Z-Score Calculator

Calculate z-scores, percentiles, and normal distribution probabilities with a curve visualization

Check Z-Score Calculator

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Formula result

Copy-ready result handoff

Use this after converting a value to a standard score or probability area.

Common searches
z-score calculatorz score formulastandard score
Input check

Confirm the mean and standard deviation come from the same distribution.

Assumption

Probability lookup assumes a standard normal curve.

Z-score interpretation note

Copy this after replacing the bracketed result with the value shown by the calculator.

Z-score result: [paste z-score, percentile, left tail, and right tail].
Inputs checked: Confirm the mean and standard deviation come from the same distribution.
Assumption: Probability lookup assumes a standard normal curve.
Use case: Statistics homework, lab sample comparison, outlier check, or score report.
Next check: Use Confidence Interval Calculator when estimating a population value from a sample.

Use calculate mode for raw values, lookup mode for a known z-score, or between mode for the area between two cutoffs. Keep the standard deviation positive and choose the tail area that matches the question.

How the result is built

Formula, Variables, And Assumptions

z = (x - mean) / standard deviation

x is the observed value, the mean is the center of the distribution, and the standard deviation sets the distance scale.

z is unitless. The observed value, mean, and standard deviation must use the same original unit, such as points, inches, seconds, or dollars.

Worked example

If x = 82, mean = 70, and standard deviation = 8, then z = 1.5. The value is 1.5 standard deviations above the mean.

Common mistakes
  • Using sample spread when the prompt gives population spread.
  • Interpreting a z-score as a probability before choosing the tail area.
  • Applying normal probabilities when the data shape is not close to normal.
Assumptions and limits
  • The standard deviation is positive and matches the distribution for x.
  • x, mean, and standard deviation are entered in the same measurement unit.
  • Probability lookups assume the standardized value follows a normal model.
  • The tail area must match the question: left, right, two-tailed, or between.
Related checks
Standard deviationStatisticsConfidence intervalsSample sizeProbability

Z-Score Formula, Assumptions, And Mistakes

Choose the right z-score mode

Use calculate mode when you know a raw value, mean, and standard deviation. Use lookup mode when you already have a z-score and need a percentile or tail probability. Use between mode when the question asks for the area between two standardized cutoffs.

How to interpret the result

A z-score tells how many standard deviations a value is from the mean. For a normal model, z = 1.5 means the value sits 1.5 standard deviations above center, with the displayed percentile showing the area to the left of that point. A negative z-score is below the mean; the distance from zero still shows how unusual the value is.

Normal model assumptions

Probability output assumes the standardized value follows a normal distribution. If the data is strongly skewed, has large outliers, or comes from a bounded scale, treat percentile and tail probabilities as model-based approximations. Use the Statistics Calculator to inspect center, spread, and shape before relying on a normal curve.

Useful companion calculations

Use the Standard Deviation Calculator when the spread is not already known, the Confidence Interval Calculator when z critical values feed an interval, and the Sample Size Calculator when margin of error planning depends on confidence level.

References & Sources

Common result checks

Questions about this tool

What does a z-score of 0 mean?
A z-score of 0 means the value is exactly at the mean. Positive z-scores are above the mean, and negative z-scores are below the mean.
Can I use this calculator if my data is not normal?
You can standardize a value with z = (x - mean) / standard deviation, but percentile and tail probability output assumes a normal model.
Which probability should I use: left tail, right tail, or two-tailed?
Use left tail for values below a cutoff, right tail for values above a cutoff, and two-tailed when the question asks for values at least that far from the mean in either direction.
What is the difference between percentile and z-score?
A z-score measures distance from the mean in standard deviations. A percentile reports the percentage of the normal distribution below that z-score.

Sources and references