- Calculate z-scores to explain the location of data points.
- Compare observations using z-scores and the Empirical Rule.
Standardized Value ([latex]z[/latex]-score)
We have learned about mean as a measure of center and that standard deviation is a measure of spread.
Let’s use a hypothetical typical arm span for Americans in our exploration of measuring distance from the mean. Arm span is the distance from the tip of the middle finger on one hand to the tip of the middle finger on the other hand when a person’s arms are stretched out and opened wide.
Suppose that, based on many measurements, a statistician believes that the distribution of arm spans of Americans has a mean of [latex]173.40[/latex] centimeters (cm) and a standard deviation of [latex]12.21[/latex] cm. Let’s assume these values represent the true population mean and standard deviation.
In the question above, you found observed values that are one standard deviation from the mean. What if we want to figure out how many standard deviations an observed value is from the mean, instead?
standardized value ([latex]z[/latex]-score)
A standardized value, or [latex]z[/latex]-score, is the number of standard deviations an observation is away from the mean.
[latex]z=\dfrac{\text{Observed Value}-\text{mean}}{\text{standard deviation}} = \dfrac{x-\mu}{\sigma}[/latex]
where [latex]x=[/latex] the value of the observation, [latex]\mu=[/latex] the population mean, [latex]\sigma=[/latex] the population standard deviation, and [latex]z=[/latex] the standardized value (or [latex]z[/latex]-score).
It is important to note that this distance is measured in standard deviations. Thus, a negative [latex]z[/latex]-score is an observation that is below the mean, and a positive [latex]z[/latex]-score is an observation that is above the mean.
The “unit” for a [latex]z[/latex]-score is standard deviations.