qcalculate

Z-Score Calculator

Find how many standard deviations a value is from the mean, or turn a z-score back into a value, and see the area under the bell curve.

Z-score

1.5

The z-score of 85 is 1.5: about 93.32% of a normal distribution is below it.

Below
93.32%
Above
6.68%
Within ±1.5
86.64%

Shares of a normal distribution, from the area under the bell curve.

Show me how

All steps as text

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    The z-score formula

    z=x−μσz = \frac{x - \mu}{\sigma}

    Subtract the mean μ from the value x, then divide by the standard deviation σ. To go the other way, x = μ + zσ.

    Worked example

    A test has a mean of 70 and a standard deviation of 10, and you scored 85. Your z-score is (85 − 70) ÷ 10 = 1.5: you are 1.5 standard deviations above the mean. If the scores follow a normal distribution, about 93.32% of students scored below you.

    The 68–95–99.7 rule

    For normally distributed data, about 68.27% of values have a z-score between −1 and 1, about 95.45% between −2 and 2, and about 99.73% between −3 and 3. A z-score beyond ±3 is unusual.

    A z-table lists the share below each z-score; this calculator works it out for any z. To find the mean and standard deviation of your own data, use the standard deviation calculator, which also lists a z-score for every value.

    Questions

    What does a negative z-score mean?

    The value is below the mean. A z-score of −2 means two standard deviations below.

    What is a good z-score?

    It depends on what you measure. For a test score, a positive z-score means above average; above 2 puts you in roughly the top 2.3%.

    Do the percentages always apply?

    No. The shares below and above come from the normal (bell-shaped) distribution. The z-score itself is right for any data, but the percentages only fit data that is roughly normal.