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.

Can I calculate a p-value from a z score?

Yes, when the supplied test statistic follows a standard normal distribution under the null model. Switch to From a z score. For a greater-than alternative use the upper tail; for a less-than alternative use the lower tail. A two-sided alternative uses the two tails together, P(|Z| >= |z|). At z = 1.96 the two-sided area is 0.049996 and the upper tail is half that amount. The page does not choose a test or derive a test statistic from sample data.

Can I enter a percentile or probability to find a z score?

This page goes from a value or a z score to its areas; it does not invert a supplied area. The Normal Distribution Calculator provides the inverse normal workflow. A percentile such as 95% is a left-tail area, while a two-sided tail probability describes a different area and cannot be substituted without changing the question.

Can I use this in a phone or desktop browser?

The same form runs in a phone, tablet or desktop browser. The numeric fields accept minus signs, decimal points and exponent notation such as 1.2e5; a properly grouped value such as 1,250 is also accepted. It does not need an account or native download, and it does not reproduce physical-calculator menu commands.

How do I find a z score in this calculator?

Choose From a value and enter your value, its mean and a positive standard deviation. Read z and the working beside the result. To find normal-curve areas from a known z, choose From a z score; the lower and upper tails are shown together. A tail such as 0.158655 is about 15.8655% when multiplied by 100.

Z-Score Calculator

Enter a value, mean and standard deviation to find its z score, or start with a z score to read the normal-curve areas and raw value. The results show lower, upper and two-sided tails, percentile and a marked bell curve.

Z-score1.00x 110 · mean 100 · sd 10 · lower tail 0.841345

What you have

Value (x)

Mean

Standard deviation

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.

1.00

Lower tail

0.841345

Upper tail

0.158655

Two-tailed

0.317311

Areas under the standard normal curve, computed to within 1e-12 and shown to six decimal places. z is computed in floating-point arithmetic: the value minus the mean, divided by the standard deviation.

Percentilethe percentage of the curve below the value
84.1345
Between the mean and the valueP(100 ≤ X ≤ 110)
0.341345
Within the same distance either wayP(|Z| ≤ 1.00)
0.682689
The workingthe arithmetic in full
z = (110 − 100) ÷ 10 = 1.00

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
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

Accuracy. z is calculated as the value minus the mean, divided by the standard deviation, using floating-point arithmetic. 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. The areas assume a normal distribution: z is arithmetic about your own three numbers, while every probability beside it is an area under the standard normal curve.

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.
Can I calculate a p-value from a z score?: Yes, when the supplied test statistic follows a standard normal distribution under the null model. Switch to From a z score. For a greater-than alternative use the upper tail; for a less-than alternative use the lower tail. A two-sided alternative uses the two tails together, P(|Z| >= |z|). At z = 1.96 the two-sided area is 0.049996 and the upper tail is half that amount. The page does not choose a test or derive a test statistic from sample data.
Can I enter a percentile or probability to find a z score?: This page goes from a value or a z score to its areas; it does not invert a supplied area. The Normal Distribution Calculator provides the inverse normal workflow. A percentile such as 95% is a left-tail area, while a two-sided tail probability describes a different area and cannot be substituted without changing the question.
Can I use this in a phone or desktop browser?: The same form runs in a phone, tablet or desktop browser. The numeric fields accept minus signs, decimal points and exponent notation such as 1.2e5; a properly grouped value such as 1,250 is also accepted. It does not need an account or native download, and it does not reproduce physical-calculator menu commands.
How do I find a z score in this calculator?: Choose From a value and enter your value, its mean and a positive standard deviation. Read z and the working beside the result. To find normal-curve areas from a known z, choose From a z score; the lower and upper tails are shown together. A tail such as 0.158655 is about 15.8655% when multiplied by 100.

z is calculated as the value minus the mean, divided by the standard deviation, using floating-point arithmetic. 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.