Z-Score Calculator

Calculate the z-score (standard score) to see how many standard deviations a value is from the mean.

Value (x)

Mean (μ)

Standard Deviation (σ)

Results update as you type.

Z-Score

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Deviation from Mean—

How it works

The z-score, also called the standard score, is a fundamental tool in statistics for comparing values from different distributions or understanding where a value falls relative to a group. A z-score tells you exactly how many standard deviations above or below the mean a particular value is. A z-score of +2 means the value is 2 standard deviations above the mean; a z-score of −1.5 means it's 1.5 standard deviations below the mean. Z-scores enable comparison across different scales. For example, if a student scored 85 on a math test (mean 70, SD 10) and 72 on an English test (mean 65, SD 5), their math z-score is +1.5 and English z-score is +1.4 — so they performed slightly better in math relative to their peers, even though the raw scores differ. Using the standard normal distribution table (Z-table), z-scores also map to percentile ranks. A z-score of +1.96 corresponds roughly to the 97.5th percentile. In many fields — medicine, education, finance — z-scores are used to identify outliers, normalize data, and build statistical tests. Because the calculation strips away the original units, a z-score places any measurement on one common scale, which is exactly what makes standardized comparison across different tests and datasets possible.

Formula

z = (x − μ) / σ

Subtracting the mean μ from the value x shifts the center of the data to zero, so the result tells you the raw distance above or below average. Dividing by the standard deviation σ rescales that distance into standard-deviation units, which lets you compare values from different datasets on one common scale. A positive z-score sits above the mean, a negative one below, and zero equals the mean exactly.

Source:NIST/SEMATECH e-Handbook of Statistical Methods

We review all our tools from time to time to ensure accuracy.

Worked example

Value (x):85Mean (μ):70Standard Deviation (σ):10

A test score of 85 with a class mean of 70 and standard deviation of 10 gives z = (85 − 70) / 10 = 1.5. This means the score is 1.5 standard deviations above average, placing it roughly at the 93rd percentile.

How to use this calculator

1
Enter your values
Enter value (x), mean (μ) and standard deviation (σ).
2
Adjust the options
Fine-tune any options to match your situation.
3
Read your result
Z-Score updates instantly as you type — no button to press.

Frequently asked questions

What is a good or bad z-score?

There's no universally 'good' or 'bad' z-score — it depends entirely on context. In education, a high positive z-score means scoring above average, which is typically desirable. In quality control, z-scores far from 0 (either positive or negative) might indicate defects. In medical testing, z-scores far from 0 often flag abnormal results. Generally, z-scores between −2 and +2 include about 95% of values in a normal distribution, so anything beyond ±2 is considered unusual.

How do I convert a z-score to a percentile?

Z-scores map to percentiles through the standard normal distribution. A z-score of 0 is at the 50th percentile (exactly average). A z-score of +1 is approximately the 84th percentile. A z-score of +2 is about the 97.7th percentile. A z-score of −1 is approximately the 16th percentile. For precise conversions, you'd use a standard normal distribution table (Z-table) or a statistical calculator. The general pattern: higher z-score = higher percentile rank.

What does a negative z-score mean?

A negative z-score means the value is below the mean. For example, a z-score of −1.5 means the value is 1.5 standard deviations below average. Whether this is concerning depends on context: a body temperature z-score of −2 might indicate hypothermia, while a z-score of −0.5 on a test just means slightly below average. The magnitude (absolute value) tells you how far from average; the sign tells you the direction (above or below).

Can z-scores be used with any data distribution?

Z-scores can be calculated for any numerical data, but their percentile interpretations only apply precisely when the data follows a normal (bell curve) distribution. For heavily skewed distributions or those with multiple peaks, a z-score of +2 may not correspond to the 97.7th percentile. In those cases, z-scores still measure distance from the mean in standard deviation units, which is useful for comparison and outlier detection — but interpret percentile mappings with caution.

How are z-scores used in real-world applications?

Z-scores are used across many fields. In medicine, bone density z-scores compare a patient's density to age-matched peers, with z-scores below −2 indicating clinical concern. In finance, z-scores (Altman Z-score) help predict corporate bankruptcy risk. In education, standardized tests are often reported as z-scores or derived scores. In manufacturing, z-scores measure process capability (how many standard deviations between the mean and specification limits). In machine learning, z-score normalization (standardization) scales features to have mean 0 and standard deviation 1.