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Normal Distribution Probability Calculator

Find P(a < X < b) for a normal variable

How it is calculated

Each bound is turned into a z-score, z = (x − μ) / σ, where μ is the mean and σ the standard deviation, both in the same unit as the bounds. The probability is the area between them: P = Φ(zb) − Φ(za). Φ is evaluated numerically with the Abramowitz and Stegun erf approximation, which is accurate to a few decimal places for the two-decimal display. The standard deviation must be above zero. If you enter the bounds in the wrong order the tool swaps them. The result is shown as a decimal and as a percent.

Worked example

Scores have mean 100 and standard deviation 15; find P(85 < X < 115). Lower z: (85 − 100) / 15 = −1. Upper z: (115 − 100) / 15 = 1. Φ(1) − Φ(−1) ≈ 0.8413 − 0.1587 = 0.6827. The tool shows 0.68 (68.27%), the familiar one-standard-deviation band.

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Common mistakes

  • Entering the variance instead of the standard deviation; if the variance is 225, the standard deviation to enter is 15.
  • Expecting P(X = a) for a single value: for a continuous normal variable any single point has probability zero, so use a range.
  • Using a normal model for data that is clearly skewed or bounded, such as incomes or waiting times; the answer is only as good as the assumption.
  • Treating the bounds as z-scores. They are raw values in the original units; the tool does the standardising.
  • Reading 0.68 as exact: the first number is rounded to two decimals, so use the percent for a bit more precision.

About this calculator

This calculator finds P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ), where Φ is the standard normal cumulative distribution function, μ is the mean and σ is the standard deviation — it converts your two bounds into z-scores and looks up the area under the bell curve between them.

Statisticians, quality-control engineers and students use it whenever a real-world quantity is roughly normally distributed — exam scores, manufacturing tolerances, measurement error — to answer questions like "what fraction of parts fall within spec?" or "what percentage of students scored between a B and an A," without needing z-tables or calculus by hand.

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