Z-Score Calculator: Normal Probability & Percentile
Convert a value to a Z score or enter a Z score directly. Then estimate a left-tail, right-tail or interval probability under the standard normal distribution. A Z score alone does not establish that your data follow a normal distribution.
Read How to Calculate Z Scores, Normal Probabilities and Percentiles for worked examples and help interpreting tail areas and normal-model percentile ranks.
Calculate a Z score from a value
Formula: Z = (x - mean) / standard deviation. A positive Z score is above the mean; a negative score is below it. For Mean 100, Standard deviation 15 and Value (x) 115, Z = (115 - 100) / 15 = 1.
The score describes distance from the mean in standard-deviation units. The arithmetic can describe standardized values even when data are not normally distributed. However, the probabilities reported by this tool use a normal model and need that assumption to be appropriate.
Left tail, right tail and percentile
For Z = 1, the normal-model probability to the left is approximately 84.1345%; the probability to the right is approximately 15.8655%. The left-tail percentage corresponds to the approximately 84.13th percentile under this normal model, not a ranking measured from your actual observations.
Because the normal distribution is continuous, including or excluding an exact endpoint does not change these probabilities. The probability of one exact value is zero; an interval around that value can have positive probability.
Probability between two bounds
Choose Between two bounds and enter the lower bound first. In direct-Z mode, Lower Z score -1 and Upper Z score 1 give approximately 68.2689%. In raw-value mode, both values use the same mean and standard deviation.
Equal bounds produce a zero interval probability. Reversed bounds are rejected rather than silently swapped. Each resulting Z score must be between -12 and 12; this is the calculator's supported numerical range, not a limit on all possible standardized values.
What this calculator does not assume for you
A standard deviation of zero is invalid here because standardization would divide by zero. Negative standard deviations are also invalid. If you enter a sample standard deviation, the result still uses the chosen normal model; it does not automatically become a Student's t calculation or account for uncertainty in estimated parameters.
This tool does not perform a normality test, a two-sided significance test, a confidence interval or a sample-size calculation. A test statistic must be interpreted using the correct null model and alternative hypothesis. Do not label every normal tail area a p-value.
Reference and privacy
For normal-distribution definitions, see the NIST/SEMATECH e-Handbook: Normal Distribution. This calculator numerically integrates the standard normal density. Its curve is an illustration of the selected model, not a histogram of your data.
Calculations happen in your browser. This calculator does not save or transmit your entries. The hosting website may have separate analytics and privacy practices.
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