2. z-Scores What is a z-Score? How Are z-Scores Useful? Distributions of z-Scores Standard Normal Curve

3. What is a z-Score? A z-score is a standard score that measures distance from the mean in units of standard deviations. A standard score allows comparison of scores across distributions.

4. How are z-Scores Useful? z-scores allow meaningful comparisons across different distributions. z-scores allow us to make estimates of percentages above or below any point in a normal distribution.

5. How are z-Scores Useful? Example: A client had a score of 16 on an achievement test and an IQ score of 110. Are these scores consistent with each other? Is one score relatively higher than the other? We can’t answer without standardizing.

6. How are z-Scores Useful? If the achievement test has a population  of 24 and a population  of 4, then a score of 16 is two standard deviations below the mean, so z = -2.00. If the intelligence test has a population  of 100 and a population  of 15, then a score of 110 is two thirds of a standard deviation above the mean, so z = +0.67.

7. Distributions of z-Scores Any distribution, when converted to z- scores, has / a mean of zero / a standard deviation of one / the same shape as the raw score distribution

8. Standard Normal Curve If a variable is distributed normally, converting all of the scores to z-scores results in a standard normal curve. The shape of the standard normal curve allows us to estimate proportions of the curve above or below any z-score.

9. Standard Normal Curve Really simple example of estimating a proportion from a normal curve: – Sam had an IQ score of 100 on a normally distributed IQ test with a  of 100 and  of 15. What is Sam’s percentile rank? – Because a normal curve is symmetrical, we know that 50% is below the mean, and because Sam’s score is at the mean, the percentile rank is 50%.

10. MEAN +1+2-2 68% 95% +3-3 99%