normalized scores and z-score distribution

Standardized scores and the z-score distribution

Ed 510 Applications of Educational Research

Here is some terminology that will help you master course material. Define them and create examples of each.

Raw score
Standard score
Transformed scores
Statistical norms
Z scores
Z distribution
stanines
percentiles
grade equivalent scores

When an educational researcher collects information using tests and tools, the scores that result are termed raw scores. Raw scores are scores in their natural form. Raw scores are informative because they can be arranged in distributions, and it is possible to calculate central values and measures of dispersion, such as the standard deviation. As long as a researcher is interested in exploring a set of raw scores without making comparisons between and among samples, or with other distributions of scores, raw scores can be used..

There are many occasions however when educational researchers are interested in making comparisons between groups and between two or more distributions of scores. When this is the research objective raw scores have serious limitations. These limitations result from the fact that two samples of scores may have different sample sizes, different means and even different standard deviations. When any combination of these circumstances is true, raw scores are not adequate for most research purposes Thus a researcher will want to convert raw scores to standard scores or transformed scores.

The graphic below demonstrates how scores on the Gates-McGiniity Reading Test can be interpreted in several ways. The raw score has been represented according to different statistical norms.

The median, or the 50^th percentile, is the standard against which all other percentile values are compared. Percentiles are interpreted in relation to their distance from the 50^th percentile. The greatest disadvantage of percentiles results from the fact that there will always be fewer individuals scoring at the ends of a percentile distribution and more at the middle of the same distribution. Thus comparisons made in the center of a distribution of percentile scores are more informative than comparisons made of percentiles at the extreme ends of a distribution. Comparisons between middle percentiles and those at the extremes are not based on equivalent measurements of learning or ability.

The median, or the 50^th percentile, is the standard against which all other percentile values are compared

Grade equivalent scores

Grade equivalent scores indicate how a student at the grade level of the score can be expected to perform on a particular test. Grade equivalent scores do not describe the grade level of the examinee. Therefore they do tend to be somewhat misleading.

Stanine

Stanine scores allow comparisons between individuals because raw scores cam be located within subdivisions of the distribution of scores. Each subdivision is based on 1/9 ^th of the scores. Thus once standardized as stanines, raw scores can be compared on a relative basis to other raw scores.

Stanine scores do not allow for fine discriminations between individuals because stanine scores are gross measures of relative learning or ability.

These are examples of standardized scores - standardized because they make it easier to compare individual scores using a common yardstick. Once scores are standardized any raw score can be compared to any other raw score without concerns about the sample size, the central norms or the standard deviation that pertains to the sample from which an individual raw score has been drawn.

Each of the standardized scores discussed so far are limited by the fact that comparisons tend to be inexact. The z score overcomes this problem.

Z scores

These are scores that can be directly compared to the normal curve. Each score represents a percentage of individuals who occupy a specific location on the normal curve. Because z scores express equivalent intervals on a distribution each score is proportional in value to all the other z scores. Thus the differences between z scores provide precise information about a differences in learning or ability.

When are z scores used?

There are three ways in which z-scores are used in educational research.

Comparisons between two scores achieved by the same individual on two different tests.

For example, Mary has achieved a raw score of 20 on a math test and a raw score of 30 on a reading test. Has Mary done better on the reading test. It is impossible to say because the standard deviations of the math and reading tests may have been very different. Perhaps the scores are statistically equal. Perhaps a 20 on the math test is a statistically higher value than the score of 30 on the reading test.

Comparisons between scores achieved by two individuals on the same test.

Mary may have achieved a raw score of 20 on a math test in September and a 25 on a similar version of the math test in December. Again one cannot tell which score has the statistically greater value because each test may have different statistical characteristics.

Comparisons between a single score on a test achieved by one person and the entire distribution or population of scores for the same test.

Mary may have achieved a score of 20 on a math test, and the highest possible score is a 35.. What does this really tell us about her ability. When Mary's math score is compared to the norms supplied by the publisher of the math test it may turn out that a 20 is a fairly high value-- or maybe not!

The formula for calculating z scores demonstrates how z scores allow more exact comparisons

z = (X - X) / S_x

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The formula clearly adjusts the raw score (X) in relation to the sample mean (X). It also corrects for the average amount of deviation in the sample (s).

Let's use the information in the table below to calculate individual z scores. Nine students have taken different forms of the same spelling test The raw scores have been arranged from lowest to highest, appearing to distinguish pupils according to their performance on the spelling tests Is that a correct interpretation?

Pupil Raw score Mean Standard deviation Z score

Hazel 10 40 10

Myron 20 10 20

Phillip 30 5 10

Sam 40 50 5

Carol 50 50 10

Stephen 60 45 10

George 70 50 20

Monica 80 50 50

Karen 90 70 10

If a raw score is equal to the mean score of a distribution the z score will be zero. The raw score measures average performance.

Z scores that have a negative sign indicate that the raw score was lower than the average score for the sample. But once the z-scores have been computed the raw scores have been equated.

Does the new distribution of z scores indicate that the distribution of raw scores from lowest to highest should change our understanding of the students' spelling performance?

Z scores are normally distributed. That is if all z scores from -3.00 through 0 to +3.0 were plotted for a large population of individuals, the shape of the distribution would be normal.

The relationship between z scores and standard deviations

A z score of 1 corresponds to a standard deviation of 1? Are there other correspondences like this one? The correspondences are consistent. Thus any z score can be compared to a standard deviation unit. This means by knowing an individual's z score, it is possible to determine how close to or how far from the mean a raw score will be.

Summarizing questions

How might you use the concept of z scores in your educational practice?

Are there other questions about standardized scores that you would like to ask?

return to the course schedule

Pupil	Raw score	Mean	Standard deviation	Z score
Hazel	10	40	10
Myron	20	10	20
Phillip	30	5	10
Sam	40	50	5
Carol	50	50	10
Stephen	60	45	10
George	70	50	20
Monica	80	50	50
Karen	90	70	10