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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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Comments (1)

  • Editorial team

    The tool needs two bounds, so a one-sided question such as the chance of exceeding a value is handled by putting a far-away number as the other bound. With mean 100 and standard deviation 15, the range 130 to 1000 returns 0.02, or 2.28 percent. Anything several standard deviations out is effectively the tail. Are you usually after one-sided or two-sided probabilities, and what are you measuring?

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