What does a z-score of 0 mean?

A z-score of 0 indicates that the raw value is exactly equal to the population mean. It is right in the center of the normal distribution curve.

What is the formula for calculating a z-score?

The formula to calculate a z-score is: z = (X - μ) / σ, where X is the raw value, μ is the population mean, and σ is the standard deviation.

How do you interpret a negative z-score?

A negative z-score indicates that the raw value is below the population mean. For instance, a z-score of -1.5 means the data point is 1.5 standard deviations below the mean.

Z-Score Calculator

Calculate the z-score (standard score) of a raw value, given the population mean and standard deviation. Includes probability percentile lookup.

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Parameters

R Reset

Raw Value (X)The individual data value (X) to analyze.

Population Mean (μ)The average value of the dataset or population.

Standard Deviation (σ)Standard deviation. A measure of the dispersion or spread of values.

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Calculated Z-Score

1.5000Calculated · 100%

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Cumulative Probability P(Z ≤ z)0.933193

Percentile93.32%

Percentage Above Raw Value6.68%

Result Interpretation

The Z-score is 1.5000, representing the number of standard deviations from the mean.

Mathematical Explanation

z = \frac{x - \mu}{\sigma}

A raw value of 85 in a population with mean μ = 70 and standard deviation σ = 10 results in a Z-score of 1.5000. This indicates that the value is 1.50 standard deviations above the mean. Approximately 93.32% of the population lies at or below this value (percentile), and 6.68% lies above it.

How to Use the Z-Score Calculator

Standardize your data points with our free z-score calculator. A z-score (or standard score) indicates how many standard deviations a raw data point is above or below the population mean. Simply enter your raw value, mean, and standard deviation to calculate the z-score and find its corresponding standard normal cumulative probability.

Input variables:

Raw Value (X): Bounded parameter in the mathematical model.
Population Mean (μ): Bounded parameter in the mathematical model.
Standard Deviation (σ): Bounded parameter in the mathematical model.

Normal Distribution: Bell Curve & Standard Scores

Discover why the bell curve appears everywhere in nature, how the 68-95-99.7 rule works, and how to use z-scores to interpret any normally distributed dataset.

Read Full Guide →