The z-score is a way of counting the number of standard deviations between a given data value and the mean of the data set. In this article, we will look at the formula for the z-score, how to calculate it, and a little more closely at this idea of counting standard deviations.

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The formula

To calculate the z-score, you first find the distance from the mean, and then divide by the standard deviation.

z-score-formula

If we label the value x, we could write the formula as:
z-score-formula-symbols

Calculating and interpreting the z-score

Let’s look at an example to see how to use this formula.

Example

The mean score on a standardized test was 508 with a standard deviation of 42. One test-taker’s score was 590. Find and interpret the z-score for this score.

From the example, we have the following information:

The mean: \mu = 508

The standard deviation: \sigma = 42

Therefore, the z-score is:

z = \dfrac{x-\mu}{\sigma} = \dfrac{590 - 508}{42} = \dfrac{82}{42} \approx \boxed{1.95}

The z-score counts standard deviations from the mean, and the sign gives the direction. In this case, we can say that:

The test-score of 590 is about 1.95 standard deviations above the mean.

More about interpretation

As you saw above, the value and the sign of the z-score gives you information about the location of the data value. Specifically:

  • A positive z-score means the data value is larger than the mean.

    If a data value has a z-score of 2, that tells us that this data value is 2 standard deviations larger than the mean.

  • A negative z-score means the data value is smaller than the mean.

    If a data value has a z-score of –3.1, then this data value is 3.1 standard deviations smaller than the mean.

  • A z-score of zero means that the data value equals the mean.

    For example, consider a data set with a mean of 50 and a standard deviation of 2. If a data value also is 50, then the z-score is:

    z = \dfrac{x-\mu}{\sigma} = \dfrac{50 - 50}{2} = \dfrac{0}{2} = \boxed{0}

Conclusion

In your study of statistics, you will come across the z-score in a wide variety of settings. For this reason, it is important to make sure you thoroughly understand the ideas discussed here. The best way to do this is to practice the calculation so that it simply becomes second nature! This will help make future topics much easier to understand.

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