Z-score calculator
A z-score says how many standard deviations a value sits from the mean. The percentile follows from the normal cumulative distribution.
Z-score
2.000 00
- Percentile
- 97.7250 %
- Share of the population above this
- 2.2750 %
- Two-tailed probability
- 4.550 01 %
- One in how many, two-tailed
- 22.0
Show the working
- Value = 130
- Mean = 100
- Standard deviation = 15
- Z-score = (130 - 100) / 15 = 2
- z_abs = abs(2) = 2
- z_t = 1 / (1 + 0.2316419 * 2) = 0.683 394
- z_pdf = 0.3989422804014327 * exp(-2^2 / 2) = 0.053 991
- z_tail = 0.053 991 * (0.319381530*0.683 394 - 0.356563782*0.683 394^2 + 1.781477937*0.683 394^3 - 1.821255978*0.683 394^4 + 1.330274429*0.683 394^5) = 0.022 750 1
- z_cdf = (2) >= 0 ? 1 - 0.022 750 1 : 0.022 750 1 = 0.977 25
- Percentile = 0.977 25 * 100 = 97.725
- Share of the population above this = 100 - 97.725 = 2.275 01
- Two-tailed probability = 2 * 0.022 750 1 * 100 = 4.550 01
- One in how many, two-tailed = 4.550 01 > 0 ? 100 / 4.550 01 : 0 = 21.978
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- z = (x - mean) / standard deviation
The normal CDF is computed from the Abramowitz and Stegun rational approximation, accurate to better than 7.5 x 10^-8 — two orders of magnitude finer than a printed z-table.
An IQ of 130 is z = 2, which is the 97.7th percentile.