Z-score CalculatorFree online math calculator — instant results, no sign-up
Z-score and percentile of a value.
About the Z-score Calculator
This calculator determines the standard score (z-score), cumulative probability, and percentile rank for a specific data point within a normal distribution. It is used by students, researchers, and analysts to measure how many standard deviations a value lies above or below a population or sample mean.
- Value
- The individual observation or raw score (x) being evaluated, expressed in the same units as the mean.
- Mean
- The central tendency or average (μ) of the distribution, in the same units as the value.
- Std. deviation
- The standard deviation (σ) measuring the spread of the data. Must be greater than zero.
Formula and method
z = (x - m) / s P(X < x) = Φ(z) Percentile = Φ(z) × 100
- xRaw score or observed value
- mMean of the distribution (μ)
- sStandard deviation of the distribution (σ)
- zZ-score (standard score)
- Φ(z)Standard normal cumulative distribution function evaluated at z
The tool calculates the z-score by subtracting the mean from the raw value and dividing the difference by the standard deviation. It then passes that z-score into the standard normal cumulative distribution function Φ(z) to determine the left-tail probability, rounded to 4 decimal places, and multiplies it by 100 to yield the percentile rank.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Value85
- Mean70
- Std. deviation10
Results
- Z-score1.5
- P(X < x)0.9332
- Percentile93.32%
How to use the Z-score Calculator
- 1Subtract the mean (m) from the observed value (x) to find the deviation.
- 2Divide the deviation by the standard deviation (s) to obtain the z-score.
- 3Evaluate the standard normal cumulative distribution function Φ(z) to find P(X < x).
- 4Multiply the cumulative probability by 100 to express the standing as a percentile rank.
Reading your result
A positive z-score indicates the value is above the mean, while a negative z-score means it is below the mean. The probability P(X < x) and the corresponding percentile reflect the proportion of observations in a standard normal distribution that fall below the given value.
Limitations
The cumulative probability and percentile calculations assume the underlying data follows a standard normal (Gaussian) distribution. If the data is skewed, bimodal, or has heavy tails, the z-score remains mathematically defined but the calculated percentile will not accurately reflect real-world rank.
Common questions
What does a z-score of 0 mean?+
A z-score of 0 means the observed value is exactly equal to the mean of the distribution, placing it at the 50th percentile.
Can a z-score be negative?+
Yes. A negative z-score simply indicates that the observed value falls below the distribution mean, resulting in a percentile rank below 50%.
How is the right-tail probability P(X > x) calculated from this tool?+
Because total probability equals 1, the right-tail probability P(X > x) is simply 1 minus P(X < x).
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.