Z-Score Calculator
Calculate the z-score of a value relative to a normal distribution.
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Table of Contents
Z-Score Formula
A z-score (or standard score) represents the number of standard deviations a value is from the mean of a normal distribution.
Where:
- z is the z-score
- x is the value
- μ is the mean
- σ is the standard deviation
How to Calculate Z-Score
To calculate a z-score, follow these steps:
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1Identify the value (x) you want to convert to a z-score
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2Determine the mean (μ) of the distribution
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3Find the standard deviation (σ) of the distribution
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4Apply the z-score formula: z = (x - μ) / σ
Interpreting Z-Scores
Understanding what z-scores tell you:
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1Positive Z-Score:
Indicates the value is above the mean.
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2Negative Z-Score:
Indicates the value is below the mean.
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3Magnitude:
The larger the absolute value, the further the value is from the mean.
Practical Examples
Example 1 Test Scores
A student scored 85 on a test with a mean of 75 and standard deviation of 5.
x = 85, μ = 75, σ = 5
z = (85 - 75) / 5 = 2.0
This score is 2 standard deviations above the mean.
Example 2 Height
A person is 170 cm tall in a population with mean height of 175 cm and standard deviation of 10 cm.
x = 170, μ = 175, σ = 10
z = (170 - 175) / 10 = -0.5
This height is 0.5 standard deviations below the mean.
Example 3 IQ Scores
A person has an IQ of 130 in a population with mean IQ of 100 and standard deviation of 15.
x = 130, μ = 100, σ = 15
z = (130 - 100) / 15 = 2.0
This IQ score is 2 standard deviations above the mean.