1. Z-scores

2. 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?

3. 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

4. The relationship between z-scores and locations in a distribution

5. 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

6. 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.

7. 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

8. 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

9. 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?

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

11. 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.