Z-scores indicate how many standard deviations a value is from the mean.
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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.
