Normal Distribution Calculator
Compute probabilities and Z-scores for a standard normal distribution.
Normal Distribution Parameters
Understanding Normal Distribution
The normal distribution, often called the bell curve, is a common probability distribution in statistics. It is symmetrical around its mean, and its shape is determined by the mean (μ) and standard deviation (σ).
Z-score Formula:
Where:
- Z = Z-score (standard score)
- X = Data point
- μ = Mean of the population
- σ = Standard deviation of the population
Empirical Rule (68-95-99.7 Rule)
| Range from Mean | Percentage of Data |
|---|---|
| μ ± 1σ | ~68% |
| μ ± 2σ | ~95% |
| μ ± 3σ | ~99.7% |
Frequently Asked Questions
What is a normal distribution?
The normal distribution, also known as the Gaussian distribution or bell curve, is a 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. It is widely used in natural and social sciences.
What is a Z-score?
A Z-score (or standard score) measures how many standard deviations an element is from the mean. It is calculated as Z = (X - μ) / σ, where X is the data point, μ is the mean, and σ is the standard deviation. Z-scores allow for comparison of data from different normal distributions.
What is the Empirical Rule (68-95-99.7 Rule)?
The Empirical Rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.