Z-Score Calculator Formula

Understand the math behind the z-score calculator. Each variable explained with a worked example.

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Formulas Used

Z-Score

z_score = (x - mu) / sigma

Absolute Distance (in SDs)

abs_z = abs((x - mu) / sigma)

Raw Deviation

raw_deviation = x - mu

Variables

VariableDescriptionDefault
xValue (x)85
muMean (mu)70
sigmaStandard Deviation (sigma)10

How It Works

How to Calculate a Z-Score

Formula

Z = (X - mu) / sigma

The z-score expresses how far a data point is from the population mean in units of standard deviations. A positive z-score means the value is above the mean; a negative z-score means below. Z-scores allow comparison across different scales and distributions.

Worked Example

A student scores 85 on a test where the class mean is 70 and SD is 10.

x = 85mu = 70sigma = 10
  1. 01Z = (X - mu) / sigma
  2. 02Z = (85 - 70) / 10
  3. 03Z = 15 / 10 = 1.5
  4. 04The score is 1.5 standard deviations above the mean

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