Z-Score Calculator: Standard Score, Percentile, and Raw Score Finder
Use this z-score calculator to find a standard score from a raw value, mean, and standard deviation. You can also convert a z-score to a percentile, convert a percentile back to a z-score, find a raw score from z, or paste raw data and choose sample or population standard deviation.
- Standard score finder Convert a raw value, mean, and standard deviation into a z-score with percentile and tail areas.
- Z-score to percentile Enter a z-score and see the cumulative percentile, right-tail area, and two-tail area.
- Raw score from z Convert a z-score, mean, and standard deviation back into the original raw score.
- Percentile to z-score Convert a percentile or cumulative probability into the matching standard normal z-score.
- Raw data helper Paste a data set, choose sample or population standard deviation, and calculate a value z-score.
Calculate a Z-Score
Choose the conversion you need. For a raw score, enter the value, mean, and standard deviation. For a data set, paste numbers and choose whether to calculate the standard deviation as a sample or population. This calculator is for z-scores and percentiles; use the z-test calculator when your assignment asks for a formal hypothesis test.
Inputs
Results
A raw score of 85 with mean 70 and standard deviation 10 has z = 1.5000.
The shaded area shows the percentile below z = 1.5000.
| Formula | z = (85 - 70) / 10 = 1.5000 |
|---|---|
| Probability | Phi(1.5000) = 0.9332 |
| Interpretation | The value is 1.50 standard deviations above the mean. |
Z-Score Formula
A z-score standardizes a value by measuring its distance from the mean in standard deviation units.
| Raw score to z | z = (x - mean) / standard deviation |
|---|---|
| Raw score from z | x = mean + z * standard deviation |
| Percentile | Percentile = Phi(z), the standard normal cumulative probability |
| Left-tail probability | P(Z <= z) = Phi(z) |
| Right-tail probability | P(Z > z) = 1 - Phi(z) |
| Sample data | Use sample standard deviation when your data are a sample from a larger population. |
Common Mistakes
- Using a sample standard deviation without saying that the z-score is estimated from sample data.
- Entering the standard deviation as 0 or as a negative number.
- Confusing a z-score with a z-test. A z-score standardizes one value; a z-test evaluates a hypothesis.
- Treating a percentile as a percent in some places and a decimal in others.
- Using the normal percentile interpretation when the underlying distribution is strongly non-normal.
Z-Score Calculator Examples
These examples cover common statistics homework wording: raw score to z, z to percentile, percentile to z, and z-scores from raw data.
| Question type | Setup | Answer |
|---|---|---|
| Find z from a raw score | x = 85, mean = 70, standard deviation = 10 | z = 1.5000, percentile = 93.32% |
| Find percentile from z | z = 1.96 | Percentile = 97.50%, right tail = 2.50% |
| Find z from percentile | 95th percentile | z = 1.6449 |
| Find z from raw data | Data: 62, 67, 70, 72, 75, 81, 85, 90; x = 85; sample SD | Mean = 75.2500, sample SD = 9.4680, z = 1.0298 |
How to Calculate a Z-Score Step by Step
A z-score tells you how far a value is from the mean in standard deviation units. For example, if a score is 85, the mean is 70, and the standard deviation is 10, then z = (85 - 70) / 10 = 1.50.
- Write down the raw score, mean, and standard deviation.
- Subtract the mean from the raw score.
- Divide the difference by the standard deviation.
- Interpret the sign and size: positive z-scores are above the mean, negative z-scores are below the mean.
- Use the percentile result if your homework asks what percentage of values are below the score.
When This Calculator Helps
Statistics homework
Use it when your assignment asks how unusual a value is, what percentile a value falls in, or how to standardize scores before using a normal table.
Data checks
Use it to compare values from different scales, spot unusual observations, or explain whether a score is above or below the average in standard deviation units.
Z-Score FAQs
What does a z-score tell you?
A z-score tells how many standard deviations a value is above or below the mean. Positive z-scores are above the mean, and negative z-scores are below the mean.
What is the z-score formula?
The basic formula is z = (x - mean) / standard deviation. If the mean and standard deviation come from a sample, say that the z-score is estimated from sample statistics.
Is a z-score the same as a percentile?
No. A z-score is measured in standard deviation units. A percentile tells the percentage of normally distributed values below that z-score.
Should I use sample or population standard deviation?
Use population standard deviation only when the data describe the whole population. Use sample standard deviation when the data are a sample from a larger population.
Is this calculator a z-score finder or a z-test calculator?
This page is a z-score finder. It standardizes one value and shows the related percentile. If you are testing a hypothesis about a mean or proportion, use a z-test calculator instead.
Can I convert a z-score back to a raw score?
Yes. Use x = mean + z * standard deviation. This calculator includes a raw-score-from-z mode for that conversion.
Related Statistics Resources
If your z-score problem turns into a hypothesis test, confidence interval, or written interpretation, these pages are good next steps.
Need Help Explaining a Z-Score?
Statskan can help you calculate the z-score, choose the right standard deviation, connect it to a percentile, and write the interpretation clearly for your statistics assignment.