Normal Distribution Calculator
Find the cumulative probability below a value for a normal distribution.
Normal distribution probability
The calculator standardizes x into a z-score and estimates the cumulative normal probability P(X ≤ x).
Understanding the Normal Distribution
The normal distribution is a symmetric bell-shaped probability model defined by a mean and standard deviation. It is commonly used to describe standardized measurements and sampling behavior.
Normal Distribution Example
If a variable has mean 100 and standard deviation 15, a value of 115 is one standard deviation above the mean. Converting the value to a z-score helps estimate the proportion of the distribution below or above it.
How to Use the Result
Normal probabilities can help with percentiles, standardized scores and statistical inference. Use the model only when a normal approximation is reasonable for the data being analyzed.
Frequently Asked Questions
What does the mean control?
The mean sets the center of the distribution.
What does standard deviation control?
It determines how spread out values are around the mean.
Is every data set normally distributed?
No. Some data are skewed or otherwise not well represented by a normal curve.
Normal Distribution and Z Scores
A normal distribution is symmetric and defined by its mean and standard deviation. Standardizing a value with a z score expresses its distance from the mean in standard-deviation units. Real datasets are not automatically normal, so distributional assumptions should be checked when they matter to an analysis.
Methodology, Assumptions and Interpretation
Normal-distribution probabilities are commonly calculated by standardizing x to z=(x−μ)/σ. The model is symmetric and continuous, but real data should not be assumed normal solely because mean and standard deviation are available.