Normal Distribution Calculator

Find probabilities for a normal distribution, or work backward from a percentile to a value. Enter your values below, and see the formula, a worked example, and ready-to-use Python code underneath.

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Mean (μ)

Std. deviation (σ)

Find

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Result

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What the normal distribution tells you

The normal distribution is the classic symmetric, bell-shaped curve defined entirely by its mean (center) and standard deviation (spread). Once you know μ and σ, you can find the probability of landing in any range of values, or work in reverse to find the value at any given percentile.

If you just need to standardize a single value into a z-score and percentile, the z-score calculator is a quicker, more focused tool. This calculator is for working with ranges and the inverse direction.

Normal distribution formula

z = (x − μ) / σ
P(X ≤ x) = Φ(z), where Φ is the standard normal cumulative distribution function

Every normal-distribution question reduces to converting values to z-scores and reading probabilities off the standard normal distribution.

Python code

from scipy.stats import norm
mu, sigma = 100, 15

norm.cdf(130, mu, sigma)              # P(X \u2264 130)
1 - norm.cdf(130, mu, sigma)          # P(X \u2265 130)
norm.cdf(115, mu, sigma) - norm.cdf(85, mu, sigma)  # P(85 \u2264 X \u2264 115)
norm.ppf(0.90, mu, sigma)             # value at the 90th percentile

Worked example

IQ scores are modeled as normal with μ = 100 and σ = 15. What proportion of people score above 130 ("gifted" range)?

1
Standardize the value.
z = (130 − 100) / 15 = 2.00
2
Look up the cumulative probability.
Φ(2.00) = 0.9772
3
Find the upper-tail probability.
P(X ≥ 130) = 1 − 0.9772 = 0.0228

About 2.28% of people score above 130. For context, the classic 68–95–99.7 rule shows up here too: about 68.27% of scores fall between 85 and 115 (within 1 SD of the mean).

Frequently asked questions

What is the 68-95-99.7 rule?

In a normal distribution, about 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2 standard deviations, and about 99.7% fall within 3 standard deviations. It's a quick way to sanity-check normal distribution results.

How is this different from the z-score calculator?

The z-score calculator converts one specific value into a standardized z-score and percentile. This calculator works more generally: probabilities for ranges of values, and the reverse direction — finding the value that corresponds to a given percentile.

What real-world data tends to follow a normal distribution?

Many naturally occurring measurements are approximately normal, such as heights, standardized test scores, and measurement errors. Many phenomena are not normal, though — income and city population sizes, for example, are typically right-skewed — so it's worth checking the shape of your actual data before assuming normality.

Can the normal distribution produce negative values?

Mathematically, yes — the normal distribution extends from negative infinity to positive infinity. Whether negative values are realistic depends on what you're modeling; a normal model of, say, weight or height is only an approximation, since those quantities can't truly go below zero.