Turn a value, mean and standard deviation into a z-score, or turn a z-score or percentile back into a value, with left, right and between probabilities. Runs on your device.
A z-score is (x − μ) ÷ σ: how many standard deviations a value x sits above or below the mean μ, where σ is the standard deviation. The page opens on x = 130 with a mean of 100 and a standard deviation of 15. That gives z = (130 − 100) ÷ 15 = 2, so the value is 2 standard deviations above the mean. For a normal distribution, the share of values below it is 0.9772, the share above is 0.0228, and the chance of landing at least this far from the mean in either direction is 0.0455. A negative z-score means the value is below the mean: 70 on the same scale is z = −2. A z-score of 0 is exactly the mean. Use a mean of 0 and a standard deviation of 1 if you already have a z-score and only want its probabilities.
The left probability P(X < x) is the area under the bell curve to the left of the value. It is the same number as the percentile, shown as a percentage: z = 2 is the 97.72nd percentile. The right probability is one minus that, and it is worked out on its own from the far tail, so a z-score of 8 shows a right-tail probability of about 6.2e-16 instead of rounding to zero. The two-tailed probability is twice the smaller tail. It is the p-value usually quoted for a z-test of a result at least as far from the mean as yours, in either direction. Probabilities come from a calculation of the normal cumulative distribution (a series and a continued fraction for the error function), not from a printed z-table, so z = 1.64 and z = 1.645 give different answers instead of the same table row.
Choose Probability between two values, then give a lower and an upper value. The page converts both to z-scores and subtracts: P = Φ(z₂) − Φ(z₁). With a mean of 100 and a standard deviation of 15, the range 85 to 115 is z = −1 to z = 1, and the probability is 0.6827. That is the first number of the 68-95-99.7 rule: within 2 standard deviations is 0.9545 and within 3 is 0.9973, and the table under the answer lists all three for your mean and standard deviation. A range that starts above the mean, such as 130 to 145, is z = 2 to z = 3 and has a probability of 0.0214, the 2.275% above 130 minus the 0.135% above 145. The upper value must be larger than the lower one.
To go the other way, x = μ + z × σ. A z-score of 1.96 on a mean of 100 and a standard deviation of 15 is x = 100 + 1.96 × 15 = 129.4. For a percentile, the page first finds the z-score whose left-tail probability equals it, then converts to x. The 95th percentile is z = 1.6449, which is x = 124.6728 on that scale. The familiar two-sided 95% cut-off, 1.96, is the 97.5th percentile, because 2.5% is left in each tail. Percentiles from 0.000001 to 99.999999 are accepted. Results are shown to 4 decimal places like a z-table, and you can pick any number from 1 to 10.
Every probability assumes the values follow a normal distribution. For skewed data such as incomes or waiting times, a z-score still says how far a value is from the mean, but the probabilities are not right. This page uses the population standard deviation you type. If your σ was estimated from a small sample, a Student t distribution is the usual choice instead, and the confidence interval calculator on this site uses it. The standard deviation must be more than 0. Z-scores beyond 40 in either direction are treated as probability 0 or 1.
Subtract the mean from the value and divide by the standard deviation: z = (x − μ) ÷ σ. With x = 130, μ = 100 and σ = 15, z = 30 ÷ 15 = 2. Type the three numbers and the page shows the z-score, the probabilities and the working with your numbers.
A negative z-score means the value is below the mean. A test score of 70 on a scale with a mean of 100 and a standard deviation of 15 is z = −2, which is the 2.28th percentile. The size of the number is the distance in standard deviations, and the sign is the side.
It depends on the question. For the middle 95% of a normal distribution the cut-offs are z = ±1.96. For the 95th percentile (95% of values below) the z-score is 1.6449. Choose Percentile to value, type 95, and the page shows the second one; type 97.5 to get 1.96.
Subtract the smaller cumulative probability from the larger: P(a < Z < b) = Φ(b) − Φ(a). For z = −1 and z = 1 that is 0.8413 − 0.1587 = 0.6827. To do it here, choose Probability between two values and set the mean to 0 and the standard deviation to 1, then type the z-scores as the lower and upper values.
About 4.55% of values in a normal distribution are at least 2 standard deviations from the mean in either direction, so one in about 22 values. That is why 2 is often used as an informal cut-off, and 1.96 as the usual rounded cut-off for 5%. Whether that is unusual enough depends on your field and on the data really being close to normal.
No. The numbers are processed in your browser and are not uploaded. They are not stored on your device either, and reloading the page puts the 130, 100 and 15 example back.