Z-scores

Z-score or standard score A statistical techniques that uses the mean and the standard deviation to transform each score (X) into a z-score Why z-scores are useful?

Z-scores and location in a distribution The sign of the z-score (+ or –) The numerical value corresponds to the number of standard deviations between X and the mean

The relationship between z-scores and locations in a distribution

Transforming back and forth between X and z The basic z-score definition is usually sufficient to complete most z-score transformations. However, the definition can be written in mathematical notation to create a formula for computing the z-score for any value of X. X – μ z = ──── σ Deviation score Standard deviation

What if we want to find out what someone’s raw score was, when we know their z-score? Example: Distribution of exam scores has a mean of 70, and a standard deviation of 12.

Transforming back and forth between X and z (cont.) So, the terms in the formula can be regrouped to create an equation for computing the value of X corresponding to any specific z-score. X = μ + zσ

Distribution of z-scores shape will be the same as the original distribution z-score mean will always equal 0 standard deviation will always be 1

Using z-scores to make comparisons Example: You got a grade of 70 in Geography and 64 in Chemistry. In which class did you do better?

z-Scores and Samples It is also possible to calculate z-scores for samples.

z-Scores and Samples Thus, for a score from a sample, X – M z = ───── Using z-scores to standardize a sample also has the same effect as standardizing a population.