Normal Distribution Calculator
Calculate normal-distribution probabilities, z-scores and percentile position for a value or interval
Probability
How we calculated it
About the Normal Distribution Calculator
Normal Distribution Calculator converts values to z-scores or finds probabilities under a bell-shaped normal model. It is useful for standardized measurements, quality control and statistical inference when normality is a reasonable approximation.
Not every data set is normally distributed. Strong skew, heavy tails or multiple clusters can make normal-model probabilities misleading.
How to Use the Normal Distribution Calculator
- Enter the mean and standard deviation of the normal distribution.
- Enter the x value, z-score or probability target.
- Choose whether you need left-tail, right-tail or interval probability.
- Calculate and interpret the result only if a normal model is appropriate.
How the Calculation Works
Standardization uses z = (x−μ)/σ, converting the value to standard-normal units.
Probability is then obtained from the cumulative standard normal distribution Φ(z). Interval probabilities are differences between cumulative probabilities at the two endpoints.
Worked example
Normal Distribution Calculator Worked Example
x=85, mean 70, standard deviation 10
z=(85−70)/10=1.5. Under a standard normal model, the cumulative probability below z=1.5 is about 0.9332.
Result guide
Understanding Your Normal Distribution Calculator Result
A cumulative probability of 0.9332 means about 93.32% of the modeled normal distribution lies below that value, not that the individual observation has a 93.32% chance of being correct.
Before you rely on the answer
Assumptions and Limitations
- Standard deviation is positive.
- The normal distribution is an appropriate model.
- Mean and standard deviation describe the same population or scenario as the value being evaluated.
Practical tip
Using This Result in Practice
Use a histogram, Q-Q plot or domain knowledge before applying normal probabilities to real data. A convenient bell curve is not evidence that the data are normal.
Related calculations
Useful Calculations to Continue With
If you only need to standardize an observation, use the Z-Score Calculator.
If normal critical values are being used for interval estimation, use the Confidence Interval Calculator.
Common questions
Normal Distribution Calculator FAQ
What is a normal distribution?
A symmetric bell-shaped probability distribution defined by its mean and standard deviation.
What does a z-score represent?
The number of standard deviations a value lies above or below the mean.
Can I assume every large data set is normal?
No. Sample size does not automatically make the raw data distribution normal.