Normal Distribution Calculator
Calculate probabilities, z-scores, and percentiles for normal and standard normal distributions
Calculator Parameters
Results
Enter values and click Calculate to see results
What is the Normal Distribution?
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about its mean. It describes many natural phenomena and is fundamental to statistics and probability theory.
Key Properties
Bell-Shaped Curve
The distribution forms a symmetric bell curve where most values cluster around the mean, with fewer values at the extremes.
68-95-99.7 Rule
Approximately 68% of values fall within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.
Standard Normal Distribution
A special case of the normal distribution with mean = 0 and standard deviation = 1. Any normal distribution can be converted to standard form using z-scores:
Where z is the z-score, X is the raw score, μ is the mean, and σ is the standard deviation.
How to Use This Calculator
Step 1: Choose Your Variable Type
Raw Score: Use when working with actual values from your distribution (e.g., test scores, heights, temperatures).
Z-Score: Use when working with standardised values or when you want to use the standard normal distribution.
Step 2: Select Calculation Type
P(X ≤ x): Find the probability that a randomly selected value is less than or equal to your specified value.
Find X given Probability: Determine what value corresponds to a specific cumulative probability.
Step 3: Enter Parameters
For raw scores, specify the mean and standard deviation of your distribution. For z-scores, these are automatically set to 0 and 1 respectively.
Example: Light Bulb Lifespan
A manufacturer produces light bulbs with an average lifespan of 1000 hours and a standard deviation of 100 hours. What’s the probability that a randomly selected bulb lasts 1200 hours or less?
Solution: Set Random Variable to “Raw Score”, Mean = 1000, Standard Deviation = 100, X Value = 1200. The result shows approximately 97.7% probability.
Frequently Asked Questions
What’s the difference between a raw score and a z-score?
A raw score is the original measurement from your data (e.g., exam score of 85). A z-score tells you how many standard deviations that raw score is from the mean. Z-scores allow you to compare values from different normal distributions.
When should I use the standard normal distribution?
Use the standard normal distribution when you’re working with z-scores, comparing standardised values, or using statistical tables. It’s also useful for hypothesis testing and confidence intervals.
What does cumulative probability mean?
Cumulative probability P(X ≤ x) represents the probability that a random variable takes a value less than or equal to x. It’s the area under the normal curve to the left of your specified value.
How accurate are these calculations?
The calculator uses mathematical approximations that are accurate to several decimal places, suitable for most practical and academic applications.
Applications of Normal Distribution
Education
Test scores, IQ measurements, and academic performance often follow normal distributions, making this calculator useful for grade analysis and standardisation.
Quality Control
Manufacturing processes use normal distribution to monitor product quality, set control limits, and calculate defect rates.
Finance
Stock returns, risk assessment, and portfolio analysis frequently assume normal distributions for modelling market behaviour.
Science & Research
Measurement errors, experimental results, and natural phenomena often follow normal patterns, making this distribution central to scientific analysis.