Not sure if your score is “good” compared to the average?
A z-score turns any score into a common scale. It tells you how far your value is from the mean, in standard deviations. Use it to compare test scores, heights, or any data that roughly follows a bell curve. Then use the percentile and probabilities to connect to normal distribution questions.
Click Calculate to get the z-score, and optionally the percentile and probabilities.
A z-score tells you how many standard deviations your value (X) is from the mean.
How to use this in homework: You can write: “The score is z standard deviations above (or below) the mean.”
Assumptions (quick):
- X and the mean use the same units.
- SD is correct and greater than 0.
- Percentiles and probabilities assume a normal model.
Show steps / formulas
z = (X - mean) / SD
What each symbol means
- X = your value
- mean = average
- SD = standard deviation
- z = standardized score
- Compute X - mean.
- Divide by SD.
- The result is your z-score.
If you turn on “Percentile,” the tool converts z into a percentile using the standard normal model (Phi(z)).
- Using SD = 0 (z-score is undefined).
- Mixing units (X in one unit, mean in another).
- Using the wrong SD (sample vs population) compared to your class convention.
- Interpreting percentile outputs without checking if a normal model is appropriate.
- Rounding too early before finishing the calculation.
Next, practice “area under the curve” problems using our Normal Distribution Probability Calculator (you will use it for probabilities between two z-values).
What this calculator does
This z-score calculator computes:
- Z-score: how many standard deviations your value is from the mean
- Percentile (optional): the percent of values below your z-score (normal model)
- P(Z ≤ z) (optional): probability below your z
- P(Z ≥ z) (optional): probability above your z
Key idea (easy rule):
- Positive z means your value is above the mean.
- Negative z means your value is below the mean.
- z = 0 means your value equals the mean. (OpenStax, 2019)
When to use it (and when not to)
Use it when
- You have one value (X), plus the mean and standard deviation (SD).
- The data is roughly bell-shaped (normal-ish), or your teacher says to use a normal model.
- You need to compare scores from different tests or different units.
- You are preparing for normal probability and later z-tests (OpenStax, 2019; NIST/SEMATECH, 2012–2023).
Do not use it when
- The problem is about a sample mean and uses standard error (that is a different tool).
- The distribution is strongly skewed or has extreme outliers and your class is not using a normal model.
- You are doing t-tests but only have sample data and small n (your teacher may want t, not z). (OpenStax, 2019)
How it works (simple explanation)
A z-score standardizes a value.
Formula: \(z=\frac{X-\text{mean}}{\text{SD}}\)
What each part means
- X: your value
- mean: the average
- SD: typical spread (how spread out the data is)
- z: how many SDs away X is from the mean (OpenStax, 2019)
One-line meaning:
If z = 1.50, your value is 1.5 standard deviations above the mean.
Step-by-step example (static example in text)
Problem: A test score is X = 86. The class mean is 75. The SD is 8. Find the z-score.
Step 1: Subtract the mean
86 − 75 = 11
Step 2: Divide by SD
11 ÷ 8 = 1.375
Answer:
z = 1.375, so the score is 1.375 SD above the mean.
How to write it in homework (copy-ready):
“Given X = 86, mean = 75, and SD = 8, the z-score is \(z=\frac{86-75}{8}=1.375\). Therefore, the score is 1.375 standard deviations above the mean.” (OpenStax, 2019)
Common mistakes
- Using SD = 0 (z-score is undefined).
- Mixing units (X and mean must use the same unit).
- Using the wrong SD type (sample SD vs population SD) if your teacher is strict about notation.
- Rounding too early (keep more digits until the end).
- Assuming “percentile” always applies even when the problem does not use a normal model. (OpenStax, 2019; NIST/SEMATECH, 2012–2023)
FAQs
What does my z-score result mean?
It tells how far your value is from the mean in SD units. Positive means above the mean. Negative means below the mean. (OpenStax, 2019)
What is a “good” z-score?
It depends on context. In many classes, |z| ≥ 2 is “far” from the mean. But your teacher may set different cutoffs. (OpenStax, 2019)
Is percentile the same as probability?
It is closely related. A percentile is a percent rank. A probability is a number from 0 to 1. For example, 90th percentile is about 0.90 probability below. (NIST/SEMATECH, 2012–2023)
Why is my answer different from my teacher’s?
Common reasons:
Your teacher used a z-table with rounding.
You rounded too early.
Your teacher used sample SD vs population SD (or a different SD value).
The problem did not assume a normal model, but the calculator shows normal-model outputs. (OpenStax, 2019)
Which test should I use: z-test or t-test?
Many classes use z when population SD is known or when n is large and a normal model is assumed. They use t when SD is estimated from the sample and n is small. Follow your course rule. (OpenStax, 2019)
Can I use this calculator to find probability between two z-scores?
Not on this page. Use the Normal Distribution Probability Calculator (area between two values). That is the next step after this tool.
What if my z-score is above 4 or below −4?
That is very far in the tails. Some tables stop at ±3.99. A calculator can still compute it, but the probability may be extremely close to 0 or 1. (NIST/SEMATECH, 2012–2023)
Related BrainMatters Learning statistics tools
Sideways/forward calculators
- Standard Deviation Calculator (build the descriptive base)
- Normal Distribution Probability Calculator (next bridge step; areas between two z-values)
- Percentile from Z Calculator (quick converter)
- Standard Error Calculator (bridge to inference)
- Confidence Interval Calculator (inference)
- Hypothesis Test / P-value Calculator (inference)
Related tools
- Descriptive Statistics Calculator
- Standard Deviation Calculator
- Normal Distribution Probability Calculator (coming soon)
- Percentile from Z Calculator (coming soon)
- Standard Error Calculator
- Confidence Interval Calculator (coming soon)
- Hypothesis Tests Hub (coming soon)
References
NIST/SEMATECH. (2012–2023). e-Handbook of statistical methods: Normal distribution. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/
OpenStax. (2019). Introductory statistics (2nd ed.). OpenStax, Rice University. https://openstax.org/details/books/introductory-statistics-2e
Triola, M. F. (2018). Elementary statistics (13th ed.). Pearson. https://www.scirp.org/reference/referencespapers?referenceid=885289