AP Stats Z-Score Calculator

Find how many standard deviations a value is from the mean, and what percentile it falls at. Enter a value, the mean, and the standard deviation, and this calculator gives you the z-score and the percentile from the normal distribution.

What Is a Z-Score?

A z-score tells you how many standard deviations a value is away from the mean of its distribution. A z-score of 1.20 means the value sits 1.20 standard deviations above the mean, and a z-score of -0.50 means it sits half a standard deviation below the mean. Z-scores let you compare values from different distributions, because they put every value on the same standard scale.

How to Use This Calculator

1) Type the value you want to place, x. 2) Type the mean of the distribution. 3) Type the standard deviation, which must be greater than 0. 4) Press Calculate Z-Score. Read the z-score first, then the percentile: a percentile of 88 means about 88% of values fall below your value.

How to Calculate a Z-Score

The formula is: z = (x - mean) / standard deviation. A positive z-score means the value is above the mean, a negative z-score means it is below the mean, and a z-score of 0 means it equals the mean. The percentile is the area under the normal curve to the left of that z-score.

Worked example: x = 62, mean = 50, standard deviation = 10. z = (62 - 50) / 10 = 1.20. The percentile is about 88.5%, so a value of 62 is at about the 88th percentile.

Tips and Common Mistakes

• Use the standard deviation, not the variance. If you are given variance, take its square root first.
• Do not mix up sample and population standard deviation when the question names one of them.
• A z-score is not a probability. To get a probability or percentile, use the normal CDF area for that z-score.
• For a value below the mean, expect a negative z-score and a percentile below 50.

Frequently Asked Questions

What does a z-score of 1.5 mean?

It means the value is 1.5 standard deviations above the mean.

What percentile is a z-score of 0?

The 50th percentile, because half of a normal distribution falls below the mean.

Can a z-score be negative?

Yes. A negative z-score means the value is below the mean, and its percentile is below 50.

What is the difference between a z-score and a percentile?

A z-score measures distance from the mean in standard deviations. A percentile measures what percentage of values fall below that value.

When do I use a z-score instead of a t-score?

Use a z-score when the population standard deviation is known or the distribution is normal and the sample is large. Use a t-score when you estimate the standard deviation from a small sample.

What if my standard deviation is 0?

A standard deviation of 0 means every value is the same, so a z-score cannot be calculated. Check your data before calculating.

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