Z-Score Calculator
Convert a raw value into a z-score using the mean and standard deviation of its distribution, and see the corresponding standard normal probabilities.
Results from this calculator are estimates provided for general informational purposes only, based on formulas, rates, and standards commonly accepted as of 2026. Figures may differ slightly from other calculators or professional sources due to rounding methods, differing assumptions, or regional regulations, and rules may change over time. Always consult a qualified professional — such as a financial advisor, healthcare provider, or other relevant specialist — before making decisions based on these results.
Z-Score
1.4
Example
A value of 72 from a distribution with mean 65 and standard deviation 5 has a z-score of 1.4, meaning 91.92% of values fall below it and 8.08% fall above it.
P(x < Z) — Left Tail
91.924%
P(x > Z) — Right Tail
8.076%
P(0 to Z)
41.924%
P(-Z < x < Z)
83.849%
Standard normal curve, shaded left of Z
What Is a Z-Score?
A z-score is the signed number of standard deviations a value sits above or below the mean of its distribution. A z-score of 0 means the value equals the mean; positive scores sit above the mean, negative scores sit below it. Because z-scores are unitless, they let you compare values from completely different scales — like an exam score and a height measurement — on the same standardized footing.
The z-score is calculated as z = (x − μ) / σ, where x is the raw value,
μ is the population mean, and σ is the population standard deviation. Once
converted, the z-score maps directly onto the standard normal distribution, letting you read off
probabilities from the classic z-table.
Reading the Probabilities
P(x < Z) is the area under the curve to the left of your z-score — the proportion of the population below that value. P(x > Z) is the mirror image, the proportion above it. P(0 to Z) is the area between the mean and your z-score, and P(-Z < x < Z) is the middle band symmetric around the mean out to your z-score's distance in both directions.
Common Z-Score Milestones
A z-score of ±1 covers about 68% of a normal distribution; ±2 covers about 95%; and ±3 covers about 99.7% — the well-known "68-95-99.7 rule." Values beyond z = ±3 are rare enough that they're often flagged as statistical outliers.
Uses in Hypothesis Testing
Z-scores underpin standard hypothesis tests: a result is often considered "statistically significant" at the common 95% confidence level when its z-score falls beyond about ±1.96, meaning less than a 5% chance the result occurred by random variation alone if the null hypothesis were true.
Population vs. Sample Z-Scores
This calculator assumes you know the true population mean and standard deviation. When you only have a sample, the same formula using the sample mean and sample standard deviation is still called a z-score if the sample is large, though smaller samples often call for a t-score instead, which accounts for the added uncertainty.
Example — Your Current Inputs
A value of 72 from a distribution with mean 65 and standard deviation 5 has a z-score of 1.4, meaning 91.92% of values fall below it and 8.08% fall above it.
Additional Example — Exam Scores
On an exam with a mean of 75 and a standard deviation of 8, a score of 91 has a z-score of 2.0 — meaning that score sits at roughly the 97.7th percentile, better than about 97.7% of test takers.
About These Parameters
- Value (x)
- The specific data point you want to standardize — a test score, a measurement, or any single observation from the distribution.
- Mean (μ) & Standard Deviation (σ)
- These describe the overall distribution the value comes from — the average and the typical spread around that average. Both must be known or estimated from the full population or a representative sample.
Frequently Asked Questions
What does a negative z-score mean?
It means the value is below the distribution's mean — the more negative the z-score, the further below average the value sits.
How accurate is the probability calculation?
This calculator uses a numerical approximation of the standard normal distribution accurate to about seven decimal places — more than sufficient for any practical statistics application.
Can z-scores be used for non-normal data?
The z-score itself can be computed for any distribution, but the probability interpretations shown here assume the underlying data is approximately normally distributed — for skewed data, those probabilities won't be accurate.