onlinecalculator.me

Free online z-score and normal distribution calculator

A z-score measures how many standard deviations a value is above or below the mean. This calculator converts between raw scores, z-scores, and percentiles for any normal.

About this calculator

How to use

  1. Select a mode — convert a raw value to a z-score, look up a z-score’s percentile, or find the z-score for a given percentile.
  2. Fill in the required fields.
  3. Click Calculate to see the full output.

Formulas

Raw value to z-score:

z = (x − μ) / σ

where x is the raw value, μ is the mean, σ is the standard deviation.

Cumulative probability (CDF):

P(Z ≤ z) = 0.5 × (1 + erf(z / √2))

Error function approximation (Hart):

erf(x) ≈ 1 − (a₁t + a₂t² + a₃t³) × exp(−x²)
where t = 1 / (1 + 0.47047|x|)
a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556

Two-tailed p-value:

p = 2 × min(P(Z ≤ z), 1 − P(Z ≤ z))

Worked example

A student scores 75 on a test with mean 70 and standard deviation 10:

z = (75 − 70) / 10 = 0.50
P(Z ≤ 0.50) ≈ 0.6915 → 69.15th percentile
two-tailed p = 2 × 0.3085 ≈ 0.617

Standard critical values:

z = 1.645 → 95th percentile (one-tailed 5%)
z = 1.960 → 97.5th percentile (two-tailed 5%)
z = 2.576 → 99.5th percentile (two-tailed 1%)

Frequently asked

What is a z-score?
A z-score (also called a standard score) is a signed number of standard deviations a data point is from the population mean. Formula: z = (x − μ) / σ.
How is the percentile calculated from a z-score?
The percentile is the cumulative probability P(Z ≤ z) from the standard normal distribution, expressed as a percentage. The calculator uses Hart's rational approximation to the error function (erf), which is accurate to about 4 decimal places.
What is a p-value?
The two-tailed p-value is 2 × min(P(Z ≤ z), P(Z > z)). It represents the probability of observing a z-score at least as extreme as the one computed, in either tail of the distribution.
What does "percentile to z-score" mode do?
Given a percentile (0–100), it returns the z-score that corresponds to that cumulative probability. For example, the 97.5th percentile corresponds to z ≈ 1.96, the critical value used in a two-tailed 95% confidence interval.
Is this calculator accurate for extreme z-scores?
The approximation used is accurate to roughly 4 decimal places for z in the range −5 to +5. At extreme values (|z| > 5), the probabilities are extremely close to 0 or 1 and small absolute errors matter little.
How do I share my calculation?
Click Share with my numbers to copy a URL that restores your selected mode and inputs.