Normal Distribution Calculator

z = (value − mean) / standardDeviation → below = Φ(z), above = 1 − Φ(z), within = 2Φ(|z|) − 1

value

The measurement you are asking about — one observation, not an average of several. It is compared against the mean and the standard deviation of the distribution it is being read against

mean

The centre of the normal distribution, written μ. Every probability on the panel is stated relative to it, so a mean of 0 turns the value itself into the z-score whenever the spread is 1

standardDeviation

The spread of the distribution, written σ, and it must be greater than zero. It is the unit the distance is measured in, which is what lets one z-score table serve distributions of any size

z

How many standard deviations the value sits from the mean. Positive above the mean, negative below, and zero when the value is the mean itself

Φ

The cumulative distribution function of the standard normal distribution: the share of the curve that lies to the left of a given z. It has no elementary closed form, so it is computed rather than looked up

Use it whenever the question is about position rather than about the raw number: how unusual a measurement is for the process that produced it, what share of a population falls above a cut-off, or what proportion of a specification window a centred process covers. It is also the standard route into a normal table — converting to a z-score first is what makes one table apply to every distribution. Reach for the z-score calculator next door instead when the z-score itself is the point and the probabilities are not needed; reach for this one when you need the areas, since a z-score alone says how far out a value is but not how much of the curve is out there with it.

1. One standard deviation above the mean

   1. Subtract the mean: 115 − 100 = 15
   2. Divide by the standard deviation: 15 / 15 = 1, so the z-score is 1
   3. The share of a normal curve below z = 1 is about 84.13%, which leaves 15.87% above it
   4. The share within one standard deviation on both sides is about 68.27%, and the two one-sided figures are simply what is left over at each end

   The three numbers are the same fact said three ways, and the middle one is the one people misread: 84.13% below means a value one standard deviation above the mean beats about five sixths of the distribution, while 15.87% of it still sits above. Note also that within is not the sum of the two one-sided probabilities — it is the middle band only, so it is always the smallest of the three for a value near the mean and grows toward 100% as the value moves out into the tail.

2. A value exactly at the mean

   1. Subtract the mean: 100 − 100 = 0
   2. Dividing by any positive spread leaves 0, so the z-score is 0
   3. A normal curve is symmetric, so exactly half of it lies below the mean and half above
   4. A band of zero width around the mean contains none of the distribution, so within is 0%

   The within figure of 0% is not a rounding of something small — it is exact, and it is the one case where the width of the band is genuinely nothing. That makes this example the cleanest check on the direction of the band: within measures the distance out to the value on both sides, so at the mean there is no distance at all. It is also the case that shows the symmetry the rest of the panel relies on, since the two one-sided probabilities split exactly in half.

3. A value far out in the upper tail

   1. Subtract the mean: 200 − 100 = 100
   2. Divide by the standard deviation: 100 / 15 ≈ 6.67 standard deviations above the mean
   3. The share above z = 6.67 is about 1.3 × 10⁻¹¹ — one in tens of billions
   4. At two decimal places that share is 0.00%, and the share below is 100.00%

   The 0.00% above is the one number on this page that looks like a bug and is not: the true share is about 1.3 in a hundred billion, which two decimal places of a percentage cannot print. Two things follow from it. Read 0.00% as "smaller than the precision shown" rather than as impossible, and remember that the panel carries more information than the printed digits — the tail is computed on its own path precisely so that this case gives a clean zero instead of the 0.01 or 0.02 that subtracting from the lower tail would produce. Also note that within has reached 100%, since a band reaching 6.67 standard deviations out each way now covers essentially the whole curve.

4. One standard deviation below the mean

   1. Subtract the mean: 85 − 100 = −15
   2. Divide by the standard deviation: −15 / 15 = −1, so the z-score is negative
   3. The share below z = −1 is about 15.87%, and the share above it is about 84.13%
   4. The band within one standard deviation is the same 68.27% as for z = +1, because the band is symmetric

   Comparing this with the first example is the quickest way to see what the sign does and does not do: the two one-sided probabilities swap over, and the within figure does not move at all. That is because within measures a distance from the mean in both directions, so it cannot tell which side the value is on — which is exactly what you want when the question is how much of a distribution lies close to its centre, and exactly why the panel also prints the one-sided figures rather than the band alone.

Every number here is a property of a normal distribution, not of your data. A z-score can be computed for any set of values whatever its shape, but the probabilities are only the ones above if the distribution really is normal — with a long-tailed or skewed spread, values that sit far out are more common than the normal curve says, and the tail estimates are the ones that suffer first. A mean and a standard deviation do not make a distribution normal, and they cannot tell you whether it is. Two further cautions. Very small probabilities cannot be printed at two decimal places, so a value far into the tail shows 0.00% where the truth is a small positive number. And the value entered is treated as a single observation read against a known mean and spread, so the numbers do not account for uncertainty in the mean or the standard deviation, which matters when those were estimated from a small sample.

What does the z-score add if I already have the probability?

It is the only one of the four numbers that travels between distributions. A probability is tied to the particular curve you entered, while a z-score of 1 means the same distance from the centre in every normal distribution — exam marks with a spread of 15, machine parts measured in hundredths of a millimetre, or a measurement with an arbitrary unit. That is what makes a single z-table usable everywhere: convert to a z-score first, then read the area off the standard curve.

Why does the probability above show 0.00% for a very large value?

Because two decimal places of a percentage cannot print how small the number really is. A value 6.67 standard deviations above the mean has about 1.3 × 10⁻¹¹ of the curve above it — roughly one in tens of billions — which is a real and positive share, just far below the precision shown. Read 0.00% as "smaller than this display can show" rather than as impossible. The panel computes that tail on its own path rather than by subtracting from the lower tail, which is why it gives a clean zero instead of the small nonzero value that subtraction would produce.

Is this the same as the empirical rule?

Yes, and the table on this page is that rule written as distances rather than as percentages. The empirical rule says about 68% of a normal distribution falls within one standard deviation of the mean, about 95% within two, and about 99.7% within three; flipping each of those into a distance from the centre is exactly what the rows do. One difference is worth noting: the rule is usually quoted as a rounded set of three percentages, while the shares here are computed from the same cumulative function the calculator uses, so a value sitting exactly on a boundary gets a consistent answer from both.

What happens if the standard deviation is zero?

It is rejected rather than answered. Dividing by a spread of zero is not a probability that happens to be 1 — it describes a distribution with no spread at all, where every observation is the mean and the notion of a distance from the centre stops meaning anything. The panel reports an error instead of returning a plausible-looking 0% or 100%. Negative spreads are rejected for the same reason: a standard deviation is a distance, so it cannot be negative, and silently using its absolute value would hide a data error.

Can I use this on data that is not normal?

You can compute the z-score for any set of values, because a mean and a standard deviation exist for any data at all. The probabilities are the part that needs the assumption: they are areas under a normal curve, and a skewed or long-tailed distribution will not match them. The mismatch shows up at the tails first, where the true share beyond a large distance is usually bigger than the normal curve claims, which is why the empirical rule's 0.3% beyond three standard deviations should be treated as a property of the normal model and not as a promise about your measurements.

Why is the value treated as one observation rather than a sample mean?

Because that is what the three probabilities are: the share of the distribution below, above and around a single value. Reading a sample mean against the same mean and spread is a different calculation, and the spread you would need is the standard error of the mean rather than the spread of the individual observations — the two differ by a factor of the square root of the sample size, so using one in place of the other gives an answer that looks reasonable and is wrong. If your value is itself an average, enter the standard error as the standard deviation.

• 1.3.6.6.1. Normal Distribution — e-Handbook of Statistical Methods (the density, the role of the mean and standard deviation, and the standard normal as the reference case) — National Institute of Standards and Technology (NIST)
• 1.3.6.7.1. Cumulative Distribution Function of the Standard Normal Distribution — e-Handbook of Statistical Methods (the cumulative function behind every probability on this page, and the z-table it is read from) — National Institute of Standards and Technology (NIST)
• 6.1 The Standard Normal Distribution — Introductory Statistics 2e (the z-score as a standardised value, and the areas a normal table records) — OpenStax, Rice University
• 6.2 Using the Normal Distribution — Introductory Statistics 2e (reading probabilities off a normal curve, including the complements this page reports as below and above) — OpenStax, Rice University