Normal Distribution and Z-Score Calculator

What this calculator answers

It finds areas under a normal curve: the share of values below a value, above it, between two values or outside them. It also gives the z-score and percentile of a value, and works backward from a probability to the value at a percentile, the cut-off for a top share or the middle range that holds a given share. Each answer comes with the steps, the matching Excel, Google Sheets and TI-84 commands, and a shaded curve that shows the values and their z-scores on one axis.

How to use it

• Find: pick the question in words. “Value from a probability” is the inverse normal, the same job as invNorm or NORM.INV.
• Mean and Standard deviation: the center and spread of the distribution, in the units of your values. Leave both blank for the standard normal (mean 0, SD 1); the values you type are then z-scores. Enter the standard deviation, not the variance.
• Value (x), or Lower value (a) and Upper value (b). If the two values are in the wrong order, they are swapped and a note says so.
• Where the area is and Probability (inverse only): below x for a percentile, above x for a cut-off such as the top 5%, or the middle share around the mean. Type 0.9 or 90%. A % sign, or any number of 1 or more, is read as a percentage, and the line under the box shows how it was read.

The Try chips load the cases worked on this page. Under the steps, a table lists the ranges that hold 68% to 99.7% of values for your mean and SD.

Find the probability below, above or between two values

Turn each value into a z-score, then read the area to its left from the standard normal cumulative distribution function Φ:

• below x: $P(X < x) = \Phi(z)$
• above x: $P(X > x) = 1 - \Phi(z)$
• between a and b: $P(a < X < b) = \Phi(z_b) - \Phi(z_a)$

The probability outside a and b is 1 minus the probability between them. Φ has no formula you can evaluate by hand, which is why z-tables, calculators and this page compute it numerically.

Z-score formula: how many standard deviations from the mean

A z-score counts how many standard deviations a value lies from the mean:

where $x$ is the value, $\mu$ the mean and $\sigma$ the standard deviation. Values above the mean have positive z-scores and values below it negative ones. A 1,030 g bag from the filling line in the worked example below has $z = (1{,}030 - 1{,}012) \div 8 = 2.25$, which puts it at the 98.78th percentile.

Worked example: flour bags between 1,000 g and 1,020 g

A filling line puts a mean of 1,012 g of flour in each bag, with a standard deviation of 8 g. What share of bags weigh between 1,000 g and 1,020 g?

1. Standardize both weights: $z_a = (1{,}000 - 1{,}012) \div 8 = {-1.5}$ and $z_b = (1{,}020 - 1{,}012) \div 8 = 1$.
2. Read the areas below them: $\Phi({-1.5}) = 0.0668072$ and $\Phi(1) = 0.841345$.
3. Subtract: $0.841345 - 0.0668072 = 0.774538$. About 77.45% of bags weigh between 1,000 g and 1,020 g.
4. The rest lie outside: 0.0668072 below 1,000 g plus 0.158655 above 1,020 g is 0.225462, or 22.55%.

In Excel or Google Sheets the same area is =NORM.DIST(1020, 1012, 8, TRUE) - NORM.DIST(1000, 1012, 8, TRUE), and on a TI-84 it is normalcdf(1000, 1020, 1012, 8). The “Below 1,000 g” chip gives 0.0668072: 6.68% of bags are under 1,000 g, and the other 0.9331928 are heavier.

Convert a z-score to a percentile, and a percentile to a z-score

A z-score’s percentile is the area below it times 100. For z = 1.96, Φ(1.96) = 0.9750021, so the value is at the 97.5th percentile (the “Percentile of z = 1.96” chip). Going the other way, the z-score at a percentile is the inverse, $z = \Phi^{-1}(p)$: the 90th percentile is at z = 1.2816.

Inverse normal: find x from a probability

To find the value with a given area below it, look up the z-score for that area and convert it back to your units:

For the flour bags, the weight that 90% of bags stay under is $1{,}012 + 1.281552 \times 8 = 1{,}022.2524$ g, or =NORM.INV(0.9, 1012, 8) in a spreadsheet and invNorm(0.9, 1012, 8) on a TI-84. For a top share, use the area above: the top 5% of the standard normal starts at z = 1.6449, which is the 95th percentile (the “z for the top 5%” chip). For a middle share, half of what is left goes in each tail: the middle 95% of bag weights runs from $1{,}012 - 1.959964 \times 8 = 996.3203$ g to 1,027.6797 g (the “Middle 95%” chip).

Critical z-values for 90%, 95% and 99%

Two-tailed, 90%, 95% and 99% use z = 1.6449, 1.96 and 2.5758; one-tailed, they use 1.2816, 1.6449 and 2.3263. A two-tailed cut-off splits the remaining area between both tails, so 95% leaves 2.5% above +1.96 and 2.5% below −1.96. A one-tailed cut-off puts all of it in one tail, 5% above +1.6449. That is why “the z for 95%” gets two different answers.

The 68–95–99.7 rule

About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. The exact shares are 68.27%, 95.45% and 99.73%. For the flour bags those ranges are 1,004 g to 1,020 g, 996 g to 1,028 g and 988 g to 1,036 g. The table under the steps lists the same ranges for your mean and SD, with the middle 80%, 90%, 95%, 98% and 99% between them.

How accurate are these results?

To at least 12 significant digits for any z from −40 to 40: that is how closely areas and z-scores match 50-digit reference values in this page’s tests, well beyond the 6 digits shown. Areas use W. J. Cody’s rational approximations to the normal distribution, the method R’s pnorm is based on, and values from a probability use M. J. Wichura’s algorithm AS 241, the basis of R’s qnorm. Each tail is computed on its own, so a small tail keeps its digits instead of coming out as 1 minus a number close to 1.

Results show 6 significant digits of the smaller tail: 0.0668072, but 0.9331928 for its complement. The smallest positive number JavaScript can store is 5E-324, so computed directly, the area above z = 40 would come out as 0. Here, tails below 1E-300 are worked out as powers of ten and shown to 4 significant digits: P(Z > 40) is 3.656E-350, and P(Z < 40) is shown as 1 − 3.656E-350. Values more than 1,000,000 standard deviations from the mean are refused, because they almost always mean the SD was entered in the wrong units.

The limit on typed probabilities is at the other end. A number such as 0.999999999999 is stored with only about 4 correct digits of its distance from 1, so for a far tail, enter the small area on its own side instead: 1e-12 above x rather than 0.999999999999 below it. The page warns when a probability is that close to 1.

Reading the result

• Headline: the probability you asked for as a decimal, with the percentage under it. The inverse shows the value, or the two ends of the middle range.
• Tiles: the z-scores, the complement (the other side of the curve) and each value’s percentile rank. The inverse adds the matching value on the other side of the mean.
• Curve: the shaded area is the answer. The top row of the axis shows values in your units, the bottom row their z-scores. Chart data lists the marked points in a table.
• Steps: each z-score, the Φ values and the subtraction, then the spreadsheet and TI-84 commands.
• Intervals around the mean: ranges from ±1 SD to the middle 99% for your mean and SD, with the share inside and in each tail.
• Other percentiles (inverse only): the value at common percentiles, top shares or middle shares for the same mean and SD.

Assumptions and limitations

• The calculator assumes the values follow a normal distribution with exactly this mean and SD. Real measurements are only roughly normal, and they usually differ most in the tails, which is where small probabilities come from.
• The mean and SD are treated as exact. If they are estimates from a small sample, the uncertainty in them is not included.
• It is for continuous measurements. For whole-number counts, such as successes in a fixed number of trials, use a binomial distribution calculator.
• Values up to ±1,000,000,000,000 and probabilities down to 1E-300 are accepted.

Common mistakes

• Reading a percentile as a top share. The top 10% starts at the 90th percentile, so enter 90% below x, or choose “Above x” and enter 10%. Entering 10% below x gives the bottom 10%.
• Using the one-tailed z for a two-tailed question, or the reverse: 1.6449 and 1.96 both get called “the 95% value”.
• Entering the variance as the standard deviation. With a variance of 64 g², the SD is 8 g. Typing 64 as the SD makes the 1,030 g bag’s z-score 0.28 instead of 2.25.
• Using a table of the area from 0 to z as the area below z. Some printed tables, including NIST’s, give the area between the mean and z; add 0.5 for a positive z to get Φ(z).
• Typing a percentage as a decimal: 0.5 is read as 50%. For half a percent, type 0.5% or 0.005.