Normal (Bell Curve) Area Calculator

The area under a normal (bell) curve between two points equals the probability of a value falling in that range — a core calculation in statistics, standardized testing, and quality control. This calculator finds that area between two z-scores directly, or between two raw values once you supply the mean and standard deviation (it converts them to z-scores for you).

The same two modes also cover left-tail and right-tail areas — simply use a very large negative or positive number (such as −99 or 99) to stand in for negative or positive infinity as one of your two bounds.

What the Area Under the Curve Represents

The total area under any normal curve is always 1 (100%), no matter its mean or standard deviation. The area between any two points on the curve is the probability that a randomly drawn value from that distribution falls between them — so an area of 0.95 means a 95% probability (or, equivalently, 95% of all values fall in that range).

By Z-Scores

Area = Φ(upper) − Φ(lower)

Where:
Φ = the standard normal cumulative distribution function (CDF), the area under the curve from negative infinity up to a given z-score
Lower, Upper = the two z-scores bounding the region

Worked Example: The 95% Range

Area between z = −1.96 and z = 1.96: Φ(1.96) ≈ 0.9750, Φ(−1.96) ≈ 0.0250. Area = 0.9750 − 0.0250 = 0.9500, or 95.00% — matching the familiar “95% of values fall within about 1.96 standard deviations of the mean” result used throughout statistics.

By Raw Values (Mean & Standard Deviation)

z = (x − μ) ÷ σ, then Area = Φ(z-upper) − Φ(z-lower)

Where:
x = a raw value from the distribution
μ = the distribution’s mean
σ = the distribution’s standard deviation

This mode converts both raw values to z-scores first, then computes the area exactly as the Z-Scores mode does — useful when you have real measurements (test scores, product weights, reaction times) rather than already-standardized z-scores.

Worked Example: Test Scores

A test has a mean score of 100 and a standard deviation of 15. Area between 85 and 115 points: z(lower) = (85 − 100) ÷ 15 = −1, z(upper) = (115 − 100) ÷ 15 = 1. Area = Φ(1) − Φ(−1) ≈ 0.6827, or about 68.27% of test-takers — the well-known “about 68% of values fall within one standard deviation of the mean” result.

Finding Left-Tail or Right-Tail Areas

Because a normal curve’s area trails off toward zero but never technically reaches it, a tail area is calculated the same way as any other area — just with one bound set to a very large negative or positive number (such as −99 or 99) standing in for negative or positive infinity.

Worked Example: Right-Tail Area

The area to the right of z = 1.645 (a common cutoff for a 5% significance level): enter Lower Z-Score = 1.645 and Upper Z-Score = 99. Area ≈ 0.9999999… − Φ(1.645) ≈ 0.0500, or about 5.00% — the area in the right tail beyond z = 1.645.

Frequently Asked Questions

How do I find the area under the normal curve between two z-scores?

Use the By Z-Scores mode: enter the lower and upper z-scores, and the calculator computes Area = Φ(upper) − Φ(lower) using the standard normal CDF.

What z-scores correspond to 95% of the area (the empirical rule)?

Roughly z = −1.96 to z = 1.96 captures 95% of the area under a standard normal curve. The broader empirical (68-95-99.7) rule says about 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.

How do I convert a raw score to a z-score?

Subtract the mean from the raw value, then divide by the standard deviation: z = (x − μ) ÷ σ. The By Raw Values mode does this conversion automatically for both bounds before computing the area.

How do I find a left-tail or right-tail area instead of a middle area?

Use a very large negative number (like −99) as the lower bound for a left-tail area, or a very large positive number (like 99) as the upper bound for a right-tail area — these stand in for negative and positive infinity, since the curve’s area beyond about z = ±5 is negligibly close to zero.