Z-Score Calculator
Calculate the Z-score from a raw score, population mean, and standard deviation. Visualizes the result on a standard normal distribution curve with shaded probability area.
Z-Score
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Interpretation
Moderately above average
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Z-Score
+1.5
z = (85 − 70) / 10 = 1.5
P(Z ≤ z)
0.983053
Percentile
98.305263%
Above
1.694737%
Empirical Rule (68-95-99.7)
±1σ: ~68% of data falls within 1 standard deviation
±2σ: ~95% of data falls within 2 standard deviations
±3σ: ~99.7% of data falls within 3 standard deviations
The Formula
The Z-score measures how many standard deviations a data point is from the population mean. A Z-score of 0 means the value is exactly at the mean. Positive Z-scores are above the mean; negative Z-scores are below. The standard normal distribution has a mean of 0 and a standard deviation of 1.
Variable Definitions
Raw Score
The individual data point being standardized. Any value from the original distribution.
Population Mean
The mean of the entire population. The center of the distribution.
Standard Deviation
The standard deviation of the population. Must be greater than 0.
Z-Score
Number of standard deviations from the mean. Positive = above mean, negative = below mean, zero = at the mean.
How to Use This Calculator
- 1
Enter the raw score (x), population mean (μ), and standard deviation (σ).
- 2
The Z-score is calculated and plotted on the standard normal distribution curve.
- 3
View the cumulative probability P(Z ≤ z), percentile rank, and practical interpretation text.
- 4
The bell curve visualization shows exactly where the score falls in the distribution relative to the mean.
- 5
Use the "above this score" percentage to understand how rare or common the score is.
The Z-score measures how many standard deviations a value is from the mean
Understanding the Concept
The Z-score is one of the most important concepts in statistics. It transforms any normal distribution into the standard normal distribution (mean = 0, σ = 1), allowing comparison across different scales. For example, an SAT score of 650 with mean 500 and σ 100 gives a Z-score of 1.5, meaning it is 1.5 standard deviations above average. Z-scores are used for standardized testing (SAT, IQ tests, GRE), quality control (Six Sigma methodology), medical reference ranges (bone density, growth charts), and identifying outliers. The empirical rule (68-95-99.7 rule) states that approximately 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean. A Z-score of 1.96 corresponds to the 97.5th percentile — this is the critical value used for 95% confidence intervals, which is why it is the most common Z-value in inferential statistics. The interpretation text in the results provides a plain-English description of what the Z-score means.
Frequently Asked Questions
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