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Z-Score Calculator with Graph

Statistics & ProbabilityFreeStep by stepGraph

A z-score tells you how many standard deviations a value is above or below the mean. A z-score of 1.5 means one and a half standard deviations above average. This calculator finds the z-score exactly, then uses the normal distribution to give the percentile: the share of values below it.

Try:

How to use the z-score calculator

  1. Enter the value x.
  2. Enter the mean μ and the standard deviation σ.
  3. Press Calculate to see the z-score and percentile.

Formula

z=x−μσz = \frac{x - \mu}{\sigma}

Worked example: x = 82, μ = 70, σ = 8

  • Value x: 82
  • Mean μ: 70
  • Standard deviation σ: 8

✓ Answer checked

z=1.5z = 1.5
Percentile (area below)
93.319%
Area above
6.6807%
Meaning
82 is 1.5 standard deviations above the mean.
4050607080901000.010.020.030.040.05
The shaded area is the proportion below 82: about 93.32%.
Step-by-step working (3 steps)

  1. Subtract the mean

    Find how far the value is from the mean.

    82−70=1282 - 70 = 12

  2. Divide by the standard deviation z = (x − μ)/σ

    Express that distance in standard deviations.

    z=128=32z = \frac{12}{8} = \frac{3}{2}

  3. Find the percentile

    Use the standard normal CDF Φ(z) for the area below.

    Φ(1.5)≈0.933193\Phi(1.5) \approx 0.933193
Formulas used
z=x−μσz = \frac{x - \mu}{\sigma}
How this was checked
  • ✓ μ + zσ = 70 + (3/2)(8) = 82.

Frequently asked questions

What does a negative z-score mean?

The value is below the mean. For example, z = −1.5 means one and a half standard deviations below average.

What z-score is the 97.5th percentile?

About 1.96. Roughly 97.5% of a normal distribution lies below z = 1.96.

Does the percentile assume a normal distribution?

Yes. The z-score itself works for any data, but the percentile is only accurate if the data is roughly normal.

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