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How It Works
A Z score measures how many standard deviations a data point is from the mean using z = (x − μ) / σ. A score of 0 means the value equals the mean; positive scores are above, negative scores are below. You can rearrange the formula to solve for any of the four variables.
Example Problem
A test has mean 100 and standard deviation 15. A student scores 130. What is their Z score?
- z = (130 − 100) / 15 = 30 / 15 = 2.0
- The student scored 2 standard deviations above the mean (~97.7th percentile).
Frequently Asked Questions
What does a Z score of 1.5 mean?
A Z score of 1.5 means the value is 1.5 standard deviations above the mean. In a normal distribution, roughly 93.3% of values fall below this point.
How to interpret a negative Z score?
A negative Z score means the value is below the mean. A z of −2 means the observation is 2 standard deviations below average, placing it around the 2.3rd percentile.
When should you use Z scores instead of raw values?
Z scores standardize data so you can compare values across different scales. For instance, comparing a test score (mean 500, SD 100) with a GPA (mean 3.0, SD 0.5) requires converting both to Z scores first.
What Z score is considered an outlier?
Values with |z| > 3 are commonly flagged as outliers. In a normal distribution, only about 0.3% of data falls beyond 3 standard deviations from the mean.
Related Calculators
- Statistics Calculator — compute mean, median, standard deviation, and variance.
- Interpolation Calculator — estimate values between known data points.
- Percent Error Calculator — measure how far a value deviates from the expected result.
- Logarithm Calculator — compute log transformations common in statistical normalization.
- Natural Log Calculator — evaluate ln functions used in probability distributions.