Normal distribution calculator

Find probabilities under a normal curve, or the value for any percentile, with a shaded graph and every step of the working.

  • Free, no sign-up
  • Step-by-step working
  • Runs in your browser
What do you want to find?

Enter a decimal like 0.9 or a percentage like 90%.

How to use the normal distribution calculator

  1. Enter the mean and standard deviation of the normal distribution, in the units of your data.
  2. To find a probability, choose below, above, between or outside, and enter the value or values.
  3. To find a cutoff instead, switch to A value (percentile), choose which area you know and enter it, such as 0.90 for the 90th percentile.
  4. Click Calculate to see the answer, the shaded curve and the z-scores behind it.
  5. Read the plain-English summary and download the graph if you need it.

Normal distribution formulas

Every normal probability is found by converting to the standard normal distribution (mean 0, SD 1):

z = x − μσ

P(X < a) = Φ(za)    P(X > a) = 1 − Φ(za)

P(a < X < b) = Φ(zb) − Φ(za)

Inverse: x = μ + zσ , where Φ(z) = area to the left

X
A normally distributed random variable
μ
The mean, the center of the curve
σ
The standard deviation, which sets the width of the curve
z
The z-score: how many standard deviations a value is from the mean
Φ(z)
The standard normal cumulative probability: the area under the curve to the left of z
a, b
The values that bound the area you want

The probability density function behind the curve is f(x) = 1σ√(2π) e−(x − μ)² / (2σ²), but you never need it to find probabilities: the areas come from Φ.

Worked example

A battery's life is normally distributed with a mean of 500 hours and a standard deviation of 40 hours. What proportion of batteries last more than 560 hours?

  1. Convert 560 hours to a z-score
    z = 560 − 50040 = 1.5
  2. Find the area to the left of z = 1.5
    Φ(1.5) = 0.9332
  3. "More than" means the area to the right
    P(X > 560) = 1 − 0.9332 = 0.0668

What it means: about 6.7% of batteries, roughly 1 in 15, last longer than 560 hours. Going the other way, the manufacturer could find the life that only the best 5% of batteries exceed: z = 1.645 for an area of 0.05 to the right, so x = 500 + 1.645 × 40 ≈ 565.8 hours.

How to interpret normal probabilities

  • A probability is a proportion. P(X > 560) = 0.0668 means about 6.7% of all values exceed 560, and it is also the chance that one randomly chosen value does.
  • The 68-95-99.7 rule is a quick sanity check: about 68% of values are within 1σ of the mean, 95% within 2σ and 99.7% within 3σ.
  • Small tail areas mean unusual values. A value with less than 2.5% in its tail is often called unusual; one with less than 0.15% is extremely rare.
  • Check that the normal model fits. The answers are only as good as the assumption. Skewed data, like incomes, need a different approach; a histogram will show you the shape.

Frequently asked questions

What is the normal distribution?

The normal distribution is the symmetric, bell-shaped curve that describes many natural measurements, such as heights, test scores and measurement errors. It is defined by two numbers: the mean μ, which sets the center, and the standard deviation σ, which sets the spread. The total area under the curve is 1, and areas under it are probabilities.

What is the 68-95-99.7 rule?

Also called the empirical rule: in a normal distribution, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. For IQ scores with mean 100 and SD 15, roughly 68% of people score between 85 and 115.

How do I find the probability between two values?

Convert both values to z-scores with z = (x − μ) / σ, look up the cumulative probability Φ(z) for each, and subtract the smaller from the larger: P(a < X < b) = Φ(zb) − Φ(za). Choose “Between two values” in the calculator to see every step.

Does it matter whether I use &lt; or ≤?

Not for a continuous distribution like the normal. The probability of landing exactly on one value is zero, so P(X < a) and P(X ≤ a) are the same. It matters only when a normal curve approximates a discrete count; then a continuity correction (adding or subtracting 0.5) improves accuracy.

How do I find the value for a given percentile?

Switch to “Find a value”. For the 90th percentile, enter an area of 0.90 to the left. The calculator finds the z-score with that area to its left (the inverse of Φ) and converts it back with x = μ + zσ. For the top 5%, use an area of 0.05 to the right.

How is this different from the z-score calculator?

The z-score calculator focuses on one value: how many standard deviations it is from the mean and its percentile. This calculator works directly in your units for any area: below, above, between or outside two values, and it also runs in reverse to find cutoffs.

Last reviewed November 11, 2026