Normal Distribution Calculator

Quick Answer

Last updated: 3 November 2026 ·How we calculate this: the standard normal cumulative distribution function

How This Normal Distribution Calculator Works

Enter the mean and standard deviation, then choose what you want to find: the area below a value, above it, between two values or outside them, or the value that has a given area below it. The calculator turns each value into a z-score, looks up the standard normal curve, and shows the probability, the z-scores and the steps.

How to Use the Normal Distribution Calculator

1. Enter the mean and standard deviation.
2. Choose the area or inverse you want.
3. Enter the value or values, or the probability for the inverse.
4. Press Calculate.
5. Read the probability, z-scores and steps.

Core Formulas

• z-score: z = (x − μ) ÷ σ
• Below a: P(X < a) = Φ(z)
• Above a: P(X > a) = 1 − Φ(z)
• Between a and b: Φ(z_b) − Φ(z_a)
• Inverse: a = μ + z × σ, where Φ(z) = p

Key Terms

• Normal distribution: a symmetric, bell-shaped probability distribution.
• Mean (μ): the center of the distribution.
• Standard deviation (σ): the spread of the distribution.
• Φ (phi): the cumulative distribution function of the standard normal.
• Percentile: the share of values below a given value.

Worked Example

Heights are normal with mean 170 cm and SD 8 cm. What share is taller than 186 cm?

• z = (186 − 170) ÷ 8 = 2
• P(X > 186) = 1 − Φ(2) ≈ 0.0228
• So about 2.28% are taller than 186 cm

Real-Life Examples

• Test scores: with mean 100 and SD 15, about 95.4% score between 70 and 130.
• Quality control: find the share of parts outside the limits.
• Cutoffs: the top 10% of a mean-500, SD-100 test starts at about 628.
• Risk: estimate the chance of a value beyond a threshold.

Normal vs Standard Normal

• Normal: any mean and standard deviation.
• Standard normal: mean 0 and standard deviation 1, written Z.
• Link: the z-score turns any normal value into a standard normal value.

Standard Normal Areas

The 68-95-99.7 Rule

• Within 1 SD: about 68.27% of values.
• Within 2 SD: about 95.45%.
• Within 3 SD: about 99.73%.
• Use it: for quick estimates before exact calculation.

Is My Data Normal?

The normal model works best when data are roughly symmetric, with most values near the middle and fewer far away. Before you rely on these probabilities, take a quick look at your data.

• Histogram: a single hump that looks about the same on both sides is a good sign.
• Mean and median: if they are close, the data are probably not strongly skewed.
• Outliers: a few extreme values can make the standard deviation too large.
• Natural limits: data that cannot go below zero, like waiting times, are often skewed.

Many real measurements, such as heights, test scores and measurement errors, are close to normal. Others, like incomes or house prices, are usually skewed, so a normal model can give misleading answers for them.

Common Mistakes to Avoid

• Using variance instead of SD. σ is the square root of the variance.
• Mixing up below and above. P(X > a) = 1 − P(X < a).
• Applying it to skewed data. The normal model fits roughly symmetric data.
• Entering a percent for the inverse. Enter 0.95, not 95.
• Forgetting continuity. P(X = a) is 0 for a continuous distribution.

Frequently Asked Questions

References & Sources