MultiConvers

Z-score Calculator

Turn a raw reading into a standard score, or a standard score back into the original units. You supply the mean and a positive spread; the page does not invent a percentile.

How to use

  1. Pick Value → z-score to standardize, or z-score → value to undo it.
  2. Enter the reference mean μ and a standard deviation σ greater than zero.
  3. Type the raw x, or the z you already have, depending on the direction.
  4. Read how many SDs you sit from the mean — not a rank, not a tail probability.

How it works

A z-score is a change of units. Instead of “115 on this test,” you say “one standard deviation above the mean of 100 when the SD is 15.” The formula is z = (x − μ) / σ. The inverse, for when someone hands you a z and you need the original scale, is x = μ + zσ. Both directions live on this page. σ must be positive: a zero or negative spread is rejected rather than dividing by zero. If every observation was identical, a standard score is undefined.

What the page does not do matters as much as what it does. It does not look up a normal table. It does not print “you are in the 84th percentile.” That step needs a distributional assumption (usually a Normal curve) that this calculator never makes. A z of 1 is simply “one SD above the mean.” On a Normal curve that happens to be near the 84th percentile, but plenty of real lists are skewed, truncated or discrete. Treat z as a standard score unless you separately justify a table lookup.

Whether the σ you type is a sample s or a population σ is your decision. The algebra does not change. Compute spread on the standard-deviation page if you only have a list, then bring the number here with the mean. The z-score tool never sees the raw list — only the three (or two) numbers you enter.

Negative z means below the mean. z = 0 is exactly the mean. A z of −2 with μ = 100 and σ = 15 is a raw 70. Large |z| values are not errors; they just say the point sits far from the centre in SD units. They become suspicious only if you also believe the data cannot wander that far.

Use this when two tests or two labs use different scales and you want a common language. Use the confidence-interval page when you have a sample mean and you want a range for a population mean — that is a different z, the critical value for a coverage level, not a standard score for one observation.

Formula

z = (x − μ) / σ
x = μ + zσ

Worked examples

  • A test score

    x = 115, μ = 100, σ = 15 → z = 1

  • Back to the raw scale

    z = 1, μ = 100, σ = 15 → x = 115

Useful notes

  • Whether σ is a sample s or a population σ is your choice; the algebra is the same.
  • This is a standard score, not a percentile and not a p-value.
  • σ must be greater than 0. A zero spread cannot be used as a divisor.

FAQ

What does a z-score of 1 mean?

The value is one standard deviation above the mean. For μ = 100 and σ = 15, that is 115.

Can a z-score be negative?

Yes. Values below the mean have negative z.

Why is a zero standard deviation rejected?

You cannot divide by zero. If every observation is the same, a z-score is undefined.

Does a z-score of 1 mean the 84th percentile?

Only if you assume a Normal distribution and look that z up in a table. This calculator does not do that lookup. It reports the standard score and stops.

Can I convert a percentile into a z-score here?

No. Enter a raw value and a mean and SD, or a z and a mean and SD. Percentile-to-z needs a distribution this page does not assume.

Related tools

See all Statistics tools.