Calcylator
Standardization

Z-score formula:
(value − mean) ÷ standard deviation

Compute a z-score, read what it says about where a value sits, compare scores from different tests, and use a quick percentile table.

Calcylator Editorial Team

Updated · 5 min read

The z-score formula

The z-score formula is z = (value − mean) ÷ standard deviation. It tells you how many standard deviations a value sits from the average. A mark of 82 in a class with mean 70 and standard deviation 8 has z = (82 − 70) ÷ 8 = 1.5, which is one and a half standard deviations above the mean.

Z-score =z = (x − μ) ÷ σ
x:
the value you are checking
μ:
the mean of the data
σ:
the standard deviation, which must be above zero
This is the calculation of the Calcylator z-score calculator: value, mean and standard deviation in, z-score out.

  • Value

    82

  • Mean

    70

  • Standard deviation

    8

  • Calculation

    (82 − 70) ÷ 8

Z-score

1.5000

Positive, so the mark is above the class average.

How to read a z-score

  • z = 0 means the value equals the mean.
  • Positive z means above the mean; negative z means below it.
  • The size tells you how unusual the value is: the further from zero, the rarer.
  • Because z has no units, you can compare heights, marks or prices on one scale.

To go backwards, value = mean + z × standard deviation. With a mean of 70, a standard deviation of 8 and z = 1.96, the value is 70 + 1.96 × 8 = 85.68.

Z-score to percentile table

If your data are roughly bell-shaped, a z-score maps to the share of values below it. The figures below are the usual standard normal values.

Approximate percentage of values below a given z-score
z-scorePercentage below
-2.02.28%
-1.015.87%
0.050.00%
0.569.15%
1.084.13%
1.593.32%
1.9697.50%
2.097.72%
2.599.38%
3.099.87%

The familiar rule of thumb follows from this: about 68 percent of values lie within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. The table only applies when the data are approximately normal.

Comparing scores from different tests

Z-scores let you compare apples and oranges. A student scores 82 on a test with mean 70 and standard deviation 8, so z = 1.5. On another test the same student scores 64, with mean 50 and standard deviation 6, so z = (64 − 50) ÷ 6 = 2.3333.

The second mark is lower in raw terms but much stronger relative to the group, since 2.33 standard deviations above the mean is rarer than 1.5. In the first test about 93 percent scored lower; in the second more than 99 percent did.

Common mistakes

  • Using the variance in place of the standard deviation.
  • Mixing a sample standard deviation with the population one when you compare with published results.
  • Reading a percentile from the table when the data are heavily skewed.
  • Forgetting the sign: a negative z-score is below the mean, not an error.
  • Treating a z-score beyond 3 as proof of an error. It is unusual, and worth a look, but it can be real.

In practice, z-scores appear in quality checks too. If a packing line fills bags with a mean of 500 g and a standard deviation of 4 g, a bag weighing 509 g has z = (509 − 500) ÷ 4 = 2.25. That is far enough out to prompt a look at the machine, even though no single bag proves a fault.

Common questions

What is the z-score formula?

The formula is z = (x − mean) ÷ standard deviation, where x is the value you are checking. It shows how many standard deviations the value is from the mean. A mark of 82 with mean 70 and standard deviation 8 gives z = 1.5.

How do I calculate a z-score?

Subtract the mean from your value, then divide by the standard deviation. For example, with a value of 64, a mean of 50 and a standard deviation of 6, the answer is 14 ÷ 6 = 2.3333. Positive means above the mean.

How do I convert a z-score to a percentile?

Look up the z-score in a standard normal table, or use the quick table in this article. A z-score of 1.0 is about the 84th percentile and 1.96 is about the 97.5th. This only holds when the data are roughly bell-shaped.

Can a z-score be negative?

Yes. A negative z-score means the value is below the mean. For instance, a mark of 58 with mean 70 and standard deviation 8 gives z = -1.5, which is about the 7th percentile on a bell-shaped distribution.

What z-score is considered unusual?

Many analysts look more closely at values beyond 2 or 3 standard deviations from the mean. For bell-shaped data, about 5 percent lie beyond 1.96 either way and about 0.3 percent beyond 3. It is a prompt to investigate, not an automatic error.

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