Z-Score Calculator

What is a Z-Score?

A z-score (also called a standard score) indicates how many standard deviations an element is from the mean. A z-score can be positive (above the mean) or negative (below the mean). Z-scores are used to standardize values from different distributions for comparison.

Z-Score Formula

z = (x - mean) / standard deviation

Where:

• x = raw score (observed value)
• mean = population mean
• standard deviation = population standard deviation

To Find Raw Score from Z-Score

x = mean + (z * standard deviation)

Interpreting Z-Scores

• z = 0: Value is exactly at the mean
• z > 0: Value is above the mean
• z < 0: Value is below the mean
• |z| = 1: Value is 1 standard deviation from mean
• |z| = 2: Value is 2 standard deviations from mean
• |z| = 3: Value is 3 standard deviations from mean

Z-Scores and Percentiles

For a normal distribution:

• z = -3: approximately 0.13th percentile
• z = -2: approximately 2.28th percentile
• z = -1: approximately 15.87th percentile
• z = 0: 50th percentile (median)
• z = 1: approximately 84.13th percentile
• z = 2: approximately 97.72nd percentile
• z = 3: approximately 99.87th percentile

Example Calculation

Given: Test score = 85, Mean = 75, Standard Deviation = 10

z = (x - mean) / std dev
z = (85 - 75) / 10
z = 10 / 10
z = 1.0

Interpretation: The score of 85 is 1 standard 
deviation above the mean, placing it at 
approximately the 84th percentile.

Applications of Z-Scores

Education

Comparing test scores across different tests with different scales and difficulties.

Quality Control

Identifying outliers in manufacturing processes. Values with |z| > 3 are often considered outliers.

Finance

Analyzing stock returns, risk assessment, and portfolio analysis.

Research

Standardizing data for statistical analysis, comparing results across studies.

The Empirical Rule (68-95-99.7)

For normally distributed data:

• 68% of data falls within z = -1 to z = 1
• 95% of data falls within z = -2 to z = 2
• 99.7% of data falls within z = -3 to z = 3