Z-Score Calculator
Calculate the z-score (standard score) that tells you how many standard deviations a data point is from the mean. Solve for z-score, raw value, mean, or standard deviation using z = (x - μ) / σ.
The z-score is one of the most important concepts in statistics. It standardizes values from different distributions, allowing direct comparison.
The Formula: z = (x - μ) / σ
- z = z-score (number of standard deviations from the mean)
- x = raw value (the data point)
- μ = population mean
- σ = population standard deviation
Interpreting Z-Scores:
- z = 0: Value is at the mean
- z = 1: Value is 1 standard deviation above the mean
- z = -1: Value is 1 standard deviation below the mean
- z = 2: Value is 2 standard deviations above the mean
The Empirical Rule (for normal distributions):
- ~68% of data falls within z = -1 to z = +1
- ~95% falls within z = -2 to z = +2
- ~99.7% falls within z = -3 to z = +3
Common Application:
IQ scores have μ = 100 and σ = 15. An IQ of 130 has z = (130 - 100) / 15 = 2.0, meaning it is 2 standard deviations above the mean, placing it at approximately the 97.7th percentile.