Z-Score Calculator

Z-Score is evaluated from Data Value, Population Mean and Standard Deviation. The calculation reports Z-Score, Percentile and Percent Above This Value.

Results

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About the Z-Score Calculator

Z-Score is treated here as a quantitative relation between Data Value, Population Mean and Standard Deviation and Z-Score, Percentile, Percent Above This Value and Interpretation.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
z = (x - mu) / sigma
Percentile = Phi(z) x 100
Phi(z) = standard normal CDF (area to the left of z)

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

z = (x - mu) / sigma
Percentile = Phi(z) x 100
Phi(z) = standard normal CDF (area to the left of z)

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: SAT Score Percentile

Inputs

x_val: 1350 mean_val: 1060 sd_val: 195
Z-Score: 1.4872. Percentile: 93.15%. Percent Above This Value: 6.85%. Interpretation: Within 2 SDs - moderately unusual (~95% within this range)

With Data Value = 1,350, Population Mean = 1,060 and Standard Deviation = 195 as the stated inputs, the result is Z-Score = 1.4872, Percentile = 93.15% and Percent Above This Value = 6.85%. Each value corresponds to the declared output fields.

Example 2: Outlier Detection — Quality Control

Inputs

x_val: 10.85 mean_val: 10 sd_val: 0.05
Z-Score: 17. Percentile: 100%. Percent Above This Value: 0%. Interpretation: More than 3 SDs - extreme outlier (less than 0.3% of data)

With Data Value = 10.85, Population Mean = 10 and Standard Deviation = 0.05 as the stated inputs, the result is Z-Score = 17, Percentile = 100% and Percent Above This Value = 0%. Each value corresponds to the declared output fields.

Example 3: Body Weight Percentile — Adult Male

Inputs

x_val: 205 mean_val: 195 sd_val: 30
Z-Score: 0.3333. Percentile: 63.06%. Percent Above This Value: 36.94%. Interpretation: Within 1 SD - typical value (~68% of data falls here)

With Data Value = 205, Population Mean = 195 and Standard Deviation = 30 as the stated inputs, the result is Z-Score = 0.3333, Percentile = 63.06% and Percent Above This Value = 36.94%. Each value corresponds to the declared output fields.

Example 4: Comparing Two Tests (Standardization)

Inputs

x_val: 78 mean_val: 65 sd_val: 8
Z-Score: 1.625. Percentile: 94.79%. Percent Above This Value: 5.21%. Interpretation: Within 2 SDs - moderately unusual (~95% within this range)

With Data Value = 78, Population Mean = 65 and Standard Deviation = 8 as the stated inputs, the result is Z-Score = 1.625, Percentile = 94.79% and Percent Above This Value = 5.21%. Each value corresponds to the declared output fields.

Common Use Cases

  • Compare test scores from different exams
  • Determine how far a value is from the mean
  • Convert raw scores to percentile ranks
  • Detect outliers in a dataset