Z-Score Calculator
Calculate a z-score from a value, mean and standard deviation, and learn how to read it as a percentile. Includes a worked example and common mistakes.
Results
Z-score
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Educational estimates · not financial advice · free, no signup
Formula / how it works
z = (x − μ) ÷ σ, where x is the value, μ the mean and σ the standard deviation. For a normal distribution, the area below z is the percentile (for example z = 1.5 is about the 93.3rd percentile).
Detailed guide
What it is, how to use it, how to read the numbers, common mistakes, a worked example, formulas, and FAQs.
What this calculator is
A z-score tells you how far a value is from the average, measured in standard deviations. A z-score of 0 means the value equals the mean, +1 means one standard deviation above, and −2 means two below. Because the result is unit-free, you can compare values from different scales, such as a test score and a height, on the same footing. Students use z-scores for grading on a curve and for standardised test reports, analysts use them to flag outliers, and quality engineers use them to describe process variation. This calculator takes the value, mean and standard deviation you supply and returns the z-score. To turn it into a percentile you need the normal distribution table or a CDF, which the guide below explains.
When to use / who it helps
- You want to compare how unusual a score is relative to its group, such as an exam mark against the class mean.
- You need to compare results measured on different scales, such as two tests with different averages and spreads.
- You are checking for outliers; values beyond about ±3 standard deviations are often flagged for review.
- You are working through a statistics assignment on the normal distribution and need a quick check of the standardisation step.
- You need the z-value to look up a probability in a z-table or to compute a confidence interval.
How to use
Steps
- Enter the value you want to standardise as x.
- Enter the mean of the group or population as μ.
- Enter the standard deviation as σ; it must be positive.
- Read the z-score, noting its sign: positive means above the mean, negative below.
- To get a percentile, look up the z-score in a standard normal table or use a spreadsheet function.
- If you only have a sample, use the sample mean and sample standard deviation and say so in your write-up.
How to interpret results
- The magnitude shows distance from the mean in standard deviations; the sign shows direction.
- For roughly bell-shaped data, about 68% of values fall within ±1, 95% within ±2 and 99.7% within ±3 standard deviations.
- A z-score of 1.5 corresponds to about the 93rd percentile of a normal distribution.
- z-scores describe position relative to the group, not whether a value is good or bad in absolute terms.
- For heavily skewed data the percentile reading is not accurate, because the normal approximation fails.
Common mistakes & gotchas
- Using the variance instead of the standard deviation in the denominator.
- Mixing the population and sample standard deviation and then comparing to a textbook answer that used the other one.
- Entering a standard deviation of zero or a negative number, which makes the ratio undefined.
- Reading a z-score as a percentile directly; z = 1.5 is not the 1.5th percentile.
- Applying normal-distribution percentages to data that are not roughly bell-shaped.
Worked example
Worked example — a score of 85 when the class mean is 70 and σ is 10
- Subtract the mean from the value: 85 − 70 = 15.
- Divide by the standard deviation: 15 ÷ 10 = 1.5.
- Interpret: the score is 1.5 standard deviations above the mean.
- Look up z = 1.5 in a standard normal table: the area below is about 0.9332, so the score is near the 93rd percentile if the scores are normally distributed.
- A score of 55 would give (55 − 70) ÷ 10 = −1.5, which is the mirror image at about the 7th percentile.
Result: z = 1.5, roughly the 93rd percentile in a normal distribution.
To load the form's own demo values, tap Try example above the Calculate button.
How to calculate / formula
z = (x − μ) ÷ σ, where x is the value, μ the mean and σ the standard deviation. For a normal distribution, the area below z is the percentile (for example z = 1.5 is about the 93.3rd percentile).
Frequently asked questions
What does a negative z-score mean?
The value is below the mean. A z-score of −1 is one standard deviation below average.
How do I convert a z-score to a percentile?
Look up the cumulative probability in a standard normal table or use a function such as NORM.S.DIST in a spreadsheet. For z = 1.5 it is about 93.3%.
What is a good z-score?
It depends on the context. For test marks a high positive z is good; for error rates or defect counts, a low value is better.
What z-score is considered an outlier?
A common rule of thumb flags values beyond ±3, and some analyses use ±2.5, but the right threshold depends on the data and the purpose.
Can I use a z-score with a small sample?
You can compute it, but the percentile interpretation needs the data to be roughly normal. For inference with small samples, a t-based approach is usually appropriate.
What is the difference between a z-score and a t-score?
A z-score uses the known or population standard deviation and the normal distribution. A t-score uses a sample estimate and the t-distribution, which has heavier tails for small samples.