How do I calculate a normal distribution probability?

Turn each bound into a z score, then read the area under the standard normal curve between them. For a mean of 100 and a standard deviation of 15, the bound 85 gives z = (85 - 100)/15 = -1.00 and the bound 130 gives z = (130 - 100)/15 = 2.00, so the area between them is 0.8185946. That working is printed under the answer with your own numbers in it, and the same three regions are available one at a time: below a value, above it, or between two bounds.

How do I find the value at a given percentile?

Pick the Find value mode and type the area to the left. The tool inverts the curve instead of searching it: for a mean of 100 and a standard deviation of 15, an area of 0.95 gives z = 1.64485 and a value of 124.6728. A probability may be written as a percentage, so 95% and 0.95 are the same request, and an area of 0.5 returns the mean itself, exactly, because half the curve lies on each side of it.

Can I use this as a standard normal table?

Yes. Set the mean to 0 and the standard deviation to 1 and the bounds are z scores, so the classic lines come straight out: the area between -1 and 1 is 0.6826895, the area between -1.96 and 1.96 is 0.9500042, and the area below 2 is 0.9772499. Two of the three cases under the answer follow whichever distribution is on the page, and the third is the standard normal band either way, so they act as a running check on it.

What is the difference between the density and the probability here?

The density is the height of the curve at one value and the probability is the area beneath a stretch of it. A single point has no width, so its probability is exactly 0, which is what the tool returns when both bounds are the same number. The density is still useful: it is printed at each bound, and it is what tells you how quickly the area is changing there.

How accurate are these numbers?

The z scores are exact arithmetic. The areas come from Cody's rational Chebyshev approximation to the error function, published in Mathematics of Computation in 1969, and the quantile from Acklam's rational inverse refined against that same curve. Both were measured against a 900-digit reference implementation rather than against another calculator: the worst relative error found was 1.7e-13 for an area and 4e-16 for a quantile, so the page quotes a bound of 1e-12 out to 37 standard deviations.

Normal Distribution Calculator

Q: Why is a small probability shown as 6.2210e-16 instead of 0.0000000?

Because it is not zero. Seven decimal places run out long before the tail does, so any area under a millionth is printed with its exponent instead. The tail is also computed as a tail rather than as one minus the area below: for a mean of 100 and a standard deviation of 15, the area above 220 is 6.2210e-16, and subtracting the area below from one gives an answer about 7% too large, because the digits were spent on the leading 1. Below about 1e-308 even that runs out, and the page says so rather than showing a zero.

Most normal distribution pages hand back one probability and nothing to check it against. This one shades the region on the curve, prints the z score of each bound with the arithmetic that produced it, and gives the density there as well, so the answer can be read rather than trusted. For a mean of 100 and a standard deviation of 15, the area between 85 and 130 is 0.8185946, and the same page runs backwards: the 95th percentile of that distribution is 124.6728. The far tail is computed directly rather than as one minus the area below, a subtraction that is off by about 7% eight standard deviations out.

Mean 100SD 15Between85 to 130

mean 100   SD 15   model N(100, 15)

P(85 < X < 130)   0.8185946
  z low    (85 - 100)/15  = -1.00
  z high   (130 - 100)/15 = 2.00
  density  f(85) = 0.01613138   f(130) = 0.003599398

standard normal  P(-1 < Z < 1) = 0.6826895
inverse          95th percentile = 124.6728

Area you want

Mean

Standard deviation

From

The area between two values, with the z score of each bound shown. Bounds on the same side of the mean are integrated rather than subtracted.

P(between)

0.8185946

z low

-1.00

z high

2.00

Method

Rational approx.

That is 81.86% of the curve, for a mean of 100 and a standard deviation of 15. The areas come from Cody's rational Chebyshev approximation to the error function (Mathematics of Computation, 1969) and the quantile from Acklam's rational inverse, refined here against that same curve.

P(-1 < Z < 1)standard normal
0.6826895
P(X < 130), N(100, 15)z = 2.00
0.9772499
95th percentile, N(100, 15)inverse
124.6728

Accuracy. The z scores are exact arithmetic: the bound minus the mean, divided by the standard deviation. The probabilities and the quantile come from the published rational approximations named on the page, measured against an arbitrary-precision reference at better than 1e-12 relative error out to 37 standard deviations, and the inverse of 0.5 returns the mean exactly. It reports probabilities for the distribution you specify and draws no conclusion from them. The areas are areas under the normal curve you named: whether your data is normal is a question about the data, not about this arithmetic.

Common questions

How do I calculate a normal distribution probability?: Turn each bound into a z score, then read the area under the standard normal curve between them. For a mean of 100 and a standard deviation of 15, the bound 85 gives z = (85 - 100)/15 = -1.00 and the bound 130 gives z = (130 - 100)/15 = 2.00, so the area between them is 0.8185946. That working is printed under the answer with your own numbers in it, and the same three regions are available one at a time: below a value, above it, or between two bounds.
How do I find the value at a given percentile?: Pick the Find value mode and type the area to the left. The tool inverts the curve instead of searching it: for a mean of 100 and a standard deviation of 15, an area of 0.95 gives z = 1.64485 and a value of 124.6728. A probability may be written as a percentage, so 95% and 0.95 are the same request, and an area of 0.5 returns the mean itself, exactly, because half the curve lies on each side of it.
Why is a small probability shown as 6.2210e-16 instead of 0.0000000?: Because it is not zero. Seven decimal places run out long before the tail does, so any area under a millionth is printed with its exponent instead. The tail is also computed as a tail rather than as one minus the area below: for a mean of 100 and a standard deviation of 15, the area above 220 is 6.2210e-16, and subtracting the area below from one gives an answer about 7% too large, because the digits were spent on the leading 1. Below about 1e-308 even that runs out, and the page says so rather than showing a zero.
Can I use this as a standard normal table?: Yes. Set the mean to 0 and the standard deviation to 1 and the bounds are z scores, so the classic lines come straight out: the area between -1 and 1 is 0.6826895, the area between -1.96 and 1.96 is 0.9500042, and the area below 2 is 0.9772499. Two of the three cases under the answer follow whichever distribution is on the page, and the third is the standard normal band either way, so they act as a running check on it.
What is the difference between the density and the probability here?: The density is the height of the curve at one value and the probability is the area beneath a stretch of it. A single point has no width, so its probability is exactly 0, which is what the tool returns when both bounds are the same number. The density is still useful: it is printed at each bound, and it is what tells you how quickly the area is changing there.
How accurate are these numbers?: The z scores are exact arithmetic. The areas come from Cody's rational Chebyshev approximation to the error function, published in Mathematics of Computation in 1969, and the quantile from Acklam's rational inverse refined against that same curve. Both were measured against a 900-digit reference implementation rather than against another calculator: the worst relative error found was 1.7e-13 for an area and 4e-16 for a quantile, so the page quotes a bound of 1e-12 out to 37 standard deviations.

The z scores are exact arithmetic: the bound minus the mean, divided by the standard deviation. The probabilities and the quantile come from the published rational approximations named on the page, measured against an arbitrary-precision reference at better than 1e-12 relative error out to 37 standard deviations, and the inverse of 0.5 returns the mean exactly. It reports probabilities for the distribution you specify and draws no conclusion from them.