1. Standard Scores Standard scores, or “z- scores” measure the relation between each score and its distribution.

2. Equation of z-score of X i

3. Example: Suppose the Mean is 100 and the Standard Deviation is 15: (a) Suppose X i = 70, find z-score (b) Suppose X i = 115, find z-score of this value.

4. Answers To find a z-score, subtract the mean and divide by the standard deviation. In this example, we subtract 100, and divide the difference by 15: (a) z = (70 – 100)/15 = –30/15 = –2. (b) z = (115 – 100)/15 = 15/15 = 1.

5. More Problems We might know the z-score and need to solve for the “raw” score; That is, we know z and we find X. If the mean is 100 and s X is 15: Suppose z = 2; find X i.

6. Solutions (a) If z = (X i – Mean)/s X = 2 Then (X i – 100)/15 = 2 Multiply both sides by 15: (X i – 100) = (2)(15) = 30. Add 100 to both sides: X i = 100 + 30 = 130.

7. Properties of standard scores z- scores always have a mean of zero.z- scores always have a mean of zero. z-scores always have a variance and standard deviation of 1.z-scores always have a variance and standard deviation of 1. If X is above the mean, its z- score is positive; if X is below its mean, its z-score is negative.If X is above the mean, its z- score is positive; if X is below its mean, its z-score is negative.

8. Next Topic: Standard Normal Distribution z-scores are useful to simplify many problems.z-scores are useful to simplify many problems. One use is to convert any normal distribution to the standard normal distribution, which is the next topic.One use is to convert any normal distribution to the standard normal distribution, which is the next topic.