Normal Distribution Calculator
Calculate z-scores, probabilities, and values for normal distributions
Z = (X - μ) / σ
Z-Score Formula
Quick Examples
Z-Score
Percentile:
Raw Score (X)
Percentile:
Probability
Z-Score:
Bell Curve Position
Mean (μ)
Std Dev (σ)
PDF Value
Probability
Common Z-Scores Reference
| Z-Score | Percentile | Meaning |
|---|---|---|
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About Normal Distribution Calculator
What is Normal Distribution?
Normal distribution (also called Gaussian distribution or bell curve) is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent than data far from the mean.
Key Concepts
Z-Score (Standard Score)
The z-score measures how many standard deviations a value is from the mean.
Formula: Z = (X - μ) / σ
Where:
- X = raw score
- μ = population mean
- σ = population standard deviation
Standard Normal Distribution
A special case where μ = 0 and σ = 1. Any normal distribution can be converted to standard normal using z-scores.
Probability Density Function (PDF)
The PDF describes the likelihood of a random variable taking on a given value.
Formula: f(x) = (1 / (σ√(2π))) × e^(-(x-μ)²/(2σ²))
Common Z-Score Values
| Z-Score | Percentile | Meaning |
|---|---|---|
| -3.0 | 0.13% | 3 SD below mean |
| -2.0 | 2.28% | 2 SD below mean |
| -1.0 | 15.87% | 1 SD below mean |
| 0.0 | 50.00% | At the mean |
| 1.0 | 84.13% | 1 SD above mean |
| 2.0 | 97.72% | 2 SD above mean |
| 3.0 | 99.87% | 3 SD above mean |
The Empirical Rule (68-95-99.7)
For any normal distribution:
- 68% of data falls within 1 standard deviation of the mean
- 95% of data falls within 2 standard deviations
- 99.7% of data falls within 3 standard deviations
Practical Applications
- Education - Standardized test scores (SAT, IQ tests)
- Quality Control - Manufacturing tolerances
- Finance - Stock returns and risk analysis
- Science - Measurement errors and experimental data
- Healthcare - Blood pressure, height, weight distributions
Note: This calculator provides approximations. For critical applications, consult a statistician.