Z-Scores

Histogram

• A bar chart of a frequency distribution.0 — 21 — 32 — 03 — 24 — 45 — 36 — 37 — 58 — 39 — 210—2

Normal Distribution

With some frequency distributions the histograms have lots of examples in the middle and fewer that are higher and lower.

Normal Distribution

Measurements based on an underlying random phenomena distribute normally if you have enough data points.

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Normal Distribution

6

8

10

12

Normal Distribution

A Normal Curve is the theoretical line that represents all of the responses in a normal distribution. The area under the curve encloses the frequency distribution of the normally distributed phenomena.

4

0

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Characteristics of Normal Curves

A line connecting the tops of the bars of a histogram.Mean, median and mode appear at the same point.Lots of responses in the middle fewer in the extremes.

One More Interesting Characteristic of Normal Curves

1 Standard DeviationMean

1 Standard DeviationMean

50.0

One More Interesting Characteristic of Normal Curves

1 Standard DeviationMean

50.0 34.13

One More Interesting Characteristic of Normal Curves

2 Standard DeviationsMean

34.1350.0

13.59

One More Interesting Characteristic of Normal Curves

A test score

Cognitive abilities are normally distributed.

A test score

Then, if the tests are designed carefully the assessment of cognitive ability should be normally distributed as well.

A test score

Results from a standardized test are, by definition, normal distributed.

Mean

-1-2-3 1 2 3

z Scores

SD

If you know the percentage of scores that are lower than the target score you will know the Percentile Rank of the target score.

A test score

Normal Curves2 Standard DeviationsMean

34.1350.0

13.59

1. The relationship between SD and percentage of the area under the curve is constant regardless of the distribution.

2. If the mean and SD of the distribution are known then the percentage of scores lower than every possible score can be computed.

3. In other words, percentile rank of every score can be computed.

z Score Computation

• Subtract the mean of the distribution from the target score.

• Divide by the standard deviation.

• Look up the z score on the z table.

-1-2-3 1 2 3

Target Score(28)Mean

(25.69)

Number of SD away from the meanPercentage of scores lower

than the target score

SD

Subtract the mean of the distribution from the target score.

Divide by the standard deviation.

Look up the z score on the z table.

1 SD(2.72)

z Score

• Target score (28)

• Score - Mean (28 - 25.69 = 2.31)

• Result / SD (2.31 / 2.72 = .85)

• Look up on z table (0.85)

• z = 0.85; area = .80

• Score is in the 80th percentile

-1-2-3 1 2 3

28

z Scores

25.69

.85 SD

80 % of scores

SD

Mean

-1-2-3 1 2 3

Target Score

z Scores

SD

-1-2-3 1 2 3

z Scores

SD

25.6923

-.99 SD

16 % of scores

z Score

• Target score (23)

• Score - Mean (23 - 25.69 = -2.69)

• Result / SD (-2.69 / 2.72 = -.99)

• Look up on z table (-.99)

• Z = -0.99; area = .16

• Score is in the 16th percentile

-1-2-3 1 2 3

z Scores

SD

25.6923

-.99 SD

16 % of scores

Percentile Rankings

• z Scores

• Assume a normal distribution

• Based on knowing everyone in the population

• Allows comparison of individualto the whole

What is Molly’s percentile ranking on the test? What is Karl’s

• 34

• 56

• 22

• 31

• 44

• 47

• 37

• 42

• 51

• 37

• 40

• 29

• 36

• 39

• 52

• 39

• 29

• 51

• 45

• 47

• 42

Molly

Karl

Bobby and Sally take a standardized test that has 160 questions

• Bobby gets a raw score of 140 and has a percentile rank of 52. Sally gets a raw score of 142 and has a percentile rank of 67.

• This doesn’t make sense to Bobby’s mother. First, how can 140 out of 160 be 52nd percentile. And second, why should just a couple of points on the test make such a huge percentile rank difference? What are you going to tell her?