Assessment-Alex

Standardized Tests

A few notes

You may wonder why you should learn these statistical concepts and procedures. You may think that you will never use these kinds of math in the rest of your life. When I was an undergraduate student, I also looked at math in this way. However, when someday you do works regarding research or administration, it will be inevitable for you to come across the following statistical concepts. You may not need to compute them yourself, but at least you should be able to interpret them.

As you know the first half of this class is devoted to alternative assessment models, as opposed to the conventional bell-curve approach. It is up to you to stand with either side. However, if you want to criticize the bell-curve approach, you should understand how it works first.

The materials in these two lessons are simple basics. Instead of assigning you to read statistical textbooks, I have already simplified the materials for you (The lecture contents are based upon Kubiszyn & Borich's Educational testing and measurement, Kubiszyn & Borich's Educational testing and measurement, and Crocker & Algina's Introduction to classical and modern test theory).The test will cover the fundamental information only.

Table of Contents

Comparisons betwwen Standardized and Teacher-made tests
Standard Scores
Formula for Z Scores
Characteristics of Z score
Standard Normal Distribution
Percentiles
Summary

A Comparison of Standardized and Teacher-made Tests
Standardized Teacher-Made
Learning Outcomes Measurement general outcomes and content appropriate to the majority of schools in the U.S. They are tests of general skills and understanding tend not to reflect specific or unique emphases of local curricula.

Well adapted to the specific and unqiue outcomes and contents of a local curriculum; they are adaptable to various sizes of work units, but tend to neglect complex learning outcomes.

Quality of test items Qualty of items generally is high. Items are written by specialists, pretested, and selected on the basis of the results of a quantitative item analysis.

Quality of items is often unknown. Quality is typically lower than that of standardized tests due to the limited time of the teacher.

Reliability Reliability is high, commonly between .80 and .95, and frequently above .90.

Reliability is usually unknown, but can be high if items are carefully constructed.

Administration and scoring Procedures are standardized; specific instructions are provided.

Uniform procedures are possible, but usually are flexible and unwritten.

Interpretation of scores Scores can be compared to norm groups. Test manual and other guides aid interpretation and use.

Score comparisons and interpretations are limited to local class or school situation. Few, if any, guidelines are available for interpretation and use.

Standard Scores

To determine how well a student did in a test or to compare scores from different tests, you need to interpret the scores in terms of the means and standard deviations of the respective distributions. Essentially, each score must be evaluated in terms of its relative standing, or position, in the distribution; then these relative standings must be compared.

A score alone cannot tell how well a student did. For exmple, is 90 a good score? Well, perhaps the majority gets above 95 and 90 is considered an F! Is 65 a poor score? If the mean is 40, then 65 may be an A! Because the first test score may be 1 standard deviation below the mean and the second score is 1.5 standard deviations above the mean. Doing this for each score in its respective distribution gives us measures of relative standing called Z scores.

Formula for Z Scores
To find out how many standard deviations a score is from the mean, subtract the mean from the score and divide by the standard deviation. So Z scores are defined by...

A general verbal formula for a Z score is a score minus its mean divided by its standard deviation. This formula is appropriate for describing the relative position of an original raw score in a sample and can be used to compare two or more scores from the same or different distributions. For example, you can compare your score on the first quiz in a given course to another person's score on that quiz or to your performance on the second quiz. So Z scores computed for each raw score allow you to compare relative performances. To describe a score in a population, the following formula would be used:

Characteristics of Z score

Notice that Z scores are expressed in terms of standard deviation units. In other words, standard deviations are the units of measure for Z scores. For an IQ score with Z=1.5, the score is 1.5 standard deviation units above the mean. For a person's height which has Z=2.7, the height is 2.7 standard deviations above the mean. Notice that Z scores not only give the distance of the score from the mean in standard deviation units, but also the direction of the score from the mean by using the sign of the Z score.

The mean of a set of Z scores is 0 and the variance and standard deviation of a set of Z scores are 1.

A final characteristic of z scores is that the transformation to z scores does not change the shape of the distribution from that found for X. If the distribution of X is positively skewed, then the distribution of z scores computed from the X scores is also positively skewed. Whatever the shape of the distribution of X, the distribution of z will have the same shape. Examine the Figure 1 in the next page for the graph of the data from Table 1 in both raw score form and as z scores.

Standard Normal Distribution

Normal Distributions are:

symmetric

continuous

unimodal

bell-shaped

asymtotic

the mean, median, and mode are all equal.

Normal distributions are important in statistics because of their wide range of applicability. They serve as good approximations for two types of distributions: the distributions of some variables, such as IQ and height, and the sampling distributions of some statistics, such as means. Also, normal distributions play an important role as the assumed distributions for scores used in some statistics and for errors in measurement theory.

Now, with these infinitely many normal distributions, how do we find the proportations of cases in any one distribution? To solve this problem, there would have to be infinitely many tables that display the proportations, or would there have to be some transformations which would not change the shape of the distribution but would give a known mean and variance. The answer is the transformation from raw to z scores. Since we are transforming any normal distribution to a standard distribution, the normal distribution with mean=0 and variance=1 is called the standard normal distribution. The standard normal distribution is available in table form and is the distribution used to find proportions of cases for any normal distribution.

In working with the standard normal distribution, it will be helpful to think of the proportion of cases and area under the curve as interchangeable concepts. The total area under the curve is 1, and the total of all proportions is 1, because area and proportion of cases are the same for theoretical distributions such as a normal distribution. Since a proportion is a relative number, it must be greater than or equal to 0 and less than or equal to 1. Relating proportions to area, we cannot have negative area, but we can have zero or any area up to the total of 1.

Percentiles

Although a standard score gives the position of a score relative to the mean in standard deviation units, it does not give information about the score relative to the rest of the scores in the distribution. Knowing that a raw score of 12 has z = 1.1 does not tell you what percentage of the scores is less than 12. The percentile rank of a score is the percentage of the distribution which lies below the score and gives direct information about position relative to the rest of the scores. As shown in the following table, the 50 percentile is the mean. A score at the 62 percentile is equivalent to a B or a C.

Summary

Z Scores: Computation of z scores gives the position, or relative standing, of a score with respect to the mean in terms of standard deviation units. And z scores have a mean of 0, standard deviation of 1, and the same shape distribution as the distribution of X. Use of z scores allows comparison of different raw scores for the same or different distributions.
Percentile Ranks: It gives the percentage of the population or sample which falls below that score and is used to describe relative position with respect to all other scores in the distribution.

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	Standardized	Teacher-Made
Learning Outcomes	Measurement general outcomes and content appropriate to the majority of schools in the U.S. They are tests of general skills and understanding tend not to reflect specific or unique emphases of local curricula.	Well adapted to the specific and unqiue outcomes and contents of a local curriculum; they are adaptable to various sizes of work units, but tend to neglect complex learning outcomes.
Quality of test items	Qualty of items generally is high. Items are written by specialists, pretested, and selected on the basis of the results of a quantitative item analysis.	Quality of items is often unknown. Quality is typically lower than that of standardized tests due to the limited time of the teacher.
Reliability	Reliability is high, commonly between .80 and .95, and frequently above .90.	Reliability is usually unknown, but can be high if items are carefully constructed.
Administration and scoring	Procedures are standardized; specific instructions are provided.	Uniform procedures are possible, but usually are flexible and unwritten.
Interpretation of scores	Scores can be compared to norm groups. Test manual and other guides aid interpretation and use.	Score comparisons and interpretations are limited to local class or school situation. Few, if any, guidelines are available for interpretation and use.