z - SCORES standard score: allows comparison of scores from different distributions z-score: standard score measuring in units of standard deviations.
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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
zx x
2.00 5
6070
z
0.50- 10
9085
z
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 and x.
• All it takes is a little calculus....• But the answer is in the back of the book.
A Really Easy ExampleSuppose your score is at the mean of a distribution, and the distribution is normal. What is your percentile rank?
Answer: 50th percentile rankThe mean = the median50% of the scores are below the median.
Another ExampleSam 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 ExampleSam 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?
500470
STEP 1: Convert to a z-score. z = (470-500)/30 = -1.00
STEP 2: Look up the z-score in the UnitNormal Table. Find the area beyond z.
area beyond z = .1587
STEP 3: Convert to a percentage.
.1587 = 16%
Working BackwardsThe of the test is 500, with a of 30. What score is at the 90th percentile?
500 X=?
90% or .9000
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 = .0987proportion above = .0987 + .5000 = .5987
What proportion is between a z of -.25 and a z of +.25?
area between mean and z = .0987proportion between = .0987 + .0987 = .1974
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