Explaining Educational Concepts Dr. Julie Esparza Brown SPED 512/Diagnostic Assessment Portland State University
Mean (Most useful) • Mean – the average of all the scores in the distribution. • Appropriate for Equal Interval and Ratio Scales. • Not appropriate for skewed distributions.
Median (Next most useful) • Median – the middle score of a distribution. • Appropriate for Ordinal, Equal Interval, and Ratio Scales. • Most appropriate when distribution is skewed, • 50% of scores are above the median, 50% of scores are below the median. • Example • Arrange scores in order from largest to smallest (or vice versa) • If N is odd, the middle score is the median. • If N is even, the average of the two middle scores is the median.
Mode (Least useful) • The Mode is the most frequently occurring score. • Appropriate for Nominal, Ordinal, Equal Interval, and Ratio Scales. • Generally used in a very rough sense to get a feel for “the peak of the mountain.”
Fifth Graders and Push-up Test • Half of the children completed 10 or more. • Half of the children completed ten or less. • The average child completed 10. • The average or mean number completed by this class of 100 5th graders is 10. • Half of the children scored above the mean score of 10. • Half of the children scored below the mean or average score of 10. • 50 percent of the children scored 10 or above. • 50 percent of the children scored 10 or below.
Fifth Graders and Push-up Test (cont.) • One-third of the children scored between 7 and 10. • One-third of the children completed between 10 and 13. • Two-thirds of the children scored between 7 to 13. • Half of the children (50 percent) completed between 8 and 12. • The lowest scoring child completed 1. • The highest scoring child completed 19.
Y axis X axis
The highest point of the bell curve on the X axis is equal to 10 push-ups. • The next most frequently obtained scores were 9 and 11, followed by 8 and 12. • This pattern continues out towards the far ends of the bell curve with the ends occurring at 1 and 19 push-ups. • Amy’s score of 10 places her at the 50% level or Amy’s percentile rank is 50 (PR = 50).
Erik’s score of 13 places him at the 84thposition or out of the 100 fifth grade children tested or 84th %ile. • Sam’s raw score of 7 placed him at the 16th %ile. 84 children earned a higher score than Sam.
Percentiles (Relative Standing) • The percent of people in the comparison group who scored at or below the score of interest. • Example: • Billy obtained a percentile rank of 42. • This means that Billy performed as well or better than 42% of children his age on the test. • Or, 42% of children Billy’s age scored at or below Billy’s score.
Advantages of Percentiles Ranks • Percentile ranks are one of the best types of score to report to consumers of a child’s relative standing compared to other children. • Scores indicate how well a student performed compared to the performance of some reference group, • Percentile ranks are Ordinal Scale (values ordered from worst to best but differences between adjacent values are unknown) data, • It is not meaningful to calculate the mean or standard deviation of percentiles.
Converting Raw Scores • Now let’s develop a weighting system to convert each raw score to a scale score so that we can compare different scores (number of push-ups, sit-ups, seconds to complete the 50 yard dash). • One way is to develop a rank order system. • The child who scores highest in an event receives a scale score of 100; the lowest receives a score of 1. The other 98 children receive their respective “rank” as their scale score. • After all raw scores are converted to scale scores, we can easily compare an individual child to the group and to all children who are the same age or in the same grade.
Composite Scores • There are difficulties with composite scores. • For example, John has good muscular strength and scored at the 70%ile in push-ups and 78%ile in sit-ups. • But, he is slow and uncoordinated and finished 2nd from last out of the 100 children or at the 2%ile. • If we average John’s scores they will average 50 (average score); however, he was not “average” in all events. • NOTE: You cannot average percentile rank scores (WHY?) – you can average standard scores or scale/subtest scores. • Moral of this story: make sure the subtest scores that create a cluster or composite score will not mislead us into believing there are no weaknesses present. • Cluster scores must be considered with caution when there is a significant difference between individual subtest scores.
Standard Deviation • Percentile ranks are computed by determining the mean score and amount of variation of all scores around the mean score. • Are the scores bunched around the number 10 in a tight uniform distribution? • Are the scores evenly distributed? • Do they peak and taper slowly? • Do they bunch at the ends with few or no scores in the middle? • Is there great variance, with the scores spread over a wide range, with two or more peaks? • Is there a normal bell curve distribution of scores?
Standard Deviation • On our push-up test, most of the 5th graders scored around 10 push-ups, with an even distribution above and below 10 push-ups. • If one-half of the children completed 5 push-ups, one-fourth completed 14 push-ups, and one-fourth completed 16, the average or mean would still be 10 – half the children scored above 10 and half below 10. • A low SD means the data points are close together. • A high SD means the data points are spread out.
SD • The standard deviation measures how much on average individual scores of a given group vary from the average (mean) for this same group. • The SD measures the spread of individual results around a mean of all results. • Let’s take a class of 40 people taking an exam. • Once it’s graded, the instructor calculates the mean. • To determine the SD, we split the total dataset, which is 100 points, into smaller, even values. • It is up to the researcher how to split it, so for example, we’ll have 10 value units, from 10 to 100.
SD • The mean score is 50. • 16 students scored between 40 and 60 which means they scored within 10 points, either higher or lower, of the average score. • This means that 40% of the entire class (16 divided by 40 and multiplied by 100) scored within one value unit of the mean score. • Another 12 students scored between 30 and 70, which means they scored within 20 points, either higher or lower, of the average score. • These students account for 30% of the class (12 divided by 40 and multiplied by 100). • Together with these students who scored within 10 points of the average score, they make up a group of 28 students or 70% of the entire class – who scored within two value units of the mean score. • We know that approximately 68% of scores in any group fall within one SD from the mean. • Based on this, 20 points is the approximate value of one SD • We know that approximately 95% of scores fall within two SDs from the mean. • What’s the range for 2 SD in this example?
Standard Scores • The average score or mean is 100. • The standard deviation is 15. • If a child had a standard score of 68, or 2.5 SDs below the mean on a writing sample, this means they scored below the 1%ile. • You can convert standard scores into percentile ranks.
Educational and Psychological Tests • These tests are designed to present normal bell curve distributions with predictable patterns of scores. • We need to know the mean and standard deviation of the test. • In most educational and psychological tests, the mean is 100 and the standard deviation is 15. • On some tests, the mean is 10 and the standard deviation is 3. • Average scores do not deviate far from the mean. • When a score falls significantly above or below the mean, it is referred to as being a distance from the mean, e.g., 1 or 2 standard deviations from the mean. • To interpret test scores, you need to know the mean and standard deviation.
Educational and Psychological Tests • One standard deviation above the mean always falls at the 84%ile. • One standard deviation below the mean always falls at the 16%ile. • Two SD’s above the mean is always at the 98%ile. • Two SD’s below the mean is always at the 2%ile.
Understanding Test Data • Sometimes, test scores are reported differently. • For example, test scores may be reported as “z scores.” • Z scores have a mean of 0 (zero) and a standard deviation of 1. • If you know a child earned a z score of -1, you know that the child scored at one deviation below the mean. • One SD below the mean is at the 16%ile. • If you convert this score into the standard score format, with a mean of 100 and a standard deviation of 15, a z score of -1 is the same as a standard score of 85.
Standard Scores (Relative Standing) Standard scores are scores of relative standing with a set, fixed, predetermined mean and standard deviation.
Understanding Test Data • Other tests report results as T scores. • T scores have a mean of 50 and an SD of 10. • A T score of 60 is the same as a Z score of +1. A child who has a T score of 60 or a Z score of +1 scored at the 84%ile. A T score of 70 is the same as a Z score of +2, a standard score of 130 and a percentile rank of 98. • A few tests report results in Stanines. In Stanine tests, the mean is 5 the SD is 2.
WISC-IV Scores • What do these mean?
Age & Grade Equivalents (Developmental Scale) • There are problems with using these scores • Identical age equivalents can mean different task performance.
Problems with Grade and Age Equivalent Scores • Systematic misinterpretation: students who earn an AE of 12.0 has answered as many questions as the average for children of 12. They have not necessarily performed as a 12 year old could. • Implication of a false standard of performance: equivalent scores are constructed so that 50% of any age or grade group will perform below or above age or grade level. • Tendency for scales to be ordinal, not equal interval: age and grade equivalent scores are ordinal, not equal interval: they should not be added or multiplied. Source: Salvia, Ysseldyke & Bolt (2009)
Maria got an age equivalent of 2-0 on a test means: Maria obtained the same number correct as the estimated mean of children 2 years and 0 months of age, It does NOT mean: Maria performed like an average 2 year old on the test. Age & Grade Equivalents (Developmental Scale)
John got a grade equivalent of 3.5 on a test means: John obtained the same number correct as the estimated mean of children 5th month of 3rd grade. It does NOT mean: John is able to do 3.5 grade level work. Age & Grade Equivalents (Developmental Scale) Bottom Line – Do not use grade or grade level scores.
The Reliability Coefficient • An index of the extent to which observations can be generalized; the square of the correlation between obtained scores and true scores on a measure. • The proportion of variability in a set of scores that reflects true differences among individuals. • If there is relatively little error, the ratio of true-score variance to obtained-score variance approaches a reliability index of 1.0 (perfect reliability) • If there is a relatively large amount of error, the ratio of true-score variance to obtained-score variance approaches .00 (total unreliability). • We want to use the most reliable tests available. • The greater the number of items, the higher the reliability coefficient. • The greater range of test scores, the higher reliability. • Moderate level of difficulty increases test reliability.
Standards for Reliability • If test scores are to be used for administrative purposes and are reported for groups of individuals, a reliability of .60 should be the minimum. The relatively low standard is acceptable because group means are not affected by a test’s lack of reliability. • If weekly (or more frequent) testing is used to monitor pupil progress, a reliability of .70 should be the minimum. This relatively low standard is acceptable because random fluctuations can be taken into account when a behavior or skill is measured often.
Standards for Reliability • If the decision being made is a screening decision, there is still a need for higher reliability. For screening devices, a standard of .80 is recommended. • If a test score is to be used to make an important decision concerning an individual student (such as special education placement), the minimum standard should be .90.
Standard Error of Measurement • SEM is another index of test error. • It is the average standard deviation of error distributed around a person’s true score. • The difference between a student’s actual score and their highest or lowest hypothetical score. • We generally assess a student once on a norm-referenced test so we do not know the test taker’s true score or the variance of the measurement error that forms the distribution around that person’s true score. • We estimate the error distribution by calculating the SEM. • The general formula SEM equals the standard deviation of the obtained scores, multiplied by the square root of 1 minus the reliability coefficient. • When the SEM is relatively large, the uncertainty that the student’s true score will fall within the range is large; when the SEM is relatively small, the uncertainty is small.
Confidence Interval • The range of scores within which a person’s true score will fall with a given probability. • Since we can never know a person’s true score, we can estimate the likelihood that a person’s true score will be found within a specified range of scores called the confidence interval. • Confidence intervals have two components: • Score range • Level of confidence
Confidence Interval • Score range: the range within which a true score is likely to be found • A range of 80 – 90 tells us that a person’s true score is likely to be within that range • Level of confidence: tells us how certain we can be that the true score will be contained within the interval • If a 90% confidence interval for an IQ is 106 – 112, we can be 90% sure that the true score will be contained within that interval. • It also means that there is a 5% chance the true score is higher than 112 and a 5% chance the true score is lower than 106. • To have greater confidence would require a wider confidence interval. • You will have a choice of confidence intervals on Compuscore. You can choose the 90 percent option but the default is set at 68%.
Validity • “The degree to which evidence and theory support the interpretation of test scores entailed by proposeed uses of tests” (APA Standards, 1999, p. 9) • Validity is the most fundamental consideration in evaluating and using tests.
Validity • “A test that leads to valid inferences in general or about most students may not yield valid inferences about a specific student…First, unless a student has been systematically acculturated in the values, behavior, and knowledge found in the public culture of the United States, a test that assumes such cultural information is unlikely to lead to appropriate inferences about that student…
Validity • Second, unless a student has been systematically instructed in the content of an achievement test, a test assuming such academic instruction is unlikely to lead to appropriate inferences about the student’s ability to profit from instruction. It would be inappropriate to administer a standardized test of written language (which counts misspelled words as errors) to a student who has been encouraged to use inventive spelling and reinforced for doing so. It is unlikely that the test results would lead to correct inferences about that student’s ability to provide from systematic instruction in spelling” (Salvia, Ysseldyke, & Bolt, 2009, p. 63.)
Types of Validity • Content validity • Criterion-related validity • Construct validity
Content Validity • A measure of the extent to which a test is an adequate measure of the content it is designed to cover; content validity is established by examining three factors: • Appropriate of type of items included • Comprehensiveness of item sample • The way in which the ietms assess the content • It is assessed by an overview of the items by trained individuals who make judgments about the relevancy of the items and the unambiguity of their formulation. • This is especially important in achievement testing and one under debate. • There is an emerging consensus that the methods used to assess student knowledge should closely parallel those used in instruction.
Criterion-related Validity • The extent to which performance on a test predicts performance in a real-life situation. • Usually expressed as a correlation coefficient called a validity coefficient. • Two types of criterion-related validity: • Concurrent validity • Predictive validity
Concurrent Validity • A measure of how accurately a person’s current test score can be used to estimate a score on a criterion measure. • We look to see if the test presents evidence of content validity and elicits test scores corresponding closely (correlating significantly) to judgments and scores from other achievement tests that are presumed to be valid, we can conclude that there is evidence for a test’s criterion-related validity.
Predictive Criterion-related Validity • A measure of the extent to which a person’s current test scores can be used to estimate accurately what that person’s criterion scores will be at a later time. • Concurrent and predictive validity differ in the time at which scores on the criterion measure are obtained. • If we are developing a test to assess reading readiness, we can ask: Does knowledge of a student’s score on the reading test allow an accurate estimation of the student’s actual readiness for instruction? How do we know that our test really assesses reading readiness? • The first step is to find a valid criterion measure and if an assessment has content validity and corresponds to another measure, we can conclude the test is valid.
Construct Validity • The extent to which a procedure or test measures a theoretical trait or characteristics. • Especially important for measures of process such as intelligence/cognition. • To provide evidence of construct validity, an author must rely on indirect evidence and inference. • To gauge construct validity a test develop accumulates evidence that the test acts in the way it would if it were a valid measure of a construct. • As the research evidence accumulates, the developer can make a stronger claim to construct validity.
The Bottom Line… “Test users are expected to ensure that the test is appropriate for the specific students being assessed.” Salvia, Ysseldyke & Bolt, 2009, p. 71