What is a Normal Distribution Calculator?
This calculator computes the z-score and percentile from a mean and standard deviation.
How it's calculated
z = (value − mean) ÷ standard deviation
z represents how many standard deviations a value is from the mean.
Step-by-step example
Test with mean 70, standard deviation 10 — what's the z-score for 85?
- Step 1 Apply the formula: z = (value−mean)÷standard deviation
- Step 2 Calculate: z = (85−70)÷10 = 1.5
- Step 3 A z-score of 1.5 corresponds to roughly the top 6.7%
Interpretation: The score is 1.5 standard deviations above the mean.
Why does this distribution show up everywhere?
Height, weight, test scores, and measurement error all tend to form a similar bell-curve shape, thanks to the Central Limit Theorem: whenever many independent factors combine, the result tends toward a normal distribution regardless of what the underlying individual distributions look like. Height, for instance, is influenced by hundreds of genes plus nutrition, sleep, and exercise — combine enough small independent influences, and a bell curve naturally emerges. That's exactly why it's called the 'normal' distribution.
Good to know
- Standardized test scoring is a direct application of z-scores — raw scores are rescaled to a common mean and standard deviation so that different test difficulties can be fairly compared.
- IQ scoring works the same way — calibrated to a mean of 100 and standard deviation of 15, so an IQ of 130 represents roughly the top 2.3%.
- Not everything follows a normal distribution. Income, city population, and earthquake magnitude follow long-tailed distributions (power laws) instead — assuming normality for this kind of data badly underestimates the probability of extreme events.
- That miscalculation is partly blamed for the 2008 financial crisis — models assuming normal distributions treated certain events as 'once in 10,000 years' when they actually occurred every few years, a theme explored in Nassim Taleb's 'Black Swan.'
- The normal distribution curve is also called the Gaussian curve, and it once appeared alongside Gauss's portrait on the German 10-mark banknote.
Frequently asked questions
Q. What does a z-score of 2 mean?
It's 2 standard deviations above the mean, corresponding to roughly the top 2.3%.
Q. What's the difference between standard score and percentile?
Standard score shows distance from the mean; percentile shows what fraction of values fall below yours.
Q. What if my data isn't normally distributed?
Z-score interpretation becomes inaccurate. Check the actual distribution shape first.