A z-score tells you how many standard deviations a value sits from the mean. Enter your data point, mean, and standard deviation into the calculator above to get your z-score, percentile rank, and probability instantly.
A z-score (also called a standard score) converts any value from a normal distribution into a standardized scale. A score of 75 on an exam with a mean of 70 and a standard deviation of 4 has a z-score of 1.25. That means the raw score sits 1.25 standard deviations above the average.
This standardized score lets you compare values across different datasets, find percentile ranks, and calculate probabilities using the standard normal distribution.
Z-Score Calculator
Use the calculator at the top of this page to find z-scores, percentiles, and probabilities. You need three inputs:
- Raw score (X): The data point you want to evaluate.
- Mean (μ): The average of the population or dataset.
- Standard deviation (σ): How spread out the values are around the mean.
The calculator returns your z-score, the corresponding percentile rank, and left-tail and right-tail probabilities. You can also enter two z-scores to find the probability between them.
What inputs do I need? Just those three values. If you are working with a sample mean instead of a single data point, you will also need the sample size. See the sample mean formula below.
How to Calculate a Z-Score
Calculating a z-score is straightforward. Subtract the mean from your raw score, then divide by the standard deviation. The result tells you how far your value is from the center of the distribution, measured in standard deviations.
Z-Score Formula
The z-score formula for a single data point is:
z = (X − μ) / σ
Where:
- X is the raw score or data point
- μ is the population mean
- σ is the population standard deviation
Example: A student scored 75 on an exam where the mean is 70 and the standard deviation is 4.
z = (75 − 70) / 4 = 5 / 4 = 1.25
That student's score is 1.25 standard deviations above the mean. Using a z-table or this calculator, a z-score of 1.25 corresponds to roughly the 89th percentile. About 89% of students scored below that student.
Z Score Formula for a Sample Mean
When you are working with a sample mean rather than a single raw data point, use the z-score formula adjusted for sample size:
z = (X̄ − μ) / (σ / √n)
Where:
- X̄ is the sample mean
- μ is the population mean
- σ is the population standard deviation
- n is the sample size
The denominator, σ / √n, is the standard error of the mean. It shrinks as sample size grows, which means a sample mean the same distance from μ produces a larger z-score with more data.
When to use which formula: Use the single-value formula when you have one raw score. Use the sample mean formula when you are comparing an average of multiple observations to the population mean. This distinction matters in hypothesis testing and quality control.
If the population standard deviation is unknown and you are working with a small sample, a t-score (not a z-score) is the better choice. The t-distribution accounts for the extra uncertainty from estimating σ with the sample standard deviation.
What Positive Z and Negative Z Scores Mean
A positive z-score means the value is above the mean. A negative z-score means the value is below the mean. A z-score of 0 means the value equals the mean exactly.
Yes, z-scores can absolutely be negative. There is nothing wrong or unusual about a negative z-score. It simply places the data point on the left side of the distribution.
| Z-Score | Meaning |
|---|---|
| +2.0 | 2 standard deviations above the mean |
| +1.0 | 1 standard deviation above the mean |
| 0 | Exactly at the mean |
| −1.0 | 1 standard deviation below the mean |
| −2.0 | 2 standard deviations below the mean |
What does a z-score of 1.5 mean? The value is 1.5 standard deviations above the average. In a normally distributed dataset, about 93.3% of values fall below this point.
What is a "good" z-score? That depends entirely on context. In hypothesis testing, z-scores beyond ±1.96 are statistically significant at the 5% level. For exam scores, a positive z-score means above average. There is no universal "good" or "bad" value.
Z-Score to Percentile
A percentile rank tells you what percentage of values in a normal distribution fall below your z-score. Converting a z-score to a percentile uses the cumulative probability from the standard normal distribution.
Common z-score to percentile conversions:
| Z-Score | Percentile |
|---|---|
| −1.645 | 5th |
| −1.28 | 10th |
| 0 | 50th |
| +1.28 | 90th |
| +1.645 | 95th |
| +1.96 | 97.5th |
What is the z-score for the 95th percentile? It is 1.645. This value appears frequently in one-tailed hypothesis tests at the 5% significance level.
What percentile is z = 1.28? Approximately the 90th percentile.
How do I convert a z-score to a percentile? Look up the cumulative probability in a z-table, or use this calculator. The cumulative probability, expressed as a percentage, is your percentile rank.
How do I convert a percentile to a z-score? Work backward. Find the cumulative probability that matches your percentile (for example, 0.90 for the 90th percentile) and identify the z-score in the table. For the 90th percentile, the z-score is approximately 1.28.
Is the percentile always accurate? The percentile assumes your data follows a normal distribution. If the data is skewed or not approximately normal, the percentile from a z-score may not match reality.
Finding a raw score from a percentile: Rearrange the z-score formula. If μ = 70, σ = 4, and you want the 90th percentile (z = 1.28):
X = μ + z × σ = 70 + 1.28 × 4 = 75.12
A score of about 75 corresponds to the 90th percentile in this example.
Calculate Probabilities With the Standard Normal Distribution
The standard normal distribution has a mean of 0 and a standard deviation of 1. Every z-score maps to this distribution, letting you find the probability that a value falls in a specific range.
Three types of probability come up most often:
- Left-tail probability: The area under the curve to the left of your z-score. This is the cumulative probability.
- Right-tail probability: The area to the right. It equals 1 minus the left-tail probability.
- Two-tailed probability: The combined area in both tails beyond ±z. Used in two-tailed hypothesis tests.
Example (right-tail): Following the earlier exam example (μ = 70, σ = 4), what proportion of students scored more than 64?
z = (64 − 70) / 4 = −1.5
The left-tail probability at z = −1.5 is about 0.0668. The right-tail probability is 1 − 0.0668 = 0.9332. About 93.3% of students scored above 64.
What does "two-tailed" mean? A two-tailed calculation looks at both extremes of the distribution. For z = ±1.96, the area outside both tails combined is about 5%. This is why ±1.96 marks the boundary for 95% confidence intervals.
What z-score corresponds to a 95% confidence level? For a two-tailed test, z = ±1.96. For a one-tailed test, z = 1.645.
What is the chance of scoring above z = 2? The right-tail probability at z = 2 is about 0.0228, or 2.28%.
Probability Between Two Z-Scores
To find the probability between two z-scores, subtract the smaller cumulative probability from the larger one.
Example: What proportion of students scored between 68 and 73 (μ = 70, σ = 4)?
- z₁ = (68 − 70) / 4 = −0.5 → cumulative probability ≈ 0.3085
- z₂ = (73 − 70) / 4 = 0.75 → cumulative probability ≈ 0.7734
- Probability between = 0.7734 − 0.3085 = 0.4649
About 46.5% of students scored between 68 and 73.
How much of the distribution lies between z = −1 and z = 1? Approximately 68.27%. This is the well-known 68-95-99.7 rule for normal distributions.
What is the two-tailed area outside ±1.96? About 5%, or 2.5% in each tail.
Standard Normal Table and Z-Score Table
A standard normal table (z-table) lists cumulative probabilities for z-scores, usually from −3.49 to 3.49. The rows represent the ones and tenths digit of the z-score. The columns represent the hundredths digit.
There are two common table formats:
- Cumulative (left-tail) z-table: Shows the probability that a value is less than or equal to z. This is the most common format.
- Negative z-score table and positive z-score table: Some references split the table into two halves. The negative z table covers z-values below 0, and the positive z table covers z-values above 0.
How to read a z-table:
- Find your z-score's row (for example, 1.2 for z = 1.25).
- Move across to the column for the hundredths digit (0.05).
- The intersection gives the cumulative probability (0.8944 for z = 1.25).
This free z score calculator does the lookup instantly, so you do not need to scan tables manually. But understanding the table helps you verify results and builds intuition about how the distribution works.
Z-Scores in a Normal Distribution
Z-scores are most useful when data is normally distributed, or at least approximately normal. In a normal distribution, values cluster symmetrically around the mean, forming the familiar bell curve.
Key reference points for z-scores in a normal distribution:
- 68% of values fall within ±1 standard deviation (z between −1 and 1).
- 95% of values fall within ±2 standard deviations (z between −2 and 2).
- 99.7% of values fall within ±3 standard deviations (z between −3 and 3).
A percentage of values beyond z = ±3 is very small (about 0.3%). This is why z-scores beyond 3 or below −3 often flag outliers.
Where are z-scores used in real life?
- Standardized testing: Comparing scores across different exams or years.
- Hypothesis testing: Determining whether an observed result is statistically significant.
- Quality control: Identifying products outside acceptable specification ranges.
- Finance: Measuring how far a return deviates from its historical average.
Remember that z-scores assume a known population mean and standard deviation, and they work best with data that is normally distributed. Heavily skewed data or small samples with unknown σ may need different methods.
Frequently Asked Questions
How Do I Calculate a Z-Score Without a Calculator?
Use the formula z = (X − μ) / σ. Subtract the mean from your value, then divide by the standard deviation. You can do this with pen and paper or a basic calculator.
For the probability or percentile, you will need a z-table. Look up your calculated z-score in the table to find the cumulative probability.
What Is the Probability for a Z of Zero?
A z-score of 0 means the value equals the mean exactly. The cumulative probability (left-tail) at z = 0 is 0.50, or 50%. Half the values in a normal distribution fall below the mean, and half fall above it.
How Do I Use This Z Score Calculator for a Test Score?
Enter your test score as the raw score (X). Enter the class or exam average as the mean (μ). Enter the standard deviation of the scores as σ. The calculator returns your z-score and percentile rank, showing where you stand relative to other test takers.
If your instructor gave you a mean of 70 and standard deviation of 4, and you scored 78, your z-score would be (78 − 70) / 4 = 2.0. That places you at roughly the 97.7th percentile.
Can This Calculator Replace a Standard Normal Table?
Yes, for most practical purposes. This calculator performs the same cumulative probability lookup that a z-table provides, with more decimal precision and no need to interpolate between rows and columns. It handles left-tail, right-tail, two-tailed, and between-two-z-scores calculations.
A printed z-table can still be useful during exams or in settings where you need to verify results independently. The underlying math is the same.