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