Normal Distribution Calculator

Find probabilities for any normal distribution: P(X < a), P(X > a), P(a < X < b) or outside a range. Or work backwards from a probability to the value of x (inverse normal, like invNorm on a TI-84).

Probability
0.8413
As a percentage
84.13%
z for a
1
115

Step-by-step working

  1. Standardise: z = (x − μ) ÷ σz = (115 − 100) ÷ 15 = 1
  2. Area to the left of zP(X < 115) = Φ(1) = 0.8413

Formula P(X < a) = Φ((a − μ) ÷ σ) x = μ + z·σ

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An AI tutor reads your inputs and results above and explains them in plain English. The numbers come from the calculator; the explanation is AI-written, so check it against your notes.

The bell curve in standard deviations

Every normal curve is the same shape once you measure in standard deviations. That’s what a z-score does: it counts how many SDs a value sits from the mean.

−1+1

68.27% of values lie within ±1 SD of the mean

That is the 68–95–99.7 rule (the empirical rule). Try a single z below: the shaded area to its left is the proportion of values below it, which is its percentile.

1.20

Worked example

IQ scores are designed to have mean 100 and SD 15. What proportion score between 115 and 130? Standardise: z₁ = (115 − 100) ÷ 15 = 1, z₂ = (130 − 100) ÷ 15 = 2. Then Φ(2) − Φ(1) = 0.9772 − 0.8413 = 0.1359, about 13.6%.

Inverse: what IQ is the 95th percentile? z = 1.645, so x = 100 + 1.645 × 15 ≈ 124.7.

Continuous vs discrete

For a continuous distribution, P(X = a) is 0, so < and ≤ give the same answer. If you are approximating a discrete count (like a binomial) with a normal curve, apply a continuity correction of ±0.5, or use the exact binomial calculator.

Frequently asked questions

What is the TI-84 equivalent?

normalcdf(lower, upper, μ, σ) for areas and invNorm(area, μ, σ) for the inverse. Use −1E99 or 1E99 for an open end.

What is the standard normal distribution?

The normal distribution with mean 0 and standard deviation 1. Any normal variable becomes standard normal after converting to z-scores.