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

Cool Things About 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

Finding Percentile Ranks with z-Scores This only works for a normal distribution! You have to know the m and sx. 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 m of the test is 500, with a s of 30. What is Sam’s percentile rank?

m 515 500

STEP 1: Convert to a z-score. 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 m of the test is 500, with a s of 30. What is Sam’s percentile rank?

m 470 500

STEP 1: Convert to a z-score. 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 m of the test is 500, with a s of 30 Working Backwards The m of the test is 500, with a s of 30. What score is at the 90th percentile?

90% or .9000 m 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 = m + zs 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