Z = 2.5 — Standard Normal Probability
Probabilities for this exact z-score. Use the calculator below to convert your own raw value, mean, and standard deviation.
Z-Score
2.5
Example
A value of 2.5 from a distribution with mean 0 and standard deviation 1 has a z-score of 2.5, meaning 99.38% of values fall below it and 0.62% fall above it.
P(x < Z) — Left Tail
99.379%
P(x > Z) — Right Tail
0.621%
P(0 to Z)
49.379%
P(-Z < x < Z)
98.758%
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 2.5 from a distribution with mean 0 and standard deviation 1 has a z-score of 2.5, meaning 99.38% of values fall below it and 0.62% 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.