Normal Distribution
PDF, CDF and Z-score
About This Calculator
The normal (Gaussian) distribution is the most important probability distribution in statistics. Its bell-shaped curve describes countless natural phenomena — from heights and test scores to measurement errors — and underlies much of statistical inference.
Formula
Z-score: Z = (x - mu) / sigma
PDF: f(x) = (1/(sigma*sqrt(2*pi))) * e^(-(x-mu)^2 / (2*sigma^2))
Empirical rule: 68% within +-1sigma, 95% within +-2sigma, 99.7% within +-3sigma
Example Calculation
Test scores: mean=70, std dev=10; find percentile for score 85
- Z = (85 - 70) / 10 = 1.5
- Look up Z=1.5 in standard normal table
- CDF(1.5) ≈ 0.9332
Score 85 is at approximately the 93rd percentile
Z-Score Cumulative Percentiles
| Z-score | Cumulative % | Z-score | Cumulative % |
|---|---|---|---|
| -3.0 | 0.13% | +0.5 | 69.15% |
| -2.0 | 2.28% | +1.0 | 84.13% |
| -1.0 | 15.87% | +1.5 | 93.32% |
| -0.5 | 30.85% | +2.0 | 97.72% |
| 0.0 | 50.00% | +3.0 | 99.87% |
Frequently Asked Questions
What does a Z-score tell you?
A Z-score measures how many standard deviations a value is from the mean. Z=0 is the mean; Z=1 is one standard deviation above; Z=-2 is two standard deviations below. It lets you compare values from different distributions.
What is the empirical rule (68-95-99.7)?
For a normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. This provides quick probability estimates without needing a table.
Why is the normal distribution so common in nature?
The Central Limit Theorem explains why: the average of many independent random variables, regardless of their individual distribution, tends toward a normal distribution. This makes it the natural model for sums of random effects.
What is the difference between PDF and CDF?
The PDF (probability density function) gives the relative likelihood at a specific value. The CDF (cumulative distribution function) gives the probability of being at or below a specific value. CDF = integral of PDF from negative infinity to x.