Normal Distribution Calculator
Calculate the Z-score and cumulative probability for any value in a normal distribution with given mean and standard deviation.
How to Use the Normal Distribution Calculator
- Enter the mean (μ) and standard deviation (σ) of the distribution.
- Enter the value (x) you want to evaluate.
- Click Calculate to see the Z-score and cumulative probability.
Quick Reference
| From | To |
|---|---|
| Z = 0 | 50th percentile |
| Z = 1 | 84.13th percentile |
| Z = 1.96 | 97.5th percentile |
| Z = 2 | 97.72nd percentile |
| Z = 3 | 99.87th percentile |
Use Cases
- •Finding percentiles in standardized test scores.
- •Evaluating probabilities in quality control processes.
- •Computing confidence intervals in statistical analysis.
Formula
Z = (X − μ) / σ. CDF gives P(X ≤ x). Uses the Abramowitz-Stegun approximation for Φ(z).
Frequently Asked Questions
What is a Z-score?
The number of standard deviations a value is from the mean. Z = 0 means the value equals the mean.
What does the CDF tell me?
The probability that a random value from the distribution is less than or equal to x.
What is the 68-95-99.7 rule?
In a normal distribution, ~68% of data falls within ±1σ, ~95% within ±2σ, and ~99.7% within ±3σ.