Normal Distribution Calculator

Calculate z-scores, percentiles, and probabilities with an interactive bell curve visualization.

Loading...
1. Enter the mean and standard deviation that define your normal distribution. 2. Input a raw value to convert it to a z-score and find its corresponding percentile. 3. Alternatively, enter a percentile or z-score to find the matching raw value in your distribution. 4. Select a probability mode - below, above, or between two values - to shade the bell curve area. 5. Read the computed probability and observe the shaded region on the interactive bell curve visualization.

About This Tool

The normal distribution - also called the Gaussian distribution or bell curve - is the most important probability distribution in statistics. It describes phenomena ranging from human heights and test scores to measurement errors and stock returns. This calculator lets you work with any normal distribution by specifying the mean and standard deviation, then computes z-scores, cumulative probabilities, and percentiles with an interactive bell curve that visually highlights the area under the curve.

Enter a raw value along with the mean and standard deviation to find its z-score (how many standard deviations it is from the mean) and the corresponding percentile. Or work in reverse - enter a percentile or probability and find the z-score and raw value. The calculator also computes the probability of values falling between two points, above a threshold, or below a threshold, covering all the common scenarios encountered in statistics courses and practical applications.

The interactive bell curve visualization shades the relevant area as you adjust inputs, making abstract probability concepts concrete and intuitive. This tool is invaluable for students studying statistics, researchers performing hypothesis tests, quality control engineers setting tolerance limits, and anyone who needs to convert between z-scores, percentiles, and probabilities quickly and accurately.

Formula / How It Works

z = (x - mu) / sigma | P(X < x) = Phi(z) | Phi(z) = (1/2)(1 + erf(z / sqrt(2)))

Frequently Asked Questions

A z-score measures how many standard deviations a value is from the mean. A z-score of 0 means the value equals the mean. A z-score of 1.0 means the value is one standard deviation above the mean. Negative z-scores indicate values below the mean. Z-scores let you compare values from different normal distributions on a common scale.
Convert both values to z-scores using z = (x - mean) / standard deviation. Look up the cumulative probability for each z-score. The probability between the two values is the difference between these cumulative probabilities. This calculator handles the conversion and lookup automatically.
The empirical rule states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule provides quick estimates without needing a z-table or calculator.
A percentile indicates the percentage of values in the distribution that fall below a given value. The 90th percentile means 90% of all values are lower. In a standard normal distribution, the 50th percentile corresponds to a z-score of 0 (the mean), and the 97.5th percentile corresponds to a z-score of approximately 1.96.
Many natural measurements approximate a normal distribution when sample sizes are large, thanks to the Central Limit Theorem. Common examples include heights, weights, test scores, and measurement errors. However, income data, survival times, and count data are typically not normal. Always check your data with a histogram or normality test before assuming normality.

Related Tools

Standard Deviation Calculator

Calculate mean, variance, and standard deviation for both population and sample data sets instantly.

Statistics and Data

Probability Calculator

Calculate basic, conditional, and Bayesian probability with visual aids and step-by-step solutions.

Statistics and Data

Sample Size Calculator

Determine the required sample size for surveys given confidence level, margin of error, and population.

Statistics and Data

Mean Median Mode Calculator

Find the mean, median, mode, and range of any data set with a frequency table and distribution display.

Statistics and Data

GPA Calculator

Calculate your semester and cumulative GPA on a 4.0 or 5.0 scale with support for weighted and unweighted grades.

Education and Academic

Combination and Permutation Calculator

Calculate nCr and nPr with step-by-step factorial breakdowns. Understand when order matters.

Statistics and Data