Normal Distribution Calculator
Calculate probabilities, percentiles, and z-scores for the normal distribution. Enter the mean, standard deviation, and the value(s) of interest to compute probabilities and visualize the distribution.
Results
| Z-score(s) | - |
|---|---|
| Probability | - |
| Percentile | - |
| Value(s) | - |
About Normal Distribution Calculator
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.
How Normal Distribution is Calculated
The probability density function (PDF) of the normal distribution is given by:
f(x) = (1/√(2πσ²)) * e^(-(x-μ)²/(2σ²))
Where:
- μ is the mean of the distribution
- σ is the standard deviation
- e is the base of the natural logarithm (approximately 2.71828)
Understanding Z-Scores
A z-score (or standard score) represents the number of standard deviations a data point is from the mean. The formula for calculating a z-score is:
z = (x - μ) / σ
Where:
- x is the raw score
- μ is the mean of the population
- σ is the standard deviation of the population
Applications of Normal Distribution
The normal distribution is widely used in statistics and many real-world applications:
- Quality control: Manufacturing processes often assume normal distribution of product measurements
- Finance: Stock returns are often modeled with normal distribution (though real returns often have "fat tails")
- Social sciences: Many psychological and educational measurements follow normal distributions
- Natural phenomena: Heights, weights, blood pressure, and many other biological measurements
Standard Normal Distribution
The standard normal distribution is a special case where μ = 0 and σ = 1. Any normal distribution can be converted to the standard normal distribution using z-scores, allowing probabilities to be looked up in standard normal tables.