Z-Score Calculator
Most z-score pages stop at the score. This one shows the score, the area to the left of it, the area to the right, both tails together and the percentile, with a bell curve marked where the value falls. The areas are computed to within 1e-12 of the true normal curve, which is far tighter than the six decimal places they are shown to, and the far tail is worked out directly rather than as one minus the near tail, so the upper tail at a z of 5 reads 2.8665e-7 instead of a rounded zero. Start from a value, a mean and a standard deviation, or start from a z score and read the value back out.
z counts how many standard deviations the value sits from the mean. Positive is above the mean, negative is below, and the size is what matters.
Areas under the standard normal curve, computed to within 1e-12 and shown to six decimal places. z itself is exact arithmetic: the value minus the mean, divided by the standard deviation.
Common z values
The lines a z table is usually opened for. Every figure here is computed by the same code as the answer above, so the table and the tool can never disagree. Load one to see the rest of its numbers.
| z | Below | Above | Both tails | Often wanted for | Load |
|---|---|---|---|---|---|
| 0.50 | 0.691462 | 0.308538 | 0.617075 | half a standard deviation | |
| 1.00 | 0.841345 | 0.158655 | 0.317311 | one standard deviation | |
| 1.282 | 0.900079 | 0.099921 | 0.199843 | about 10% above | |
| 1.645 | 0.950015 | 0.049985 | 0.099970 | about 5% above | |
| 1.96 | 0.975002 | 0.024998 | 0.049996 | the 95% two-sided line | |
| 2.00 | 0.977250 | 0.022750 | 0.045500 | two standard deviations | |
| 2.326 | 0.989991 | 0.010009 | 0.020019 | about 1% above | |
| 2.576 | 0.995002 | 0.004998 | 0.009995 | the 99% two-sided line | |
| 3.00 | 0.998650 | 0.001350 | 0.002700 | three standard deviations |
Common questions
- How do I calculate a z score?
- Subtract the mean from the value and divide by the standard deviation: z = (x − mean) ÷ sd. For a value of 110 in a distribution with a mean of 100 and a standard deviation of 10, that is (110 − 100) ÷ 10, so z = 1.00 and the value sits one standard deviation above the mean. That line of arithmetic is printed under the answer, with your own numbers in it, so a figure typed into the wrong field is easy to spot.
- What percentage of values fall below a z score of 1?
- 84.1345% of them. That is the lower tail, 0.841345, written as a percentage. The upper tail is 0.158655, and the two tails together, meaning everything at least one standard deviation from the mean on either side, come to 0.317311. All three are on the first screen rather than behind a mode switch, because which one a question needs is usually clear only once you can see them side by side.
- What does a z score of 1.96 mean?
- It is the two-sided 95% line. At z = 1.96 the area outside the two tails is 0.049996 and the area below is 0.975002, which is where the familiar 5% comes from. The z that leaves exactly 0.050000 outside is 1.959964, so 1.96 is a rounding of it rather than the exact figure. The z field takes either, so the difference is easy to see: 0.049996 against 0.050000.
- Is a negative z score bad?
- No, it only says the value is below the mean. The normal curve is symmetric, so a z of −2 carries the same tail area as a z of +2: 0.022750 below in the first case, 0.022750 above in the second. Whether below the mean is good or bad depends on what is being measured, and nothing on this page decides that for you.
- What is the difference between a one-tailed and a two-tailed probability?
- One tail is the area on a single side of the value: for a z of 1 that is 0.841345 below or 0.158655 above. Two-tailed adds the matching area on the far side, giving 0.317311, and it answers a different question: how often a value lands at least that far from the mean in either direction. The page shows all three at once, so the one you need is never the one you have to work out yourself.
- Why does a standard deviation of 0 get refused?
- Because z divides by it. With no spread there is no scale to measure the distance in, so the honest answer is that z is undefined, not that it is infinite. The page says that and asks for a standard deviation above 0. Other refusals read the same way: an entry that is not a number is quoted back at you, a negative standard deviation is answered with the positive number to type instead, and a value too far from the mean to divide is named rather than shown as infinity.
- Do these probabilities hold if my data is not normally distributed?
- The z score does. It is arithmetic, and it says how many standard deviations the value sits from the mean whatever shape the data has. The tail areas do not. Every probability here is an area under the standard normal curve, so it describes your data only as far as your data is normally distributed. For a small sample, or an obviously skewed one, read them as the normal-curve answer rather than as the share of your own values.
- How accurate are the tail probabilities?
- They are within 1e-12 of the true areas, which is far tighter than the six decimal places shown. They come from an erf evaluated as an all-positive series where its argument is small and as a continued fraction where it is large, not from a lookup table and not from the Abramowitz and Stegun polynomial whose published bound is 1.5e-7. The far tail is computed directly rather than as one minus the near tail, which is why a z of −5 reads 2.8665e-7 instead of collapsing to zero.
- Can I go from a z score back to the raw value?
- Yes. Switch to From a z score, type the z, and add the mean and the standard deviation: the value is mean + z × sd, so a z of 1.96 on a mean of 100 with a standard deviation of 10 is 119.6. Leave both the mean and the standard deviation empty and the page stays in standard units, answering the areas around the z alone, which is what a z table is usually consulted for.
z itself is exact arithmetic: the value minus the mean, divided by the standard deviation. The tail probabilities come from a numerical approximation to the standard normal distribution, accurate to within 1e-12, which is stated on the page next to them. A standard deviation of zero is refused rather than returned as infinity.