Normal Distribution Calculator
Convert a value or z-score to a cumulative probability, or find the z-score for a given probability.
z = (x − μ) / σ
Example: a test score of 85 with class mean 75 and standard deviation 10 gives z = 1.0.
Z-Score
Turning Any Value Into a Z-Score
The normal (bell curve) distribution shows up constantly — test scores, heights, measurement errors — and the z-score is what lets you compare a specific value against the whole distribution on one universal scale:
Where x is your value, μ (mu) is the distribution's mean, and σ (sigma) is its standard deviation. A test score of 85 in a class with a mean of 75 and a standard deviation of 10 gives z = (85−75)/10 = 1.0 — exactly one standard deviation above average.
From Z-Score to Probability (and Back)
Once you have a z-score, the standard normal distribution tells you what fraction of values fall below it — this is the cumulative distribution function (CDF), traditionally looked up in a printed z-table. This calculator computes it directly instead:
- z = 0 → 50% (exactly at the mean, by definition).
- z = 1.96 → 97.5% (the upper bound of a 95% two-tailed confidence interval).
- z = 1.645 → 95% (the boundary for a 90% two-tailed or 95% one-tailed interval).
- z = 2.576 → 99.5% (the boundary for a 99% two-tailed interval).
The "Probability → Z-Score" tab runs this backwards — given a target probability, it returns the z-score cutoff, which is exactly what you need when setting up a confidence interval or hypothesis test threshold.
How to Use This Calculator
- Value → Probability: enter your raw value, the distribution's mean, and its standard deviation — get the z-score and the probability above/below it.
- Z-Score → Probability: already have a z-score? Enter it directly.
- Probability → Z-Score: enter a target cumulative probability (e.g. 0.95) to find the corresponding z-score cutoff.
Frequently Asked Questions
What is a z-score?
How many standard deviations a value is from the mean: z = (x − μ) / σ. A z-score of 1.0 is one standard deviation above the mean; -2.0 is two below.
How do I convert a z-score into a probability?
Look up the cumulative probability under the standard normal curve up to that z-score. z=0 gives 50%, z=1.96 gives 97.5%, z=-1.96 gives 2.5%.
Why are 1.96 and 1.645 such common z-scores?
1.96 marks the 97.5th percentile (95% two-tailed confidence interval boundary); 1.645 marks the 95th percentile (90% two-tailed or 95% one-tailed). These come from the conventional 90/95/99% confidence levels used across statistics.
What does 'inverse' mean in Probability → Z-Score mode?
It runs the lookup backwards — given a target probability, it finds the z-score below which that proportion of the distribution falls. This is what a z-table's critical-value lookup does.
Does this only work for test scores or IQ?
Any approximately normally-distributed data — test scores, height, measurement error. It doesn't apply to skewed or multi-peaked data (like income, which is typically right-skewed).