BYTETOOLS

Z Score Calculator

Convert a raw value to a z-score with z = (x − μ)/σ, or turn a z-score back into a value. Includes left, right and between probabilities from the normal curve.

2
Z-score
130
Value (x)

Normal-curve probabilities

0.9772
P(Z < z) left tail
0.0228
P(Z > z) right tail
0.0455
Two-tail
97.72%
Percentile

The value is 2 standard deviations above the mean, higher than about 97.72% of the distribution.

Probabilities use the Abramowitz–Stegun erf approximation (accurate to ~7 decimal places).

What is the Z Score Calculator?

The ByteTools Z Score Calculator converts a raw score into a standard score using z = (x − μ) / σ, telling you how many standard deviations a value sits above or below the mean.

  • z = (x − μ)/σ and the reverse x = μ + zσ
  • Left-tail, right-tail and two-tail normal probabilities
  • Percentile equivalent of the z-score
  • High-accuracy erf approximation — matches printed z-tables
  • Plain-English interpretation of the score
  • 100% private — everything runs in your browser

How to use the Z Score Calculator

  1. 1

    Choose a direction: find the z-score from a value, or find the value from a z-score.

  2. 2

    Enter the mean (μ) and standard deviation (σ) of the distribution.

  3. 3

    Enter your raw value x (or the z-score in reverse mode).

  4. 4

    Read the z-score, the raw value and the left, right and two-tail probabilities.

  5. 5

    Click Copy to grab the results.

About the Z Score Calculator

The ByteTools Z Score Calculator converts a raw score into a standard score using z = (x − μ) / σ, telling you how many standard deviations a value sits above or below the mean. It also works in reverse, recovering the raw value x from a given z-score, mean and standard deviation.

Alongside the z-score you get the associated normal-curve probabilities: the area to the left P(Z < z), the area to the right P(Z > z), and the two-tail area — the same numbers you would otherwise look up in a printed z-table. Probabilities are computed with the Abramowitz–Stegun rational approximation of the error function, accurate to about 7 decimal places.

The calculator runs 100% locally in your browser. Nothing you type is uploaded or stored, so it is private, instant and works offline — ideal for statistics homework, standardized test analysis and quality control checks.

Frequently asked questions

How do you calculate a z-score?

Subtract the mean from your value and divide by the standard deviation: z = (x − μ) / σ. For example, a score of 130 in a distribution with mean 100 and standard deviation 15 gives z = (130 − 100) / 15 = 2, meaning two standard deviations above the mean.

What does a z-score tell you?

It expresses a value's position relative to the mean in standard-deviation units. A z of 0 is exactly average, positive z is above average, negative z is below. Because it removes the units, z-scores let you compare values from completely different distributions.

How do you find the probability from a z-score?

The probability is the area under the standard normal curve. P(Z < z) is the cumulative area to the left, which this tool computes with an error-function approximation. For z = 1.96 the left area is about 0.975, matching the classic z-table value.

What is a good z-score?

It depends on context. About 68% of values fall between z = −1 and 1, 95% between −1.96 and 1.96, and 99.7% between −3 and 3. Values beyond ±2 are often considered unusual, and beyond ±3 are typically treated as outliers.

Can a z-score be negative?

Yes. A negative z-score simply means the value lies below the mean. For instance z = −1.5 means one and a half standard deviations below average. The sign shows direction; the magnitude shows distance.

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