Z-Score: How many standard deviations your raw value sits above (positive) or below (negative) the mean. A z-score of 0 means your value equals the mean exactly.
Percentile: The percentage of values in a normal distribution that fall at or below your score. A percentile of 84% means your value is higher than roughly 84 out of 100 observations.
Area Left (P ≤ z): The probability of drawing a value at or below your score from a standard normal distribution. This is the same number as the percentile divided by 100.
Area Right (P > z): The probability of drawing a value above your score. Useful for one-tailed hypothesis tests where you want to know how unusual a result is on the upper end.
How This Calculator Works
You enter a raw score, the population mean, and the standard deviation. The tool computes z = (X − μ) / σ, then feeds that z into the standard normal cumulative distribution function (CDF) using the Horner-form error function approximation. The CDF gives the left-tail probability, and one minus that gives the right tail. Results assume the data follows a normal (Gaussian) distribution — for heavily skewed or small-sample data, interpret percentiles with caution.
Quick Questions
What z-scores are considered unusual?
In many fields, a z-score beyond ±1.96 (roughly the 2.5th or 97.5th percentile) is considered statistically unusual at the 5% significance level. Beyond ±2.58 corresponds to 1% significance.
Can I use this for sample data instead of a population?
If you're working with a sample mean and sample standard deviation, the result is technically a t-score, not a z-score. For large samples (n > 30), the difference is negligible; for smaller samples, use a t-distribution table instead.
What if my standard deviation is zero?
A standard deviation of zero means every value in the data set is identical, so the z-score is undefined (division by zero). The calculator will show "N/A" in that case.
How do I convert a z-score back to a raw score?
Use X = μ + z × σ. For example, if the mean is 100, the standard deviation is 15, and z is 1.5, the raw score is 100 + 1.5 × 15 = 122.5.
Is the percentile the same as a z-table lookup?
Yes. The percentile shown here is equivalent to looking up the cumulative probability in a standard normal (z) table. The calculator uses an error-function approximation accurate to about six decimal places.