong (84th percentile or better). For things where closer to the mean is better (manufacturing tolerances, lab values), a z-score near 0 is good and values beyond ±2 are unusual — only about 5% of a normal distribution falls outside that range.
A z-score measures distance from the mean in standard deviations and can be any number, negative or positive. A percentile tells you what share of values fall below yours, from 0 to 100. They map to each other through the normal curve: z = 0 is the 50th percentile, z = 1 is about the 84th, z = 2 about the 97.7th. The z-score is linear in the raw value; the percentile is not — one extra standard deviation matters much more near the middle than in the tails.
The percentile conversion is only accurate if the data is roughly bell-shaped. It breaks down for skewed data (income, house prices, waiting times), bounded data with many values at the edge (percent scores near 100), small samples, or data with heavy tails or multiple peaks. The z-score itself — distance from the mean in standard deviations — can still be computed, but reading a percentile from it would be misleading in those cases.
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) / σ. With x = 85, μ = 70, σ = 10: z = (85 − 70) / 10 = 1.5. To go the other way, multiply the z-score by the standard deviation and add the mean: x = μ + zσ = 70 + 1.5 × 10 = 85.
Cite this calculator
APACalcBrew. (2026). Z-Score Calculator with Percentile. https://calcbrew.com/z-score-calculator
MLA“Z-Score Calculator with Percentile.” CalcBrew, 2026, calcbrew.com/z-score-calculator.
Link to https://calcbrew.com/z-score-calculator. Results are estimates; cite the underlying formula or data source for academic work.