Z Score Calculator
Enter a raw value, mean, and standard deviation to calculate the z-score, see how many standard deviations the value sits from the mean, and estimate its normal percentile.
Formula
z = (x - mean) / SD
Output
Unitless z
Curve
Normal Percentile
Live calculator
Z-score inputs
Convert a raw value into standard deviations from the mean and estimate its normal percentile.
Z-score
0.875
Percentile
80.921309%
Value
82 points
Mean
75 points
Std deviation
8 points
Z-score formula check
| Measure | Value | Check |
|---|---|---|
| Difference from mean | 7 points | Raw value minus the mean. |
| Standard deviation | 8 points | Must be greater than zero to standardize the value. |
| Z-score | 0.875 | Difference divided by standard deviation. Z-scores are unitless. |
| Normal percentile | 80.921309% | Approximate area to the left under the standard normal curve. |
Interpretation
The value is less than one standard deviation above the mean.
Formula
z = (x - mean) / SD
Output
Unitless z
Curve
Normal Percentile
Standardize a raw value
Z-score
Convert the value into standard deviations above or below the mean.
Normal percentile
Estimate the area to the left under the standard normal curve.
Formula check
Show the raw difference, standard deviation, unitless z-score, and interpretation.
Z-score formula used on this page
A z-score standardizes a value by subtracting the mean and dividing by standard deviation. The result is unitless.
Working formulas
Z-score
z = (x - mean) / standard deviation
Positive z is above the mean; negative z is below the mean.
Difference from mean
x - mean
The raw distance stays in the original dataset unit.
Normal percentile
Phi(z) x 100
The page approximates the cumulative standard normal curve.
Symbols
- x - value
- The raw observed value being standardized.
- mean - center
- The dataset or distribution average.
- SD - standard deviation
- The positive spread value used to convert raw distance into standardized distance.
Interpret this calculation before using it
A z-score standardizes a value by subtracting the mean and dividing by the standard deviation. Positive scores lie above the mean and negative scores lie below it; the magnitude states the distance in standard-deviation units. Standardization changes the scale but preserves the value's relative location within the supplied distribution summary.
Worked example
For x = 85, mean = 70, and standard deviation = 10, z = (85 - 70) / 10 = 1.5. The value is therefore 1.5 standard deviations above the mean. Reversing the calculation gives x = 70 + 1.5 x 10 = 85, which checks that the entered mean and spread were applied consistently.
Assumptions to keep
- The standard deviation is positive and uses the same units as the value and mean.
- The supplied mean and standard deviation describe the relevant population or sample context for the comparison.
- A z-score alone does not imply a percentile unless a distribution model, often normality, is separately justified.
Limits of this result
Outliers, skew, mixed populations, or a poorly estimated standard deviation can make standardized comparisons misleading. Population and sample standard deviations may also differ. Use this tool for the arithmetic transformation and state the data source; use a distribution-specific method only when its assumptions fit the data.
Useful next step: Percentile Calculator — Work directly with ranked observations when a normal-distribution percentile should not be inferred from a z-score.
Raw units kept separate from the unitless z-score
Dataset checks before answers
- Values can be pasted with commas, spaces, semicolons, pipes, or line breaks.
- Sorted-order previews make median, percentile, minimum, maximum, and range checks easier to audit.
- Odd-count, even-count, and decimal presets expose the edge cases that often cause manual mistakes.
- A custom unit label keeps scores, dollars, kilograms, centimeters, seconds, or dimensionless values clear.
Method labels on every result
- Sample and population standard deviation are separated because they use different denominators.
- Percentiles show both nearest-rank and interpolated methods instead of hiding the convention.
- Z-scores are marked unitless while the raw value, mean, and standard deviation retain the dataset unit.
- Copy and print controls help move checked results into worksheets, reports, and study notes.
Z-score support for normal-distribution practice
Students
Check homework datasets while seeing sorted values, formulas, and the method behind each output.
Teachers
Build classroom examples for center, spread, percentile, z-score, and sample-versus-population lessons.
Analysts
Quickly validate small datasets before moving values into spreadsheets, notebooks, or reports.
How it works in three quick steps.
Enter the raw value
Add the observed value that you want to compare with the mean.
Enter mean and standard deviation
Use the dataset mean and a positive standard deviation on the same measurement scale.
Read z-score and percentile
The calculator reports the standardized score and an approximate normal percentile.
Save or print z-score calculations
Copy result summary
Copy the dataset count, main statistic, and method notes into assignments or analysis notes.
Print a checked worksheet
Print inputs, result cards, formula tables, FAQs, and related tools for offline review.
Verify sorted order
Use the sorted preview to confirm the exact values used for median and percentile positions.
Frequently Asked Questions
What does a z-score mean?
Why must standard deviation be greater than zero?
Is the percentile exact?
Can I use sample standard deviation for a z-score?
Why does the result keep raw units but not z-score units?
Related statistics tools
About this z-score calculator
Toolarithm's Z Score Calculator standardizes a raw value against a mean and standard deviation. It shows the difference from the mean, the standard deviation used as the denominator, the resulting z-score, and an approximate normal percentile. The unit label stays on the raw values and standard deviation, while the z-score is displayed as unitless to make the cancellation clear.
The calculator is useful for statistics homework, normal distribution practice, score comparison, quality checks, and interpreting values relative to a known center and spread. The percentile output is intentionally labeled as an approximate normal percentile, because a z-score percentile assumes a normal curve. Related links connect the workflow back to standard deviation and percentile calculators so users can choose the right method for actual sorted data versus a modeled normal distribution.
Who maintains this page
Ownership and review
Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.
Methodology
Equations are checked by substituting worked examples into the stated relationship and comparing the result with the interactive implementation.
Read the editorial methodologyDates and sources
Review dates change only after a substantive method or content check.
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