In statistics, a z-score tells united state how plenty of standard deviations away a worth is from the mean. We usage the complying with formula to calculate a z-score:

z = (X – μ) / σ

where X is the value we space analyzing, μ is the mean, and σ is the typical deviation.

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A z-score have the right to be positive, negative, or same to zero.

A hopeful z-score suggests that a specific value is higher than the mean, a an unfavorable z-score shows that a certain value is less than the mean, and also a z-score the zero suggests that a details value is same to the mean.

A couple of examples should make this clear.

Examples: Calculating a Z-Score

Suppose we have actually the following dataset that reflects the elevation (in inches) of a details group of plants:

5, 7, 7, 8, 9, 10, 13, 17, 17, 18, 19, 19, 20

The sample typical of this dataset is 13 and the sample typical deviation is 5.51.

1. Discover the z-score for the worth “8” in this dataset.

Here is exactly how to calculation the z-score:

z = (X – μ) / σ = (8 – 13) / 5.51 = -0.91

This method that the worth “8” is 0.91 traditional deviations below the mean.

2. Find the z-score because that the value “13” in this dataset.

Here is exactly how to calculate the z-score:

z = (X – μ) / σ = (13 – 13) / 5.46 = 0

This method that the worth “13” is specifically equal come the mean.

3. Uncover the z-score for the value “20” in this dataset.

Here is exactly how to calculation the z-score:

z = (X – μ) / σ = (20 – 13) / 5.46 = 1.28

This method that the worth “20” is 1.28 typical deviations above the mean.

How to translate Z-Scores

A Z Table tells us what portion of values autumn below details Z-scores. A couple of examples have to make this clear.

Example 1: negative Z-Scores

Earlier, we discovered that the raw value “8” in ours dataset had actually a z-score of -0.91. Follow to the Z Table, 18.14% of values fall listed below this value.

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Example 2: Z-Scores equal to zero

Earlier, we uncovered that the raw value “13” in ours dataset had actually a z-score that 0. According to the Z Table, 50.00% of values fall below this value.

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Example 3: hopeful Z-Scores

Earlier, we found that the raw worth “20” in ours dataset had actually a z-score that 1.28. According to the Z Table, 89.97% of values fall below this value.

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Conclusion

Z-scores can take on any kind of value between an unfavorable infinity and also positive infinity, but most z-scores autumn within 2 standard deviations the the mean. There’s in reality a dominion in statistics known as the Empirical Rule, which states that for a offered dataset with a typical distribution:

68% of data values autumn within one standard deviation of the mean.95% of data values autumn within two standard deviations that the mean.99.7% of data values fall within three standard deviations that the mean.

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The greater the absolute worth of a z-score, the more away a raw worth is indigenous the typical of the dataset. The reduced the absolute value of a z-score, the closer a raw value is to the average of the dataset.