Z-Score Calculator
Enter a raw score with the mean and standard deviation of its distribution. The calculator standardises it and looks up the matching areas under the normal curve.
Result
- P(X < x), area to the left
- 0.933193
- P(X > x), area to the right
- 0.066807
- Two-tailed p-value
- 0.133614
- P(within |z| of the mean)
- 0.866386
- Distance from the mean
- 15
Probabilities assume the values follow a normal distribution.
Formula
- z = (x − μ) ÷ σ
- Percentile = Φ(z) × 100, where Φ is the standard normal cumulative distribution
- x = μ + z × σ
How the Z-Score Calculator works
A z-score says how many standard deviations a value lies above or below the mean. A z of 0 is exactly average, positive values are above average and negative values are below.
Standardising lets you compare scores from different scales and read probabilities from one table. In a normal distribution about 68 percent of values have a z between −1 and 1, about 95 percent between −2 and 2, and about 99.7 percent between −3 and 3.
Worked example
A score of 85 on a test with mean 70 and standard deviation 10: z = (85 − 70) ÷ 10 = 1.5. That is the 93.32nd percentile, so about 6.68 percent of scores are higher.
Frequently asked questions
What is a good z-score?
It depends on context. A z of 0 is average. Values beyond about ±2 are unusual, occurring roughly 5 percent of the time, and values beyond ±3 are rare.
What z-score corresponds to 95 percent confidence?
1.96. The middle 95 percent of a normal distribution lies within 1.96 standard deviations of the mean. For 90 percent it is 1.645 and for 99 percent it is 2.576.
How do I convert a z-score to a percentile?
Look up the cumulative probability for that z and multiply by 100. A z of 1.0 is the 84.13th percentile.
Related calculators
- Standard Deviation and Variance CalculatorCalculate sample and population standard deviation and variance for a data set, with the mean, sum of squares and standard error.
- Sample Size CalculatorCalculate the survey sample size needed for a given confidence level and margin of error, with an optional finite population correction.
- Average CalculatorCalculate the average (mean) of a list of numbers, with the sum, count, median, mode, range and geometric mean.
- Median, Mode and Quartile CalculatorFind the median, mode, range, quartiles and interquartile range of a data set, with outlier fences and the sorted list.
- Ratio CalculatorSimplify a ratio, solve for a missing value in A : B = C : D, or split a total in a given ratio. Works with whole numbers and decimals.
- Proportion CalculatorSolve a proportion a/b = c/d for the missing value by cross-multiplication. Enter any three of the four numbers.
- Factor CalculatorFind all factors of a number, its factor pairs and its prime factorization. Also tells you whether the number is prime.
- GCF Calculator (Greatest Common Factor)Find the greatest common factor of two or more numbers, with the Euclidean algorithm steps and prime factorizations. Also gives the LCM.
- LCM Calculator (Least Common Multiple)Find the least common multiple of two or more numbers, with prime factorizations. Also gives the greatest common factor.
- Rounding CalculatorRound a number to the nearest whole number, tenth, hundredth, thousandth, ten, hundred or thousand. Exact decimal rounding with no floating point errors.
- Significant Figures CalculatorCount the significant figures in a number and round it to any number of significant figures, with scientific notation.
- Modulo CalculatorCalculate a mod n, the remainder after division, with the quotient. Handles negative numbers under both the mathematical and programming conventions.
Related unit converters
Calculations run in your browser and nothing you enter is sent to a server. Results are provided for general information. Check critical figures independently before relying on them.