Using the $$z$$ score formula, a raw score can be translated into standard deviation units. For example, in a distribution of intelligence quotient (IQ) scores with a mean of $$100$$ and a standard deviation of $$15$$, an IQ score of $$110$$ would have a $$z$$ score of $$\frac{110 - 100}{15} = +0.67$$. This indicates the score is approximately two-thirds of a standard deviation above the mean. Similarly, a raw score of $$85$$ would have a $$z$$ score of $$\frac{85 - 100}{15} = -1.00$$, placing it exactly one standard deviation below the mean.

Example of a z Score

Which of the following correctly expresses the formula for computing a z score?

The $$z$$ score for a particular individual is computed by subtracting the mean of the distribution ($$M$$) from that individual's raw score ($$X$$), and then dividing that difference by the standard deviation of the distribution ($$SD$$). The formula is expressed as: $$z = \frac{X - M}{SD}$$.

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A $$z$$ score indicates how many standard deviations a particular raw score is above or below the mean of its distribution. It standardizes the distance of a score from the mean, providing a consistent way to describe a score's relative location.

z Score

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KPU Research Methods in Psychology - 4th American Edition

Statistical z Score Formula

While outliers are generally understood as extreme values within a dataset, $$z$$ scores provide a formal, quantitative method for identifying them. Commonly, outliers are defined as scores that fall more than three standard deviations away from the mean. In standardized terms, any score with a $$z$$ score strictly less than $$-3.00$$ or strictly greater than $$+3.00$$ is classified as an outlier.

Defining Outliers using z Scores

Converting raw scores to $$z$$ scores serves several important functions in statistical analysis. Primarily, they provide a standardized way to describe exactly where an individual's score is located within a distribution, making them ideal for reporting the results of standardized tests. Additionally, $$z$$ scores offer a quantitative method for defining outliers and serve as a foundational building block for understanding and computing many other advanced statistical measures.

Usefulness of z Scores

The formula for computing a $$z$$ score is: $$z = \frac{X - M}{SD}$$, where $$X$$ is the individual raw score, $$M$$ is the mean of the distribution, and $$SD$$ is the standard deviation of the distribution.

Formula for the z Score

$$z$$ scores are important because they describe the exact relative location of an individual’s score within a distribution using a standardized metric. They are frequently used to report results on standardized tests and play a crucial role in computing and understanding more advanced statistical concepts.

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