Z-Score Calculator — Calculate Z-Score, Probability & Percentile

A z-score measures how many standard deviations a value is from the mean: z = (x − μ) / σ. Enter your value, mean, and standard deviation to instantly calculate the z-score, cumulative probability, and percentile — or reverse a z-score back to the original scale.

Frequently Asked Questions

What is a z-score?

A z-score (standard score) measures how many standard deviations a data point is from the mean of its distribution. The formula is z = (x − μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. A z-score of 0 means the value equals the mean; positive scores are above the mean, negative scores are below it.

How do I calculate a z-score?

Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) / σ. For example, if a test score is 85, the class mean is 70, and the standard deviation is 10, the z-score is (85 − 70) / 10 = 1.5. This means the score is 1.5 standard deviations above the mean.

What does a z-score tell you?

A z-score tells you where a value sits relative to the rest of the distribution. A z-score of +2 means the value is in roughly the top 2.3% of the distribution. A z-score of −1 means it is in approximately the bottom 15.9%. Z-scores allow you to compare values from different distributions on a common scale.

What is a good z-score?

It depends on context. In standardized testing, higher z-scores are better. In quality control, z-scores close to 0 indicate a process is near target. In statistics, |z| > 1.96 is often considered statistically significant at the 95% confidence level, and |z| > 3 is frequently used to flag outliers.

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

Use the cumulative distribution function (CDF) of the standard normal distribution. A z-score of 0 corresponds to the 50th percentile. A z-score of 1.645 corresponds to the 95th percentile. A z-score of 1.96 corresponds to approximately the 97.5th percentile. This calculator computes the percentile automatically.

What is cumulative probability in z-score calculations?

Cumulative probability (also called the CDF value) is the probability that a randomly selected value from the distribution is less than or equal to your observed value. For example, a z-score of 1.0 has a cumulative probability of about 0.8413, meaning approximately 84.13% of values fall at or below that point.

What is the difference between z-score and standard deviation?

Standard deviation (σ) measures the spread of an entire dataset — it is a single number describing variability. A z-score applies the standard deviation to a specific data point to express how far that point is from the mean in units of standard deviation. In other words, z-score = (value − mean) / standard deviation.

Can a z-score be negative?

Yes. A negative z-score simply means the value is below the mean. For example, if a student scores 60 on a test where the mean is 70 and the standard deviation is 10, the z-score is (60 − 70) / 10 = −1.0, meaning the score is 1 standard deviation below average.

Z-Score Formula

A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of its distribution. The formula is:

Z-Score

z = (x − μ) / σ

Where x is the observed value, μ is the population mean, and σ is the standard deviation.

Example: x = 85, μ = 70, σ = 10 → z = (85 − 70) / 10 = 1.5

Value from Z-Score (Inverse)

x = μ + z × σ

Example: z = 1.5, μ = 70, σ = 10 → x = 70 + 1.5 × 10 = 85

Standard Normal Distribution

The standard normal distribution (also called the z-distribution) is a normal distribution with a mean of 0 and a standard deviation of 1. When you convert a value to a z-score, you are standardizing it onto this distribution.

The cumulative probability (also called the cumulative distribution function, or CDF) tells you what fraction of values in the distribution fall at or belowa given z-score. This calculator uses the Abramowitz & Stegun polynomial approximation to the error function (erf) for fast, accurate results.

Key property:By the empirical rule (68–95–99.7 rule):

  • ~68% of data falls within z = ±1
  • ~95% of data falls within z = ±1.96
  • ~99.7% of data falls within z = ±3

How to Interpret Z-Scores

Positive Z-Score

A positive z-score means the value is above the mean. For example, z = 2.0 means the value is 2 standard deviations above average, placing it in roughly the 97.7th percentile.

Negative Z-Score

A negative z-score means the value is belowthe mean. For example, z = −1.0 means the value is 1 standard deviation below average, placing it in roughly the 15.9th percentile.

Z-Score of Zero

A z-score of exactly 0 means the value equals the mean exactly, placing it at the 50th percentile.

Practical Interpretation

Z-scores are widely used in standardized testing, quality control (Six Sigma), finance (measuring returns relative to benchmarks), and research (identifying outliers, computing p-values). A value with |z| > 3 is often treated as a statistical outlier.

How to Use This Calculator

  1. Select a mode— choose “Find Z-Score” to convert a raw value to a z-score, or “Find Value” to convert a z-score back to the original scale.
  2. Enter your inputs— fill in the value (or z-score), the mean, and the standard deviation.
  3. Read the result instantly— the calculator shows the z-score (or value), cumulative probability, and percentile.
  4. Copy the results— click the Copy button to copy all results to your clipboard.

Common Z-Values Table

These critical z-values are widely used in hypothesis testing and confidence intervals:

Z-ScoreConfidence LevelCumulative ProbabilityPercentile
1.28280%0.8997~90th
1.64590%0.9500~95th
1.96095%0.9750~97.5th
2.32698%0.9900~99th
2.57699%0.9950~99.5th
3.00099.7%0.9987~99.9th

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