Z-Score Calculator
Compute standard z-scores and cumulative normal distribution probabilities P(X < x) and P(X > x).
Z-score
0.5
P(X < x)
0.6915
69.15%
P(X > x)
0.3085
30.85%
z = (x - μ) / σHow to Use the Z-Score Calculator
- Enter the observed value (x).
- Enter the population mean (μ).
- Enter the standard deviation (σ), where σ > 0.
- View the resulting z-score, percentile, and left/right tail probability areas.
Formulas & Principles
Z-score: z = (x - μ) / σ
Cumulative Distribution: P(X < x) = Φ(z)
Complement Probability: P(X > x) = 1 - Φ(z)
Frequently Asked Questions
What is a Z-score in statistics?
A Z-score (standard score) measures how many standard deviations an observation (x) lies above or below the mean (μ) of a distribution: z = (x - μ) / σ.
What does a z-score of 0 mean?
A z-score of 0 means the data point is exactly equal to the population mean.
What is the 68-95-99.7 rule (Empirical Rule)?
In a normal distribution, approximately 68% of data falls within |z| ≤ 1, 95% falls within |z| ≤ 2, and 99.7% falls within |z| ≤ 3.
Why this calculator matters
This calculator helps solve mathematical problems faster and with fewer manual arithmetic mistakes. It is useful for studying, research verification, technical engineering checks, and everyday problem-solving.
When to use it
Use it when you need a fast answer, a cross-check for manual calculations, or a clear step-by-step verification of your result.
Quick tips
Double-check your inputs, keep units consistent across all fields, and verify boundary conditions.
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