Survey

Transcript

z - SCORES • standard score: allows comparison of scores from different distributions • z-score: standard score measuring in units of standard deviations Comparing Scores from Different Distributions • Suppose you got a score of 70 in Dr. Difficult’s class, and you got an 85 in Dr. Easy’s class. • In relative terms, which score was better? • Suppose the M in Dr. Difficult’s class was 60 and the SD was 5. • So your score of 70 was two standard deviations above the mean. • That’s good! • In Dr. Easy’s class, the M was 90, with a SD of 10. • So your score of 85 was half of a standard deviation below the mean. • Not as good! Calculating z-scores • Your z-score in Dr. Difficult’s class was two standard deviations above the mean. That means z = +2.00. • Your z-score in Dr. Easy’s class was half a standard deviation below the mean. That means z = -.50. z - score formula z x x 70 60 z 2.00 5 85 90 z - 0.50 10 Cool Things About z-scores • Any distribution, when converted to zscores, has • a mean of zero • a standard deviation of one • the same shape as the raw score distribution Finding Percentile Ranks with z-Scores • • • • This only works for a normal distribution! You have to know the and x. All it takes is a little calculus.... But the answer is in the back of the book. A Really Easy Example Suppose your score is at the mean of a distribution, and the distribution is normal. What is your percentile rank? Answer: 50th percentile rank The mean = the median 50% of the scores are below the median. Another Example Sam got a score of 515 on a normally distributed aptitude test. The of the test is 500, with a of 30. What is Sam’s percentile rank? 500 515 STEP 1: Convert to a z-score. z = (515-500)/30 = .50 STEP 2: Look up the z-score in the Normal Curve Table. Find the area between mean and z. area between mean and z = .1915 STEP 3: Add the area below the mean. total area below = .1915 + .5000 = .6915 STEP 4: Convert the proportion to a percentage. percentile rank = 69% A Tricky Example Sam got a score of 470 on a normally distributed aptitude test. The of the test is 500, with a of 30. What is Sam’s percentile rank? 470 500 STEP 1: Convert to a z-score. z = (470-500)/30 = -1.00 STEP 2: Look up the z-score in the Unit Normal Table. Find the area beyond z. area beyond z = .1587 STEP 3: Convert to a percentage. .1587 = 16% Working Backwards The of the test is 500, with a of 30. What score is at the 90th percentile? 90% or .9000 500 X=? STEP 1: Look up the z-score. proportion beyond z = .1000 z = +1.28 STEP 2: Convert the z-score into raw score units, using x = + z x = 500 + (1.28)(30) = 500 + 38.40 = 538.40 Finding Other Proportions • What proportion is above a z of .25? area beyond z = .4013 • What proportion is above a z of -.25? area between mean and z = .0987 proportion above = .0987 + .5000 = .5987 What proportion is between a z of -.25 and a z of +.25? area between mean and z = .0987 proportion between = .0987 + .0987 = .1974