Z-Score Calculator

This calculator computes z = (Value − Mean) / Standard Deviation (line 79), then runs that z-score through an Abramowitz & Stegun (formula 7.1.26) approximation of the standard normal CDF (lines 7-25) to get Percentile Rank and Cumulative Probability, rather than reading from an exact table. Standard Deviation gets a dedicated, exact-equality branch: `if (standardDeviation === 0)` (line 32) skips the division entirely and instead compares Value to Mean directly, returning a Z-Score of exactly 0 and a Percentile of 50, 100, or 0 depending on whether Value equals, exceeds, or falls short of Mean — a deliberate guard against dividing by zero, not a claim that a zero-spread dataset has a meaningful z-score. At the defaults (Value 85, Mean 100, Standard Deviation 15), Z-Score comes to exactly −1, and nudging Mean and Value up and down by the same 10% shows Mean moving Z-Score more than Value does, purely because Mean's default magnitude (100) is larger than Value's (85), so an equal 10% nudge produces a bigger absolute swing — not because the formula weighs Mean more heavily.

Raising Value always raises Z-Score, and raising Mean always lowers it, regardless of which side of the mean Value sits on (line 79's slopes are constant), but Standard Deviation's direction is not fixed the same way: at these defaults, where Value sits below Mean, raising Standard Deviation nudges Z-Score up, toward 0; if Value instead sat above Mean, the identical change to Standard Deviation would push Z-Score down instead, because it's shrinking the magnitude of whichever-signed ratio you started with. What this does not account for: skewed or non-normal distributions, where a z-score's percentile interpretation stops applying.