- Calculate [latex]z[/latex]-scores to explain the location of data points.
- Compare observations using [latex]z[/latex]-scores and the Empirical Rule.
[latex]z[/latex]-Scores
A higher organ weight is an indicator of higher toxicity. Suppose researchers want to compare the toxicity of a randomly selected liver with that of a randomly selected spleen. How will they know if the weight is extreme? We can use the [latex]z[/latex]-score for each of these values to help us answer these questions.
- The [latex]z[/latex]-score is the number of standard deviations an observation is away from the mean.
- The [latex]z[/latex]-score has no units associated with it. It only gives relative proximity (distance and direction) from the mean of a quantitative variable. The formula for this is: [latex]z=\dfrac{x-\mu}{\sigma}[/latex], where [latex]x[/latex] is an observation value, [latex]\mu[/latex] is the population mean, and [latex]\sigma[/latex] is the population standard deviation.