About this calculator
Normal Distribution Calculator is a professional statistical analysis tool for calculating probabilities, quantiles and statistics of the normal distribution (Gaussian distribution). The normal distribution is the most important continuous probability distribution, which is a bell-shaped curve and is completely determined by the mean μ and the standard deviation σ. Many random variables in nature and social phenomena approximately obey a normal distribution, such as height, weight, test scores, measurement errors, etc. This calculator supports standard normal distribution and general normal distribution, and can calculate probability, quantiles, confidence intervals, etc. It is an essential tool for statistical analysis and data processing.
What it calculates
The normal distribution calculator finds probabilities, z-scores, or interval areas from a mean and standard deviation.
Formula
- z = (x - mu) / sigma.
- Standard normal density phi(z) = 1/sqrt(2pi) * e^(-z^2/2).
Inputs
- Mean mu.
- Standard deviation sigma, which must be greater than 0.
- Target x value, z value, or interval.
Example
| mu | sigma | x | z |
|---|---|---|---|
| 100 | 15 | 115 | 1 |
| 0 | 1 | 1.96 | 1.96 |
| 50 | 10 | 40 | -1 |
How to interpret the result
A z-score tells how many standard deviations a value is from the mean. Probability area gives the proportion within a range.
Common mistakes
- Standard deviation cannot be 0.
- Distinguish left-tail, right-tail, and interval probabilities.
- Not all data follows a normal distribution.
How to use
Use the normal distribution calculator:
1. Select a distribution type: • Standard normal distribution (μ=0, σ=1) • General normal distribution (customized μ and σ) 2. Enter parameters: • Mean μ • Standard deviation σ (σ>0) 3. Select the calculation type: • P(X<x): Probability of being less than x • P(X>x): Probability greater than x • P(a<X<b): interval probability • Quantile: Find the value of x given the probability 4. Enter the value 5. View results and distribution plots
Main features
• Two-way calculation: probability → quantile, quantile → probability • Standardization: Automatic conversion to standard normal distribution • Distribution plots: probability density curves and cumulative distributions • Confidence Intervals: Calculate common confidence intervals • Z-score: Calculates the standardized score • Percentile: Calculate any percentile • Normality test: provides test methods • Totally free: unlimited use
Use cases
• Statistical inference: hypothesis testing and confidence intervals • Quality Control: Process Capability Analysis • Exam Analysis: Score Distribution and Ranking • Risk assessment: VaR calculation • Data analysis: normality test • Scientific research: experimental data analysis • Financial Analysis: Yield Distribution • Teaching Demonstration: Explaining the Normal Distribution