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Probability and statistics

Normal distribution probability

The probability that a normally distributed value falls below, above or between given values, from the mean and standard deviation.

P(a<X<b)=Φ(b−μσ)−Φ(a−μσ)
Probability densityf(x)=1σ2πe−(x−μ)22σ2
Below bP(X<b)=Φ(b−μσ)
Above aP(X>a)=1−Φ(a−μσ)

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

55708510011513014597.7%
  • P(X < 130) ≈ 0.9772 (z from −4 to 2)
  • Ticks mark the mean and 1, 2 and 3 standard deviations either side; about 68%, 95% and 99.7% of values lie within 1, 2 and 3 of them.

What the symbols mean

P
the probability, between 0 and 1
μ
the mean
σ
the standard deviation (greater than 0)
a
the lower bound (leave it empty for "below b")
b
the upper bound (leave it empty for "above a")
Φ
the standard normal cumulative distribution: Φ(z) is the probability of a value below z

When to use it

Use it for quantities that cluster symmetrically around an average: heights, measurement errors, exam scores, manufacturing tolerances. Enter only b for "less than", only a for "more than", or both for "between". Solving for b works backwards from a probability to a value, such as the score needed to be in the top 10%. The calculator computes Φ to about 15 significant figures, so it needs no z-table.

Worked example

IQ scores are normally distributed with mean 100 and standard deviation 15. What proportion of people have an IQ below 130? (μ = 100, σ = 15, b = 130)

  1. Convert each bound to a z-score, z=x−μσ, then read the standard normal CDF Φ(z), the probability of falling below z.

  2. Upper bound:

    zb=x−μσ
    =130−10015
    =3015
    =2

  3. The probability of a value below b:

    P(X<130)=Φ(2)
    ≈0.97725

  4. As a percentage, about 97.72%.

Answer P(X < 130) ≈ 0.9772

Why it works

Every normal distribution is the standard one stretched by σ and shifted by μ, so converting to z-scores reduces every question to areas under one curve. The area below z has no formula in elementary functions, which is why tables, and here a numerical method, are used.

Common mistakes

  • Forgetting to subtract from 1 for "more than": P(X>a)=1−Φ(z).
  • Using the variance where the standard deviation belongs: if σ² = 225, then σ = 15.
  • Applying it to clearly skewed data, such as incomes or waiting times, where the normal model gives wrong tails.
  • Worrying about < versus ≤: for a continuous distribution they give the same probability.