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Standard Scores Standard scores, or “zscores” measure the relation between each score and its distribution. Equation of z-score of Xi zX i Xi X sX Example: Suppose the Mean is 100 and the Standard Deviation is 15: (a) Suppose Xi = 70, find z-score (b) Suppose Xi = 115, find z-score of this value. 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. 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 sX is 15: Suppose z = 2; find Xi. Solutions (a) If z = (Xi – Mean)/sX = 2 Then (Xi – 100)/15 = 2 Multiply both sides by 15: (Xi – 100) = (2)(15) = 30. Add 100 to both sides: Xi = 100 + 30 = 130. Properties of standard scores • z- scores always have a mean of zero. • z-scores always have a variance and standard deviation of 1. • If X is above the mean, its zscore is positive; if X is below its mean, its z-score is negative. Next Topic: Standard Normal Distribution • 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.