z-Scores

Standardized Scores

Standardizing scores

• With non-equivalent assessments it is not possible to develop additive summary statistics.– e.g., averaging the scores from assessments with

different maximum scores.

• In classrooms we solve this problem by assigning grades which represent a uniform evaluation for every assessment.– i.e., A, B, C, D, F

Standardizing scores

• Grades often have a subjective interpretation which makes comparing students using average grades problematic.

• Assuming that assessments themselves are minimally subjective (i.e., well-designed) some consistent measure of student achievement can be used across assessments.

Histogram

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

Summary Statistics

• The summary statistics used to describe interval data are mean and standard deviation.

• Standard deviation represents the average distance of observed scores from the mean.

• Any given score can be represented as the number of standard deviations from the mean.

z-Scores

• Hence:A z-score is the distance a given score is from the mean divided by the standard deviation.

(score – mean)/standard deviation = z

• A z-score is a given score translated into units of standard deviation. Common standard score for any interval assessment

z-Score Computation

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

• Divide by the standard deviation.

• Positive z-scores are above the mean.

• Negative z-scores are below the mean.

Histogram

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

Mean = 5.74 sd = 2.61

z-Score Computation

• For a score of 9:z = (9-5.74)/2.61 = 1.25 sd above the mean

• For a score of 4:z = (4-5.74)/2.61 = - 0.67 sd below the mean

• Positive z-scores are above the mean.

• Negative z-scores are below the mean.

What is Molly’s z-score 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

z-Scores

A special case with Normal Distributions

Normal Distribution

Lots of naturally occurring phenomena distribute normally if you have enough data points.

Histogram of a frequency distribution

Things that distribute normally have lots of examples in the middle of the range of possibilities and fewer examples that are farther from the middle of the range.

Normal Distribution

Histogram of a frequency distribution

Things that are normally distributed have equal numbers of examples that are above and below the average example. Normal distributions are symmetrical.

Normal Distribution

Histogram of a frequency distribution

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

Histogram of a frequency 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

Examples of Normally Distributed Phenomena

• The height of 12 year old boys

• The life of a 60 watt light bulb

• The weight of Red Delicious apples

• Recovery time of surgery patients

• Spatial problem solving ability of 10 year olds

• Scores on norm-referenced tests of anything

4

0

2

12345678910111213141516171819202122232425262728

Normal Curves

A line connecting the tops of the bars of a histogram

Normal Curves

In a normal distribution the mean, median and mode appear at the same point.

Normal Curves

1 Standard DeviationMean

Normal Curves

1 Standard DeviationMean

50.0

Normal Curves

1 Standard DeviationMean

50.0 34.13

Normal Curves

2 Standard DeviationsMean

34.1350.0

13.59

Standard Deviations 0 +1 +2 +3-2 -1-3

34.13%34.13%13.59% 13.59% 2.14%2.14%0.13%0.13%

Normal Curves

A test score

Cognitive abilities are normally distributed.

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

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.

-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 = .8023

• 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

23

-.99 SD

16 % of scores

25.69

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 = .1611

• Score is in the 16th percentile

-1-2-3 1 2 3

z Scores

SD

23

-.99 SD

16 % of scores

25.69

Percentile Rankings

• z Scores

• Assume a normal distribution

• Based on knowing everyone in the population

• Allows comparison of individualto the whole

Practice

• Open the Excel file from the course webpage: OWM Data

• What is the percentile rank of a score of 10 on the variable pre?

• What is the percentile rank of a score of 10 on the variable delayed?

Standard Deviations (z-scores) 0 +1 +2 +3-2 -1-3

34.13%34.13%13.59% 13.59% 2.14%2.14%0.13%0.13%

Percentile Equivalents1 5 10 20 40

5060

7080 90 95 99

30

Cumulative Percentages 50.0% 84.1% 97.7% 99.9%2.3% 15.9%0.01%

IQ Scores 100 115 130 14570 8555

SAT Scores (sd 209) 1026 1235 1444 1600608 817400

Normal Curves

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?

• Explain this to Bobby’s mother in a paragraph or two and send me what you have written.

• For a challenge (if you have time), what is the mean and standard deviation for this test? (solution)