Normal Distribution Calculator
Please enter a valid x value
What is a Normal Distribution Calculator?
A normal distribution calculator is a statistical tool that computes probabilities, percentiles, and other properties of the normal (Gaussian) distribution. It's essential for statistical analysis, hypothesis testing, and probability calculations.
The normal distribution is characterized by its bell-shaped curve and is defined by two parameters: the mean (μ) and standard deviation (σ). It's the most important probability distribution in statistics due to the Central Limit Theorem.
Key Features:
- Calculate probability density function (PDF) values
- Find cumulative distribution function (CDF) probabilities
- Compute probabilities for ranges of values
- Convert percentiles to x-values and vice versa
- Calculate z-scores for standardization
- Support for any mean and standard deviation
How It Works
Set Parameters
Enter mean and standard deviation
Choose Type
Select calculation method
Input Values
Enter x-values or percentiles
Calculate
Apply normal distribution formulas
Common Examples
Standard Normal (Z-score)
IQ Scores
Height Distribution
95th Percentile
Calculation Table
| Function | Formula | Purpose | Range |
|---|---|---|---|
| f(x) = (1/(σ√2π)) × e^(-½((x-μ)/σ)²) | Probability density | [0, ∞) | |
| CDF | F(x) = ∫ f(t)dt from -∞ to x | Cumulative probability | [0, 1] |
| Z-Score | Z = (X - μ) / σ | Standardization | (-∞, ∞) |
| Percentile | X = μ + σ × Z_p | Inverse CDF | (-∞, ∞) |
| Range Probability | P(a < X < b) = F(b) - F(a) | Interval probability | [0, 1] |
Frequently Asked Questions
What is the normal distribution?
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It's symmetric around the mean and is completely defined by two parameters: mean (μ) and standard deviation (σ).
What is a z-score and why is it important?
A z-score measures how many standard deviations a value is from the mean. It standardizes values, allowing comparison between different normal distributions. Z = (X - μ) / σ.
What is the 68-95-99.7 rule?
This empirical rule states that approximately 68% of values fall within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.
How do I find the probability between two values?
Use the range calculation: P(a < X < b) = CDF(b) - CDF(a). This gives the probability that a random variable falls between values a and b.
What's the difference between PDF and CDF?
PDF (Probability Density Function) gives the probability density at a point, while CDF (Cumulative Distribution Function) gives the probability of being less than or equal to a value.
How accurate are the calculations?
Our calculator uses high-precision approximations for the error function and inverse normal function, providing accuracy to 6 decimal places for most practical applications.
Can I use this for standardized test scores?
Yes! Many standardized tests (SAT, GRE, IQ tests) follow normal distributions. Enter the test's mean and standard deviation to calculate percentiles and probabilities.
What if my data isn't normally distributed?
This calculator assumes normal distribution. For non-normal data, consider data transformation or use appropriate distribution calculators for your specific data type.
How do I interpret percentiles?
A percentile indicates the percentage of values below a given point. For example, the 90th percentile means 90% of values are below this point and 10% are above.