Normal Distribution Calculator

For a normal distribution with mean μ and standard deviation σ, the z-score \(z = \frac{x – \mu}{\sigma}\) turns any value into a standard normal value. With IQ scores (μ = 100, σ = 15), 130 has z = 2, and P(X < 130) ≈ 0.9772: about 97.7% of people score below 130.

How to use the normal distribution calculator

1. Enter the mean μ and the standard deviation σ. Use μ = 0 and σ = 1 for the standard normal (z) distribution.
2. Choose what to find: a probability below, above, between or outside values, or the value for a percentile.
3. Enter the value(s) and press Calculate. The shaded area on the bell curve is your answer.

Formulas

• z-score: \(z = \frac{x – \mu}{\sigma}\)
• Area to the left: \(P(X < x) = \Phi(z)\), where \(\Phi\) is the standard normal cumulative distribution
• Area to the right: \(P(X > x) = 1 – \Phi(z)\)
• Between: \(P(a < X < b) = \Phi(z_{b}) – \Phi(z_{a})\)
• Percentile: \(x = \mu + z\sigma\), where \(\Phi(z) = p\)

The density is \(f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}\). For a continuous distribution, P(X < x) and P(X ≤ x) are equal.

Worked example

Exam scores are normal with μ = 70 and σ = 8. What share of students score between 60 and 80?

\(z_{60} = \frac{60 – 70}{8} = -1.25\) and \(z_{80} = \frac{80 – 70}{8} = 1.25\). Then \(\Phi(1.25) – \Phi(-1.25) = 0.89435 – 0.10565 = 0.78870\). About 78.9% of students score between 60 and 80.

The 68–95–99.7 rule

Common critical values: z = 1.645 (90% in the middle), 1.960 (95%), 2.576 (99%).

Common mistakes

• Using the variance instead of the standard deviation. If you know σ² = 225, then σ = 15.
• Forgetting that “above” means 1 minus the left area.
• Entering a percent (90) instead of a probability (0.9) for a percentile.
• Using a normal model for data that is clearly skewed.

Normal distribution practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you find a normal probability?

Convert the value to a z-score with z = (x − μ) ÷ σ, then use the standard normal cumulative distribution Φ(z) for the area to the left. Subtract from 1 for the area to the right.

What is the inverse normal?

It goes the other way: given a probability (area to the left), it finds the value x. For the 90th percentile, x = μ + 1.2816σ.

Does P(X &lt; x) equal P(X ≤ x)?

Yes. For a continuous distribution the probability of hitting one exact value is 0, so the two are the same.

How accurate are the results?

The calculator uses a high-precision error function, accurate to about 15 significant digits, so results match statistical tables and software.

Can I use this as a z-table?

Yes. Set the mean to 0 and the standard deviation to 1, then enter z as the value.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods: Normal distribution
• OpenStax Introductory Statistics 2e: The Normal Distribution
• Khan Academy: Normal distributions review

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