Statistics

Z-Score Calculator

Calculate Z-scores. Fast, accurate, and completely free.

Favorites

Recent Tools

Results

Z-Score

—

P(Z ≤ z)

—

Percentile

—

P-Value (2-tailed)

—

P(Z > z)

—

Calculation Steps

Common Z-Table Values

Z-Score	P(Z ≤ z)	Percentile
−3.0	0.0013	0.13%
−2.5	0.0062	0.62%
−2.0	0.0228	2.28%
−1.645	0.0500	5.00%
−1.0	0.1587	15.87%
−0.5	0.3085	30.85%
0.0	0.5000	50.00%
0.5	0.6915	69.15%
1.0	0.8413	84.13%
1.645	0.9500	95.00%
1.96	0.9750	97.50%
2.0	0.9772	97.72%
2.5	0.9938	99.38%
2.576	0.9950	99.50%
3.0	0.9987	99.87%

Mathematical Formula

Z = \frac{X - \mu}{\sigma}

Z = Standard score (number of standard deviations from the mean)

X = Raw score / observed value

μ = Population mean

σ = Population standard deviation

P(Z ≤ z) = Cumulative probability (area to the left)

How to Use this Calculator

Enter the Raw Score (X) — the value you want to standardize.
Enter the Population Mean (μ) of the distribution.
Enter the Population Standard Deviation (σ).
Click Calculate to see the Z-score, cumulative probability, percentile, and P-value.
Examine the normal distribution curve with the shaded area representing your Z-score.
Refer to the Z-table for common benchmark values.

What Is a Z-Score?

A Z-score (standard score) measures how many standard deviations a data point is from the mean of a distribution. A Z-score of 0 means the value is exactly at the mean. Positive Z-scores indicate values above the mean, while negative Z-scores indicate values below it. Z-scores standardize different distributions onto a common scale, enabling direct comparison.

The Z-Score Formula

The formula Z = (X − μ) / σ transforms any normally distributed variable into a standard normal distribution with mean 0 and standard deviation 1. This transformation is fundamental in statistics because the standard normal distribution has well-tabulated probabilities that enable hypothesis testing and confidence interval construction.

Cumulative Probability P(Z ≤ z)

The cumulative probability tells you the proportion of data that falls at or below a given Z-score. For example, P(Z ≤ 1.96) ≈ 0.975, meaning 97.5% of values in a standard normal distribution are 1.96 or fewer standard deviations above the mean. This probability is computed using the cumulative distribution function (CDF) of the standard normal distribution.

Percentile Rank

The percentile rank is simply the cumulative probability expressed as a percentage. A Z-score of 1.0 corresponds to the 84.13th percentile — meaning the value exceeds approximately 84% of all observations. Percentiles are widely used in standardized testing, growth charts, and performance benchmarking.

P-Value and Hypothesis Testing

In hypothesis testing, the two-tailed P-value is the probability of observing a value at least as extreme as the Z-score (in either direction). P-value = 2 × (1 − P(Z ≤ |z|)). A P-value below the significance level (commonly 0.05) leads to rejecting the null hypothesis. The Z-test is one of the most fundamental statistical hypothesis tests.

Interpreting Z-Scores

As a rule of thumb: |Z| < 1 is typical (about 68% of data), 1 ≤ |Z| < 2 is somewhat unusual, 2 ≤ |Z| < 3 is rare (about 5% of data in tails), and |Z| ≥ 3 is extremely rare (about 0.3%). Values with |Z| > 3 are often considered outliers in many practical applications.

Applications

Z-scores are used in quality control (Six Sigma), finance (VaR calculations), education (standardized test scores like SAT/GRE), medicine (growth percentiles), and any field where you need to compare values from different normal distributions. Understanding Z-scores is essential for statistical reasoning and data-driven decision making.

Frequently Asked Questions (FAQ)

What does a Z-score of 2 mean?

A Z-score of 2 means the value is exactly 2 standard deviations above the mean. In a normal distribution, about 97.7% of values fall below this point, placing it at approximately the 97.7th percentile.

Can a Z-score be negative?

Yes. A negative Z-score simply means the value is below the mean. For example, Z = −1.5 means the value is 1.5 standard deviations below the mean.

How accurate is the CDF approximation?

This calculator uses a highly accurate polynomial approximation (Abramowitz and Stegun) with error less than 7.5 × 10⁻⁸, which is more than sufficient for practical purposes.

What is the difference between one-tailed and two-tailed P-values?

A one-tailed P-value tests the probability in one direction only (e.g., greater than). A two-tailed P-value tests both directions and equals 2 × one-tailed. Two-tailed tests are more conservative and commonly used.

Does this calculator assume a normal distribution?

Yes. Z-scores and the associated probabilities are based on the assumption that the underlying data follows a normal (Gaussian) distribution. For non-normal data, Z-scores can still be computed but the probability interpretations may not be accurate.

What is the 68-95-99.7 rule?

For a normal distribution, approximately 68.3% of data falls within Z = ±1, 95.4% within Z = ±2, and 99.7% within Z = ±3. This empirical rule is a quick way to assess how unusual a value is.

Explore more tools you might find useful