Z Score Calculator
Raw score to z · Percentile · Tail probabilities · Critical values
📊 Enter a score, a z score, or a probability to see percentiles, tail areas, and the normal curve.
What Is a Z Score?
A z score, or standard score, measures how many standard deviations a value lies above or below the mean. A z score of 1.5 means the value is one and a half standard deviations above average; −2 means two standard deviations below; 0 means exactly average.
Standardizing removes units, so z scores let you compare values measured on different scales, find the percentile of a score in a normal distribution, flag unusual values, and look up the probabilities used in hypothesis tests and confidence intervals. This calculator handles all four tasks.
How to Use the Z Score Calculator
- Raw score: enter x, the mean μ, and the standard deviation σ to get the z score, percentile, tail areas, and two tailed p value.
- z to probability: enter a z score and pick left, right, between ±z, or outside ±z. This replaces a printed z table.
- Probability to z: enter 95 or 0.95 and the area type to get the critical z value and, with μ and σ, the raw cutoff.
- Compare: enter two scores with their own means and standard deviations to see which is stronger relative to its group.
Z Score Formula with Examples
For a sample mean, divide by the standard error σ ÷ √n instead of σ, as shown in the z test section below.
Z Score to Percentile and Probability
The percentile is the area under the standard normal curve to the left of z, written Φ(z). The other areas follow from it.
| Area | Formula | z = 1.5 |
|---|---|---|
| Left (percentile) | Φ(z) | 0.9332 |
| Right | 1 − Φ(z) | 0.0668 |
| Between −z and z | 2Φ(|z|) − 1 | 0.8664 |
| Two tailed p | 2(1 − Φ(|z|)) | 0.1336 |
Φ has no closed form, so printed z tables list values to 4 or 5 decimals. This calculator computes Φ numerically, accurate to about 7 decimal places.
The 68 95 99.7 Rule
| Range | Share of values | Outside |
|---|---|---|
| z from −1 to 1 | 68.27% | about 1 in 3 |
| z from −2 to 2 | 95.45% | about 1 in 22 |
| z from −3 to 3 | 99.73% | about 1 in 370 |
In a normal distribution, a z score beyond ±2 happens in fewer than 5% of cases and beyond ±3 in about 0.27%. That is why |z| above 3 is a common screen for outliers.
Critical Z Values
| Confidence | Two sided z | One sided z |
|---|---|---|
| 80% | 1.282 | 0.842 |
| 90% | 1.645 | 1.282 |
| 95% | 1.960 | 1.645 |
| 98% | 2.326 | 2.054 |
| 99% | 2.576 | 2.326 |
| 99.9% | 3.291 | 3.090 |
A two sided 95% interval splits the 5% outside it into 2.5% per tail, so z = Φ⁻¹(0.975) = 1.960.
How to Read a Z Table
A standard normal table lists Φ(z), the area to the left of z. Rows give z to one decimal place and columns add the second decimal.
- Split the z score: 1.53 becomes row 1.5 and column 0.03.
- Read the cell where they meet: 0.93699.
- For a right tail, subtract from 1: 1 − 0.93699 = 0.06301.
- For a two tailed p value, double the smaller tail: 2 × 0.06301 = 0.12602.
Some tables list the area from 0 to z instead of from −∞ to z. Add 0.5 to those values to get Φ(z).
Negative Z Scores
A negative z score is below the mean. Because the normal curve is symmetric, Φ(−z) = 1 − Φ(z), so many tables print only positive values.
Negative does not mean bad. For race times, golf scores, blood pressure, or error rates, a value below the mean is often the better result. Choose "Lower" in the Compare tab when smaller numbers are better.
Z Test for a Sample Mean
When you compare the average of a sample to a known population mean, the spread of that average is the standard error σ ÷ √n, not σ. Averages vary less than single values, so the same gap from the mean produces a larger z score.
Example. A machine fills bottles with a target mean of 500 ml and a known σ of 4 ml. A sample of 16 bottles averages 498 ml. The standard error is 4 ÷ √16 = 1 ml, so z = (498 − 500) ÷ 1 = −2.0. A single bottle 2 ml under target would only be z = −0.5.
The two tailed p value for z = −2.0 is 2 × Φ(−2) = 2 × 0.0228 = 0.0455, below 0.05. To get it here, compute z by hand or in the Raw score tab with σ set to the standard error (x = 498, μ = 500, σ = 1), then read the two tailed p. If σ is unknown and estimated from a small sample, use a t test instead.
How to Interpret Your Z Score
The sign shows direction and the size shows distance from the mean. The verdict badge uses these bands: |z| under 1 is typical (50th to 84th percentile for positive z), 1 to 2 is noticeably above or below average (up to the 98th), 2 to 3 is unusual (up to the 99.9th), and above 3 is very rare, so check for an outlier or an entry error.
Typical uses include ranking test scores, setting grade or selection cutoffs (x = μ + zσ), flagging outliers in data cleaning, checking whether a process average has shifted, and finding critical values for confidence intervals.
Comparing Scores and Common Mistakes
Which score is better? A student scores 85 on a math test (mean 70, SD 10) and 30 on a reading test (mean 21, SD 5). The math z score is 1.5 and the reading z score is (30 − 21) ÷ 5 = 1.8, so the reading result is stronger relative to the class, even though 30 is a smaller number. The Compare tab does this for any two scores.
- Using the variance. Divide by the standard deviation, not the variance. If σ² = 100, use σ = 10.
- Wrong tail. "At least" or "more than" needs the right tail; "at most" or "below" needs the left tail.
- Assuming normality. A z score can be computed for any data, but converting it to a percentile is only accurate when the data are close to normal.
- Sample or population SD. Use σ when it is known. With a small sample and an estimated standard deviation, a t distribution gives more accurate probabilities.
Frequently Asked Questions
Sources & References
- OpenStax. Introductory Statistics 2e, 6.1 The Standard Normal Distribution (formula, empirical rule). openstax.org
- NIST/SEMATECH. Engineering Statistics Handbook, 1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (z table, critical values). itl.nist.gov
- NIST/SEMATECH. Engineering Statistics Handbook, 1.3.6.6.1 Normal Distribution (σ ÷ √N). itl.nist.gov
Results are for educational purposes. Probabilities assume a normal distribution. Methodology reviewed: 2026.