Normal Distribution Z and Percentile Calculator

Normal distribution: z-score and percentile

The z-score standardises a value x drawn from a normal distribution with mean μ and standard deviation σ: Z = (x − μ) / σ. Its cumulative function Φ(z) gives you the percentile, which is the area under the curve sitting to the left of z. A couple of landmarks worth memorising: z = 1.96 puts Φ ≈ 0.975 (the 97.5th percentile, which underpins the 95% two-sided confidence interval), while z = 2.576 reaches 99.5%. The 68–95–99.7 rule sums it up nicely. Roughly 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. Take IQ ~ N(100, 15). At x = 120 you get z = (120 − 100)/15 ≈ 1.333, so Φ(1.333) ≈ 0.909, landing at the 90.9th percentile.

Applications

It drives hypothesis testing and confidence intervals, and it underlies IQ scales and standardized test scoring (ENEM, for instance, uses TRI with mean 500 and σ = 100). You'll also see it in six-sigma quality control, in financial risk through Value-at-Risk, and when standardizing features for machine learning.

FAQ

What is a "good" z-score? That depends entirely on what you're doing. When hunting for outliers, anything with |z| > 3 counts as unusual. In a two-sided hypothesis test at α = 0.05, |z| > 1.96 is enough to reject the null.

Why is Φ not in closed form? Because the normal CDF is the integral of e^(−t²/2), and that integral has no elementary antiderivative. In practice people fall back on tables, on numerical approximations like Abramowitz & Stegun, and on the erf function.

Z-test or t-test? Go with z when σ is actually known (which is rare) or when the sample is large enough that n > 30 and the central limit theorem kicks in. In every other case, Student's t is the right call.

What if data isn't normal? First confirm it with a Q-Q plot or a Shapiro-Wilk test. A transformation such as log or Box-Cox often pulls the data back toward normal; when nothing helps, switch to non-parametric tests.