In order to understand test results from standardized tests it is important to be familiar with a variety of terms and concepts that are fundamental to “measurement theory” - the academic study of measurement and assessment. Two major areas in measurement theory - reliability and validity were discussed in the previous chapter; in this chapter we focus on concepts and terms associated with test scores.
Basic Concepts about Standardized MeasurementEdit
A frequency distribution is a listing of the number of students who obtained each score on a test. If 31 students take a test, and the scores range from 11 to 30 then the frequency distribution might look like Table 11-2. We also show the same set of scores on a histogram or bar graph in Figure 11-5. The horizontal (or x axis) represents the score on the test and the vertical axis (y axis) represents the number or frequency of students. Plotting a frequency distribution helps us see what scores are typical and how much variability there are in the scores. We describe more precise ways of determining typical scores and variability next.
Central tendency and VariabilityEdit
There are three common ways of measuring central tendency or which score(s) are typical. The mean is calculated by adding up all the scores and dividing by the number of scores. In the example 11-2 the mean is 24. The median is the “middle” score of the distribution – that is half of the scores are above the median and half are below. The median on the distribution is 23 because 15 scores are above 23 and 15 are below. The mode is the score that occurs most often. In Table 11-2 there are actually two modes 22 and 27 and so this distribution is described as bimodal. Calculating the mean, median and mode are important as each provides different information for teachers. The median represents the score of the “middle” students, with half scoring above and below, but does not tell us about the scores on the test that occurred most often. The mean is important for some statistical calculations but is highly influenced by a few extreme scores (called outliers) but the median is not. To illustrate this, imagine a test out of 20 points taken by 10 students, and most do very well but one student does very poorly. The scores might be 4, 18, 18, 19, 19, 19, 19, 19, 20, 20. The mean is 17.5 (170/10) but if the lowest score (4) is eliminated the mean is now is 1.5 points higher at 19 (171/9). However, in this example the median remains at 19 whether the lowest sore is included. When there are some extreme scores the median is often more useful for teachers in indicating the central tendency of the frequency distribution.
The measures of central tendency help us summarize scores that are representative, but they do tell us anything about how variable or how spread out are the scores. Figure 11-6 illustrates sets of scores from two different schools on the same test for 4th graders. Note that the mean for each is 40 but in school A the scores are much less spread out. A simple way to summarize variability is the range, which is the lowest score subtracted from the lowest score. In School A with low variability the range is (45 – 35) = 10; in the school B the range is ( 55-22 = 33. However, the range is only based on two scores in the distribution, the highest and lowest scores, and so does not represent variability in all the scores. The standard deviation is based on how much, on average, all the scores deviate from the mean. In the example in Figure 11-3 the standard deviations are 7.73 for School A and 2.01for School B. In Figure 11-7 we demonstrate how to calculate the standard deviation.
The Normal Distribution Knowing the standard deviation is particularly important when the distribution of the scores falls on a normal distribution. When a standardized test is administered to a very large number of students the distribution of scores is typically similar, with many students scoring close to the mean, and fewer scoring much higher or lower than the mean. When the distribution of scores looks like the bell shape shown Figure 11-8 it is called a normal distribution. In the diagram we did not draw in the scores of individual students as we did in Figure 11-6, because distributions typically only fall on a normal curve when there are a large number of students--too many to show individually. An ideal normal distribution is symmetric, and the mean, median and mode are all the same. Real sets of test scores only approximate the ideal normal distribution, so the mean, median, and mode usually differ at least slightly.
Normal curve distributions are very important in education and psychology because of the relationship between the mean, standard deviation, and percentiles. In all normal distributions 34% of the scores fall between the mean and one standard deviation of the mean. Intelligence tests often are constructed to have a mean of 100 and standard deviation of 15 and we illustrate that in figure 11-9. In this example, 34% of the scores are between 100 and 115 and as well, 34% of the scores lie between 85 and 100. This means that 68% of the scores are between -1 and +1 standard deviations of the mean (i.e. 85 and 115). Note than only 14 % of the scores are between +1 and +2 standard deviations of the mean and only 2% fall above +2standard deviations of the mean.
In a normal distribution a student who scores the mean value is always in the 50th percentile because the mean and median are the same. A score of +1 standard deviation above the mean (e.g. 115 in the example above) is the 84tile (50% and 34% of the scores were below 115). In Figure 11-10 we represent the percentile equivalents to the normal curve and we also show standard scores.
Kinds of Test ScoresEdit
A standard score expresses performance on a test in terms of standard deviation units above of below the mean (Linn & Miller, 2005). There are a variety of standard scores:
One type of standard score is a z-score, in which the mean is 0 and the standard deviation is 1. This means that a z-score tells us directly how many standard deviations the score is above or below the mean. For example, if a student receives a z score of 2 her score is two standard deviations above the mean or the 84th percentile. A student receiving a z score of -1.5 scored one and one half deviations below the mean. Any score from a normal distribution can be converted to a z score if the mean and standard deviation is known. The formula is
- Z score = (Score – mean score)/(Standard deviation)
So if the score is 130, the mean is 100, and the standard deviation is 15, then the formula leads to this calculation:
- Z = (130 – 100)/15 = 2
If you look at Figure 11-9, you can see that this is correct; a score of 130 is 2 standard deviations above the mean. So the z score is 2.
A T-score, by definition, has a mean of 50 and a standard deviation of 10. This means that a T score of 70 is two standard deviations above the mean and so is equivalent to a z score of 2.
Stanines (pronounced stay-nines) are often used for reporting students’ scores and is based on a standard nine point scale and with a mean of 5 and a standard deviation of 2. They are only reported as whole numbers and Figure 11-10 shows their relation to the normal curve.
Grade Equivalent ScoresEdit
A grade equivalent score provides an estimate of test performance based on grade level and months of the school year (Popham, 2005, p. 288). A grade equivalent score of 3.7 means the performance is at that expected of a 3rd grade student in the 7th month of the school year. Grade equivalents provide a continuing range of grade levels and so can be considered developmental scores. Grade equivalent scores are popular and seem easy to understand however they are typically misunderstood. If, James, 4th-grade student, takes a reading test and the grade equivalent score is 6.0 this does not mean that James can do 6th-grade work. It means only that James performed on the 4th-grade test as a 6th-grade student is expected to. Testing publishers calculate grade equivalents by giving one test to several grade levels. For example, a test designed for 4th graders would also be given to 3rd and 5th graders. The resulting raw scores are plotted, and a "trend line" is established which is used to establish the grade equivalents. Note that in Figure 11-12, the trend line extends beyond the grades levels actually tested so a grade equivalent above 5.0 or below 3.0 is based solely on the estimated trend lines.
Grade equivalent scores also assume that the subject matter that is being tested is emphasized at each grade level to the same amount and that mastery of the content accumulates at a mostly constant rate (Popham, 2005). Many testing experts warn that grade equivalent scores should be interpreted with considerable skepticism and that parents often have serous misconceptions about grade equivalent scores. Parents of high achieving students may have an inflated sense of what their child’s levels of achievement.
- Linn, R. L., & Miller, M. D. (2005). Measurement and Assessment in Teaching, 9th ed. Upper Saddle River, NJ: Pearson .
- Popham, W. J. (2005). America’s “failing” schools. How parents and teachers can cope with No Child Left Behind. New York: Routledge Falmer.
- Popham, W. J. (2005). erica’s “failing” schools. How parents and teachers can cope with No Child Left Behind. New York: Routledge Falmer.