Math Tools

Z-Score Calculator

Compute z-scores to standardize values and see where they fall in a normal distribution. Enter a raw value, population mean, and standard deviation.

Raw Value (x)

Population Mean (μ)

Standard Deviation (σ)

Z-Score

1.50

Above average by 1.50 standard deviations

Percentile93.32%

Probability Below93.32%

Probability Above6.68%

What is the Z-Score Calculator?

A z-score is a standardized value that tells you how many standard deviations a data point is from the population mean. Calculated as z = (x − μ) / σ, it transforms raw measurements into a common scale (typically −4 to +4) so values from different datasets can be compared. A z-score of +2 means 2 standard deviations above the mean; −1 means 1 below.

How it works

Enter your raw value, the population mean, and standard deviation. The calculator subtracts the mean from your value, then divides by the standard deviation. The result is your z-score. The tool also computes the percentile—the percentage of the normal distribution that falls below your value—and the probability of observing values above and below.

z = (x − μ) / σ, where x is the raw value, μ is the mean, and σ is the standard deviation

Subtract the mean from your raw value to center it at zero, then divide by the standard deviation to scale it. This gives the number of standard deviations between your value and the mean. Positive z-scores are above the mean; negative are below.

Examples

Input	Result	Notes
x=85, mean=70, standard deviation=10	z = 1.5	(85 − 70) / 10 = 1.5. The value is 1.5 standard deviations above the mean, at the 93rd percentile.
x=65, mean=70, standard deviation=10	z = −0.5	(65 − 70) / 10 = −0.5. The value is half a standard deviation below the mean, at the 31st percentile.
x=100, mean=100, standard deviation=15	z = 0	(100 − 100) / 15 = 0. The value equals the mean exactly, at the 50th percentile.

How to use the Z-Score Calculator

Gather your raw value, the population mean, and the standard deviation

Enter the raw value into the first field

Enter the population mean into the second field

Enter the standard deviation into the third field

The calculator instantly computes the z-score and shows the percentile and probabilities

Interpret: positive z = above average, negative z = below average, z=0 = at mean

Benefits

Standardizes values across different scales—compare test scores, heights, and prices on one metric

Identifies outliers—z-scores beyond ±3 are rare and often suspicious

Supports quality control—z-scores help detect defects in manufacturing and processes

Enables hypothesis testing—z-scores form the basis of many statistical tests

Quick percentile lookup—instantly see what percentage of a normal distribution falls below your value

Supports academic and professional work—essential in statistics, psychology, finance, and engineering

Tips & common mistakes

Common mistakes

Using sample standard deviation instead of population—verify which applies to your data (s ≠ σ)

Entering the wrong mean or standard deviation—double-check these are for the correct population

Forgetting that z-scores assume normality—the 68-95-99.7 rule only holds for bell-curve distributions

Confusing percentile with percentage—percentile is cumulative (everything below); percentage could mean anything

Tips

Remember the 68-95-99.7 rule: roughly 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ

Use z-scores to spot extreme values—anything beyond ±3 is rare and worth investigating

Z-scores are scaleless—you can compare a test score to a weight if both are standardized

For skewed distributions, z-scores and percentiles are less reliable—check for normality first

Frequently asked questions

What does a z-score of 2 mean?

A z-score of 2 means the value is 2 standard deviations above the mean. In a normal distribution, about 97.7% of values fall below it (the 97.7th percentile).

Can z-scores be negative?

Yes. Negative z-scores mean the value is below the mean. A z-score of −1.5 is 1.5 standard deviations below the mean.

What's the difference between z-score and percentile?

Z-score is the number of standard deviations from the mean (e.g., 2.0). Percentile is the percentage of the distribution that falls below that value (e.g., 97.7th percentile). This calculator computes both.

How do I interpret a z-score of 0?

A z-score of 0 means your value equals the mean exactly. It's at the 50th percentile—half the distribution is below, half above.

Are z-scores only for normally distributed data?

Z-scores work mathematically for any data, but the percentile interpretation assumes a normal distribution. Always check if your data is bell-curve shaped.

What is the standard deviation?

Standard deviation measures spread—how far values typically stray from the mean. A large SD means values are scattered; a small SD means they're tightly clustered.