Normal Distribution Calculator Formula
Understand the math behind the normal distribution calculator. Each variable explained with a worked example.
Formulas Used
Z-Score
z_score = zPDF f(x)
pdf = (1 / (sigma * sqrt(2 * pi))) * pow(e, -0.5 * pow(z, 2))Deviations from Mean
deviation = abs(z)Variance
variance = pow(sigma, 2)Variables
| Variable | Description | Default |
|---|---|---|
x | Value (x) | 75 |
mu | Mean (mu) | 70 |
sigma | Standard Deviation (sigma) | 10 |
z | Derived value= (x - mu) / sigma | calculated |
How It Works
How to Use the Normal Distribution
PDF Formula
f(x) = (1 / (sigma * sqrt(2*pi))) * e^(-0.5 * ((x - mu)/sigma)^2)
The normal (Gaussian) distribution is the most important continuous distribution in statistics. It is defined by its mean (mu) and standard deviation (sigma). The z-score standardizes any value to units of standard deviations from the mean.
Worked Example
Test scores have mean 70 and SD 10. Find the z-score and PDF for a score of 75.
- 01Z = (75 - 70) / 10 = 0.5
- 02The score is 0.5 standard deviations above the mean
- 03PDF = (1 / (10 * sqrt(2*pi))) * e^(-0.5 * 0.25)
- 04= 0.03989 * 0.8825 = 0.03521
- 05Variance = 10^2 = 100
Frequently Asked Questions
What does the 68-95-99.7 rule mean?
About 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. This empirical rule applies to any normal distribution.
What is the difference between PDF and CDF?
The PDF gives the relative likelihood at a point (height of the curve). The CDF gives the cumulative probability up to that point (area under the curve to the left). The PDF value itself is not a probability.
Why is the normal distribution so important?
The Central Limit Theorem states that the sum (or mean) of many independent random variables tends toward a normal distribution, regardless of the underlying distribution. This makes it foundational for statistical inference.
Learn More
Guide
Understanding the Normal Distribution
Learn what the normal distribution is, why it matters in statistics, and how to use the bell curve for probability calculations, z-scores, and real-world data analysis.
Ready to run the numbers?
Open Normal Distribution Calculator