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๐Ÿ“Š Probability & Statistics

Z-Score & Normal Distribution Calculator

Convert raw scores into standard Z-scores, calculate p-values, percentiles, and visualize shaded cumulative probabilities on an interactive standard bell curve.

Presets:

Parameters

Z-Score
+1.0000
Standard score
P(Z < z) Left
0.8413
84.13% Percentile
P(Z > z) Right
0.1587
15.87% Above
Two-Tailed P
0.3173
p-value (2-tail)

๐Ÿ”” Standard Normal Distribution Bell Curve (N(0, 1))

Shaded: P(Z < 1.00)

๐Ÿ“ Statistical Step-by-Step Derivation

How to Use This Calculator

  1. Enter the raw score X, the population mean μ, and the standard deviation σ.
  2. Click Calculate Z-Score & Probability to compute z = (X − μ) / σ, the number of standard deviations X sits above or below the mean.
  3. Read the percentile and probabilities: P(Z < z) is the percentile rank of the score, and P(Z > z) is the proportion of the distribution above it.
  4. Use the two-tailed p-value when running a two-sided hypothesis test on your z-score.
  5. Try the presets (IQ, SAT, critical z) to see how different scales map onto the same standard curve.

Formula & Definition of Z-Score

The standard score, or Z-score, is the number of standard deviations by which the value of a raw score x is above or below the mean value of what is being observed. The formula is:

Z = (X - ฮผ) / ฯƒ

For samples with sample mean xฬ„ and sample standard deviation s, the formula is equivalently Z = (X - xฬ„) / s.

Worked Example

A student scores X = 115 on a test with mean ฮผ = 100 and standard deviation ฯƒ = 15. z = (115 โˆ’ 100) / 15 = 1.00, so the score is exactly 1 standard deviation above the mean โ€” the 84.13th percentile (P(Z < 1) โ‰ˆ 0.8413).

Common Questions & Answers

Z-Score Calculator FAQs

What does a Z-score represent in statistics?
A Z-score (standard score) indicates how many standard deviations an observation or raw score is above or below the population mean. A positive Z-score indicates a value above the mean, while a negative Z-score indicates a value below the mean.
What is the empirical rule (68-95-99.7 rule)?
In a normal distribution, approximately 68% of data lies within ±1 standard deviation of the mean, 95% lies within ±2 standard deviations, and 99.7% lies within ±3 standard deviations.

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Reviewed & updated: October 2026 Maintained by the OnlineResult Study Tools Team Report an issue
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