Normal distribution calculator (area under curve)

By Jude Wallis · Published July 28, 2026

A normal distribution calculator finds the area under the bell curve between your bounds, and that area is the probability a value falls in that range. Enter the mean and standard deviation, pick a region (left, right, between, or outside), and set the bounds.

Mean (mu)Standard deviation (sigma)Lower bound (a)Upper bound (b)

probability (area under the curve)

0.6827

That is 68.27% of the distribution.

Steps

1.standardize: z = (x - 100) / 15
2.z for 85: -1; z for 115: 1
3.area between: 0.8413 - 0.1587 = 0.6827

AP Statistics: Unit 2 (topics 2.11 The Normal Distribution). This calculator supports Fall 2026 Unit 2 topic 2.11, where the probability a value lies in an interval is the area under the normal curve, found using technology or z-scores and a standard normal table.

What area under the normal curve means

A normal distribution is a symmetric, bell-shaped curve set by two numbers: the mean, written $\mu$ (the center), and the standard deviation, written $\sigma$ (the spread). A normal distribution calculator finds the area under that curve between the bounds you choose, and that area equals the probability a randomly chosen value lands in your range.

The total area under any normal curve is 1, so every probability the tool returns sits between 0 and 1. That is why an area under normal curve calculator and a normal probability calculator are the same tool: over an interval, area and probability are the same quantity.

Because the curve is continuous, you always work with a range, not a single point. The probability of landing on one exact value is 0, so you ask for the area to the left of a bound, to the right of a bound, between two bounds, or outside two bounds.

The four region types

The calculator supports four ways to describe an interval. Pick the one that matches the probability you want. Here $a$ is the lower bound and $b$ is the upper bound.

Left ( $P(X < b)$ ): the area from far left up to an upper bound $b$ . This is the cumulative area.
Right ( $P(X > a)$ ): the area from a lower bound $a$ out to the far right. It equals $1$ minus the left area.
Between ( $P(a < X < b)$ ): the area trapped between a lower bound $a$ and an upper bound $b$ .
Outside ( $P(X < a \text{ or } X > b)$ ): the two tails beyond the bounds, equal to $1$ minus the between area.

How to use the normal distribution calculator

Enter the mean $\mu$ and the standard deviation $\sigma$ for your variable.
Choose a region type: left, right, between, or outside.
Type the bound or bounds in the same units as your data.
Read the probability, shown as a decimal and a percent, with the matching area shaded on the curve.

You do not need to convert to z-scores first. The calculator standardizes for you. The next section shows the same steps by hand so you can check the result or reproduce it on a test.

How this connects to z-scores and the z-table

A z-score tells you how many standard deviations a value sits from the mean. You find it with

z = \frac{x - \mu}{\sigma}

where $x$ is your value, $\mu$ is the mean, and $\sigma$ is the standard deviation. Standardizing turns any normal distribution into the standard normal distribution, which has mean 0 and standard deviation 1.

Once you have a z-score, a z-table gives the area to its left, written $\Phi(z)$ , the cumulative probability. The calculator does the same lookup internally. For a between region you subtract the two left areas; for a right region you subtract the left area from 1. See how to find a z-score for the standardizing step in full.

Checking answers with the empirical rule

When your bounds sit exactly 1, 2, or 3 standard deviations from the mean, the empirical rule gives a fast sanity check. About 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.

So for a mean of 100 and a standard deviation of 15, the range 85 to 115 is one standard deviation each way, and you should expect roughly 68%. The first worked example below confirms it.

P(85 < X < 115) for mean 100, SD 15

A score is modeled by a normal distribution with mean $\mu = 100$ and standard deviation $\sigma = 15$ , where $\mu$ is the mean and $\sigma$ is the standard deviation. Find the probability that a randomly chosen score is between 85 and 115, written $P(85 < X < 115)$ .

Standardize the lower bound: $z = \frac{85 - 100}{15} = \frac{-15}{15} = -1.00$ .
Standardize the upper bound: $z = \frac{115 - 100}{15} = \frac{15}{15} = 1.00$ .
Look up the left area for each z-score in a z-table: $\Phi(1.00) = 0.8413$ and $\Phi(-1.00) = 0.1587$ .
Subtract the smaller left area from the larger to get the area between: $0.8413 - 0.1587 = 0.6826$ .

$P(85 < X < 115) \approx 0.6826$ , or about 68%. With full precision the area is 0.6827, which matches the empirical rule for one standard deviation on each side of the mean.

P(X > 130) for mean 100, SD 15

Using the same distribution, with mean $\mu = 100$ and standard deviation $\sigma = 15$ , find the probability that a score is greater than 130, written $P(X > 130)$ .

Standardize the bound: $z = \frac{130 - 100}{15} = \frac{30}{15} = 2.00$ .
Look up the left area in a z-table: $\Phi(2.00) = 0.9772$ .
A right-tail area is 1 minus the left area: $1 - 0.9772 = 0.0228$ .

$P(X > 130) \approx 0.0228$ , or about 2.28%.

Frequently asked questions

Is the area under the normal curve the same as probability?

Yes. For a normal distribution, the area under the curve over an interval equals the probability that a value falls in that interval. The total area under the whole curve is 1, so the answer is always between 0 and 1.

Why is the probability of an exact value zero?

The normal distribution is continuous, so a single point has no width and no area. Its probability is therefore 0. You always measure probability over an interval, which is why the calculator asks for bounds.

Do I still need the z-table if the calculator does the work?

On the AP exam a graphing calculator with statistical capabilities and the provided standard normal table both work. Knowing the z-score method lets you check the tool and show your reasoning on a free-response question.