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Understanding Tests and Measurements for the Parent and Advocate

Part 1

Peter Wright and Pamela Darr Wright

(article reprinted with permission of the author)

“If something exists, it exists in some amount. If it exists in some amount, then it is capable of being measured.” Rene Descartes, Principles of Philosophy, 1644.


Most parents of special needs children know that they must understand the law and their rights. Few parents know that they must also understand the facts. The “facts” of their child’s case are contained in the various tests and evaluations that have been administered to the child. Changes in test scores over time provide the means to assess educational benefit or regression. Most important educational decisions, from eligibility to the intensity of educational services provided, are based on the results of psychological and educational achievement testing. Parents who obtain appropriate special educational programs for their children have learned what different tests measure and what the test results mean.

As an attorney who specializes in representing special education children, I know that many parents consult with me after deciding that their child’s special education program is not appropriate. These parents are often right. However, in most cases they do not have the evidence to support their belief, nor do they know how to interpret and use the evidence contained in educational and psychological tests. They need evidence to support their beliefs.

Often these parents are convinced that a special education program is not providing sufficient help for the child — that under the present special education program, the child is failing to make progress and has fallen further behind. These parents experience a sense of urgency — the child has usually received special education for several years and time is running out.

Critical educational decisions are often made, based on the subjective beliefs of parents and educators. As a parent, you may believe that your child is not making adequate progress in a special education program. The special education staff may firmly believe that he is doing as well as he can — or that your expectations are too high. Without objective information, both sides will take positions that are based upon emotions — and tempered by hopes and fears. Effective educational decision-making must be based on objective information and facts, not subjective emotional reactions and beliefs.

Before you can participate in the development of an appropriate special education program, you must have a thorough understanding of your child’s strengths and weaknesses. This information is contained in the various tests that are used to measure the child’s ability and educational achievement.

To successfully advocate for your child, you must also learn about tests and measurements — statistics. Statistics are ways of measuring progress or lack of progress, using numbers. After you analyze the scores your child obtains when tested and understand what these numbers mean, you will be able to develop an appropriate educational program for your child — a program from which the child benefits.

As you master the material contained in this article, you will understand what various tests and evaluations measure and how to use information from tests to measure academic progress. You will learn how to use graphs to visually demonstrate your child’s progress or lack of educational progress in a very powerful and compelling manner.

The United States Supreme Court
Florence County School District Four v. Shannon Carter
November 9, 1993

In Florence County School District Four v. Shannon Carter, 510 U. S.7, 114 S. Ct. 361, (1993), the United States Supreme Court issued a landmark decision. In Carter, the school system defaulted on their obligation to provide a free appropriate education to Shannon Carter, a child with learning disabilities and an Attention Deficit Disorder. Let’s look at how the courts viewed the facts and the law in the Carter case.


When Shannon was in the seventh grade, her parents talked to the public school staff and expressed concerns about Shannon’s reading and academic problems. She was evaluated by a public school psychologist who described Shannon as a “slow learner” who was lazy, unmotivated and needed to be pressured to try harder. Her parents pressured her to work harder. Despite the intense pressure, when Shannon was in the ninth grade, she failed several subjects. Her parents had her evaluated by a child psychologist. That evaluator determined that Shannon’s intellectual ability was actually above average. Educational achievement testing demonstrated that sixteen year old Shannon was reading at the fifth grade level (5.4 GE) and doing math at the sixth grade level (6.4 G.E.). Shannon had dyslexia. As she prepared to enter tenth grade, she was also functionally illiterate.

In Shannon’s case, the school district developed an IEP which proposed that after a year of special education in the tenth grade, Shannon would read at the 5.8 grade equivalent level and perform math at the 6.8 grade equivalent level. In other words, after one year of special education designed to remediate her learning disabilities, Shannon was expected to gain only four tenths of a year, as measured by her scores on the Woodcock-Johnson and KeyMath educational achievement tests, a gain from 5.4 to 5.8 and 6.4 to 6.8 grade levels in reading and math respectively.

Shannon’s parents insisted that their daughter required a more intensive program so that she could master necessary reading, writing and math skills. They felt that the proposed program was inadequate, and worried that Shannon would still be functionally illiterate when she graduated in three years. Emory Carter insisted that his daughter should be able to read, write and do arithmetic at a twelfth grade level when she graduated from high school.

Although Emory and Elaine Carter shared their concerns and wishes with the public school officials, the administrators took a “take it or leave it” position and refused to provide Shannon with a more intensive special education program that provided actual remediation in reading, writing, and arithmetic. Seeking more services for their daughter, the parents requested a special education due process hearing. The Hearing Officer ruled that the public school IEP was appropriate. The parents appealed this decision to a Review Panel and lost again.

At that point, Emory and Elaine Carter withdrew Shannon from her local public high school and enrolled her in Trident Academy. Trident is a private school in Mt. Pleasant, South Carolina that specializes in remediating children with learning disabilities, including dyslexia. Shannon’s parents then appealed the Review decision to the U. S. District Court. They asked Judge Houck to award them reimbursement for Shannon’s private school education at Trident.

When Shannon graduated from Trident Academy three years later, her reading and math scores were on a high school level.

After hearing testimony and reviewing the transcripts and documents from the Due Process and Review Hearings, U. S. District Court Judge Houck found that the school district’s IEP was “wholly inadequate” to meet Shannon’s needs. He ruled that Shannon had received an appropriate education at Trident and ordered Florence County to reimburse Shannon’s parents for the costs of her education.

On what basis did Judge Houck decide that the IEP proposed by Florence County was inappropriate? What evidence caused him to decide that Shannon received an appropriate education at Trident Academy?

Evidence and Law

The decisions in Shannon’s case, and in many special education cases, rest on the evidence provided by various tests and evaluations of the individual child. When Judge Houck wrote that the Florence County’s IEP was “wholly inadequate” to meet Shannon’s needs, he was relying on his interpretation of the results of testing. Judge Houck understood the importance of accurately interpreting test scores. He charted Shannon’s test scores and included this data as part of his U. S. District Court decision. (See also Hall v. Vance, 555 EHLR 437, (E.D. NC 1983), affirmed at 774 F. 2d 629, 557 EHLR 155, (4th Cir. 1985)) in which U. S. District Court Judge Dupree charted out James Hall’s test scores to support his 1983 decision that Vance County, North Carolina did not provide James with an appropriate education in the public school program.) When you finish this article, you will also be able to interpret and chart your child’s test scores and measure educational progress or lack of progress.

Florence County appealed Judge Houck’s decision to the U. S. Circuit Court of Appeals for the Fourth Circuit. Appeals from the U. S. District Courts in Maryland, Virginia, West Virginia, North Carolina and South Carolina are heard in the U. S. Court of Appeals for the Fourth Circuit by a three judge panel. The Fourth Circuit is composed of thirteen judges. Appeals from U. S. Circuit Courts of Appeals are filed in the U. S. Supreme Court. Occasionally a U. S. Circuit Court of Appeals will convene all Judges appointed to the Circuit to hear a case. This is called an en banc review.

A three judge panel of the Fourth Circuit affirmed Judge Houck’s decision as to the inadequacy of Florence County’s proposed IEP. Florence County then appealed to the United States Supreme Court.

On November 9, 1993, the United States Supreme Court issued a unanimous decision on Shannon’s behalf. In the Carter decision, authored by Justice Sandra Day O’Connor, the Court upheld the lower decisions, ruled against Florence County School District Four, and ordered them to reimburse Shannon’s parents for the costs of her tuition, room and board, and attorney’s fees.

Objective Measurement of Progress

IEPs must include objective means of measuring the child’s progress in a special education program. Volume 34 of the Code of Federal Regulations, Section 300.347, “Content of individualized education program,” states that an IEP must include:

(2) A statement of measurable annual goals, including benchmarks or short-term objectives, related to (i) Meeting the child’s needs that result from the child’s disability to enable the child to be involved in and progress in the general curriculum . . .


(7) A statement of (i) How the child’s progress toward the annual goals described in paragraph (a)(2) of this section will be measured; . . .

In Shannon’s case, her IEP stated that she “will be able to improve total reading level from the 5.4 grade level to the 5.8 grade level as measured by the Woodcock Reading Mastery Test . . . (and that she) will improve math skills from the 6.4 grade equivalent to the 6.8 grade equivalent as measured by the Key Math Diagnostic Test.” This IEP complied with regulation in existence at that time, (34 C.F.R. § 300.346, since modified), by including “appropriate objective criteria.” The criteria required a re-administration of the Woodcock-Johnson and KeyMath tests to measure progress.

The U. S. District Court and the Fourth Circuit found that the proposed gain of four months after a year of special education was “wholly inadequate.”

In an effort to avoid Florence County’s fate, many school districts around the country now develop IEPs that include no objective measures of the child’s progress. Instead of including educational goals where the child’s progress is measured using objective tests and measurements, as Florence County did with Shannon, many schools now propose IEPs that rely exclusively on subjective teacher observations of the child’s progress. Let’s see how this works.

We’ll look at Johnny, a child who has a learning disability that is manifested in the area of reading. Johnny is below grade level in reading. Instead of developing an IEP that will measure progress in reading on a specific objective test, the special education staff may come up with a goal such as: “Johnny will make measurable progress in reading, as measured by teacher observation and teacher made tests at 80% accuracy.”

“Objective measurement of progress” becomes the teacher’s subjective observation as to whether the child has improved in reading, writing, or arithmetic. The criteria of mastery becomes 80% of a subjective opinion. When parents object and ask for a more intense program with clear independent objective standards, they are often rebuffed or criticized.

Many school board counsel and state departments of education have advised schools to move away from using objective measurements of progress for special education children.

If you believe that the special education your child is receiving is inadequate, you must have evidence to support your position. You will find this evidence in the public school and private sector testing that has been or will be completed on your child.

After you master the material contained in this article, you will understand what the various tests and evaluations measure and how the test results are reported. You will know how to convert the scores on different tests into numbers that are easily understood. And, you will know how to measure educational progress or lack of progress, i.e. regression.


Three years ago, your eight year old son Mike began to have serious difficulties in school. By the time he reached third grade, his difficulty in reading was of great concern. His handwriting was nearly illegible and homework was a nightmare. On several occasions, you consulted with Mike’s teacher about the problems he was having. Eventually, the teacher sent Mike’s “case” to a special education committee. You attended a meeting of this committee — which recommended that Mike be evaluated through the school’s special education department. Relieved that something was going to be done, you consented to these battery of tests.

According to the evaluations, your son has a learning disability. In Mike’s case, he has visual-perceptual problems and visual-motor problems that negatively affect his ability to read and write. Based on the results of the evaluations, your son was found eligible for special education services through his neighborhood school.

After Mike was found eligible for special education, you attended a meeting to develop his Individualized Education Program (IEP). This IEP provided for Mike to receive one period of special education in an “LD Resource” class every day. It was your understanding that Mike would receive individualized help in reading and writing from a teacher who was specially trained to remediate his learning disability problems.

Three years have passed. Mike hasn’t made much progress, despite the special education help. He still has difficulty reading aloud. His spelling is poor, and his handwriting is unreadable. He is behind most of the children in his class. His attitude has changed. He is angry and depressed and says he “hates school.”

When you discussed your concerns about Mike’s lack of progress with his special education teacher, she reassured you that he was making progress and told you to be patient. You think that patience is not the issue; you are worried that your son will never master basic educational skills. What kind of future will he have?

At a recent IEP meeting, you reiterated your concerns about Mike’s lack of progress and expressed the belief that he needs more help than he is getting in the Resource program. The committee disagreed with you. One person told you that Mike was getting all the help he needs and that he was really doing quite well. Another committee member told you that your expectations were too high — and that if you didn’t accept Mike’s limitations, you would damage him emotionally.

What should you do? You know that the time in the LD resource class with several other children is not providing Mike with the individualized help he needs. The school has not focused on teaching your son how to read, write and do arithmetic. Now, the IEP team suggests more “accommodations” and “modifications.” They propose to reduce his workload, give him untimed tests, and provide him with “talking books” and a calculator. They do not propose to give him individualized help so that he will learn to read, write, and do arithmetic.

You believe that Mike’s emerging “emotional problems” are due to shame and embarrassment about not being successful in school. How can you, a parent, prove this to the staff at Mike’s school so that they will develop an appropriate educational program for him? How will you know when he is getting the help he needs?

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The Process of Educational Decision-Making

Many parents erroneously assume that interpreting test data is beyond their competence and is the responsibility of the school personnel. If parents default on their responsibility and obligation to understand this information, then the interpretation of the test data is left to the school psychologist — a person who often has very limited information about your child, aside from test scores.

The basic principles of tests and measurements are not difficult to master. As you read this article, you will see that you are already familiar with many of the concepts discussed. Statistics and statistical terms are used in many other areas of life, from business and sports to medicine. Newspaper and magazine articles use statistics to inform readers of change or lack of change. You read articles about changes in the population, the economy — even public opinion polls — that include statistical information to inform you or persuade you of a point.

Parents need to expend time and effort to develop an adequate degree of expertise in statistics. You should reread parts of this article several times. Underline, make margin notes, and use a highlighter to help you master the material. Be patient and put in the time. The time you expend will help to change your child’s life.

As you study this material, you will probably encounter some terms and concepts that seem confusing at first — terms like standard deviation, standard score, and grade and age equivalents. Other concepts will be familiar — averages, percentiles.

After you master this information, you will understand the educational and psychological tests that are administered to your child. You will be able to use this information to make wise educational decisions. You will find that your newfound knowledge and expertise exceeds that of many of the special education committee members.

When you attend your next IEP or Eligibility meeting, you will be glad you did your homework!


Katie is a fourteen year old ninth grader. She “hates school” and is failing several subjects. As a young child, Katie was bright, happy, and curious. When she entered third grade, her attitude began to change. Now, she locks herself in her room, lies on her bed, and listens to music for hours. She is sullen and angry and says she can’t wait to quit school.

In desperation, Katie’s parents took her to a child psychologist for testing. At a meeting to interpret the test results to Katie and her parents, the psychologist explained that Katie scored two “standard deviations” above the mean on the Similarities subtest of the Wechsler Intelligence Test for Children, Third Edition (WISC-III) and two and a half “standard deviations” below the mean on the spontaneous writing sample of the Test of Written Language, Third Edition (TOWL-III).

Test publishers are constantly updating and revising their tests. The Wechsler Intelligence test for children was originally known as the WISC. Later, it was revised and became known as the WISC-R. Several years ago, the next version was published as theWISC-III. The first Test of Written Language (TOWL) was replaced by the TOWL-II and was recently revised again.

The Woodcock Johnson battery of tests was known as the Woodcock Johnson Psycho-Educational Battery. The WJPEB included educational achievement testing and cognitive ability testing. Dr. Woodcock also produced the Woodcock Reading Mastery Test. Today, the current test series is called the Woodcock-Johnson Psycho-Educational Battery, Revised, (WJ-R) which is an educational achievement test that includes the Test of Cognitive Abilities.

The current version of any popular test is probably in a revision status. A competitor test publishing company is probably trying to develop a new and better version of the competitor’s product. This article will not focus on an analysis of each test’s strengths and weaknesses. Weaknesses in a current test will probably be eliminated by the next version which will be out within a couple of years.

Parents must understand that tests do not necessarily measure what they purport to measure. As you will see, a child’s score on a push-up test can be represented as an overall fitness score, a measure of arm strength, an upper body measurement score, a measure of perseveration and persistence, or a measure of a child’s motivation. A score may measure only one of the variables or it may accurately reflect all of the above.

To demonstrate this point, let’s look at tests that measure reading ability. One test that measures a child’s reading ability actually measures the child’s ability to correctly read aloud and pronounce isolated words out of context, i.e., a word recognition test. The test includes a list of words, i.e., cat, tree, dog, house, person, etc. This kind of reading test does not measure true reading and may be adversely impacted by speech or word finding problems.

Another reading test measures reading by having the child read a passage of text, then answer a series of multiple choice questions about the passage. In this case, the child’s score may be a measure of the child’s ability to intellectually eliminate certain answers of the multiple choice format, i.e., a test of reasoning, not true reading. Some very bright children may need to recognize and interpret only a few words to discern the total context. Other children have excellent word recognition abilities but cannot link or interpret the words in a body of text or passage. Another reading test has the child read a passage of text aloud (measuring oral reading) and then answer questions. The accuracy of the words read aloud and the child’s understanding of the passage makes up the reading score.

You need to know exactly how the test was administered and what it measured.

When we first discussed Katie, we saw that she scored two “standard deviations” above the mean on the Similarities subtest of the Wechsler Intelligence Test for Children, Third Edition (WISC-III) and two and a half “standard deviations” below the mean on the spontaneous writing sample of the Test of Written Language, Third Edition (TOWL-III).

Do these test scores explain the academic problems Katie is having? Do they have anything to do with her moodiness and her intense dislike of school? (Answers: Yes and Yes.) When we return to Katie’s case later in this article, you will understand the significance of her test scores. You will also understand why Katie’s self esteem has plummeted.

Remember: After you master the material contained in this article, you will understand and be able to interpret your child’s test scores. You will be able to go back to the preceding paragraph and understand the significance of Katie’s scores. You will have acquired skills that will enable you to answer questions like these:

  • How is your child functioning, compared with other children the same age ?
  • How is your child functioning, compared with others in the same grade?
  • How much educational progress has your child made (what has been learned) since the last test battery?
  • If your child is receiving special education, has the child progressed or regressed in the special education placement?
  • If your child has shown an increase in age and grade equivalent test scores, has the child actually fallen further behind the peer group?

And, you will learn how to incorporate objective measurements into your child’s IEP so that educational progress can be charted on a regular basis.

Measuring Change: Rulers, Yardsticks and Other Tools

To clarify these points, let’s change the facts. You can measure your child’s physical growth with a measuring tape and a bathroom scale. You can measure growth by charting how much height increases, as measured in inches, and how much weight increases, as measured by pounds, over a period of months or years. Using these tools, you can document his physical growth. You don’t need to be a doctor to understand that increases in these measurements prove that your child is growing.

Assume that your child’s height was five feet, three inches last year. This year, the child is five feet, six inches tall. You can report this information in several ways. You can say that last year, your child was sixty-three inches tall and is now sixty-six inches tall. Or, you can say that your child was 5.25 feet tall and is now five and a half feet tall. You can even say that a year ago, your child was 160 centimeters tall and is now 168 centimeters tall. Or, that your child was 1.75 yards tall and is now 1.83 yards tall!

If you (or your child’s pediatrician) have been measuring your child at regular intervals, you can create a chart or graph that documents changes in height or weight over time. Your child’s pediatrician has “growth charts” that you can use to compare your child’s growth with the growth of the “average” child.

Likewise, educational growth can be measured and charted. The yardsticks used for measurement are different, but the principles are the same. Measuring educational growth or progress is not much different from charting physical growth. Instead of a tape measure and a set of bathroom scales, you need psychological and educational achievement test results. Where will you find the information you need? How can you measure change?

Most school districts test their students on standardized group educational achievement tests at regular intervals. The results of these tests provide information about how well school districts are accomplishing their mission of educating children. The information contained in the group standardized tests can provide you with some basic information.

Standardized educational achievement tests are general measures. The information they provide is similar to that provided by medical screening tests. Medical screening tests can suggest that a problem exists. Additional testing is usually necessary before the problem can be accurately identified and a treatment plan developed. Children’s learning problems can be identified in a similar manner. In most public schools, specific individual ability and achievement tests to clarify learning problems are administered by school psychologists and educational diagnosticians.

What Do Evaluations Tell You?

As you continue on your advocacy journey, you must understand the exact nature of your child’s disabling condition(s). How does the disability affect her? In what areas? How serious is it? What are her strengths and weaknesses? Does she need special education? What educational issues need to be addressed? How will you know if she is making progress? How much progress is sufficient? The answers to these questions will be found in the evaluations and tests that are administered to children and adolescents.

Many parents erroneously believe that they cannot understand the tests. They believe that this information is beyond their ability to understand or comprehend. Usually, their reasoning goes like this:

Gosh. I’m just a parent. I didn’t even finish college. I don’t have any training
in education or special education so I can’t understand that stuff!


The people who did that testing on my kid went to school for years to learn how to do that. Who am I to think I can understand it? I’m not a psychologist!

If you believe that you “can’t” understand your child’s testing, it’s time to change your beliefs. You may be reading this article because your son or daughter is performing poorly in school — or has been identified with learning problems — and now believes that he or she “can’t” read or write or do arithmetic. Your child must confront and overcome these erroneous beliefs about learning new or difficult material. And, so must you.

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Statistics: General Principles

Statistics are simply ways to measure things and to describe relationships between things, using numbers. Part of the confusion that many people experience when they first begin to learn statistics is because of the unfamiliar terms and concepts. As we learned in our earlier discussion about measuring physical growth, there are several different ways to report the same information (inches, feet, yards, centimeters, etc.) In the beginning, this can be confusing.

First, let’s look at another familiar example that many of us deal with regularly — how to measure our car’s gas mileage. Remember: When using statistics, we can use several different terms to describe the same concepts. If you want to describe your car’s gas mileage, you can make any of the following statements:

  • My gas tank is half full.
  • My gas tank is half empty.
  • I am at the fifty percent mark.
  • My odometer shows that I have another 150 miles before the next fill-up.
  • My odometer shows that I have traveled 150 miles since I last filled the tank.

All of these statements accurately describe your car’s consumption of gas.

With this information, you can make decisions. When will you need to buy more gas? You know that your car has a fifteen gallon gas tank. According to the gas gauge, your tank is slightly below the halfway mark. You’ve been driving in the city. You’ll be driving on the highway for the rest of your trip. You have used a precise amount of gas and have a precise amount of gas left in your tank. You can describe and define this information in several ways — gallons used, gallons remaining, miles driven, miles to go, percentage full, and so forth. Using the information above, you can do some simple math calculations and learn that your car averages between seventeen to twenty-three miles to a gallon of gas, depending on driving conditions.

Using this information or data, you can also measure change. If you compare your car’s present or current mileage to the mileage you obtained last month, before you had your car tuned up, you can measure miles per gallon before and after the tune-up. In this way, you can measure the impact of the tune-up on your car’s gas consumption. You can also compare your car’s mileage performance to that of other vehicles.

Let’s look at another common way in which we use tests and measurements. When you last visited your doctor, you mentioned that you were feeling tired and sluggish. Your doctor asked several questions, then recommended that you have some lab work. After reviewing the test results, the doctor explained that your blood glucose level was moderately elevated.

To lower your blood glucose level, the doctor recommended a plan of treatment that included a special diet and a daily program of moderate exercise. After a month, you return for a follow-up visit. More lab work is completed. If your glucose level has returned to normal, it is unlikely that you will require additional treatment. But, if your glucose level remains high, despite the diet and exercise program, you may need more intensive treatment. By measuring change after an intervention and using “appropriate objective criteria and evaluation procedures,” you and your doctor can make rational decisions about your medical treatment.

Remember: The principles that enable you to compute your car’s gas mileage and make medical decisions will also enable you to understand educational change. When you measure educational progress (just as when you measure your gas mileage and blood levels), the test scores can be reported and compared in several different ways.

Because educational test scores are often reported in different formats and compared in different ways, it is essential for parents and advocates to understand all of the scoring methods used in measuring and evaluating educational progress, including:

  • age equivalent scores (AE)
  • grade equivalent scores (GE)
  • standard scores (SS) and standard deviations (SD)
  • and percentile ranks (PR).

Knowledge about statistics will enable you to assess your child’s progress or lack of progress in a particular educational program. Lack of progress is usually referred to as regression. Unfortunately, regression is a common educational problem that we will discuss in more detail later. You must learn how to recognize regression and reverse the downward spiral before your child is further damaged.

Statistics: Applied

Let’s turn our attention to the performance of a group of children. You must understand how an individual child scores when compared with other children who are his age or in his grade — and what this means.

First, we will examine a single component of physical fitness in a group of elementary school students. Our group or sample consists of 100 fifth grade students. These children are enrolled in a physical fitness class to prepare them to take the President’s Physical Fitness Challenge. We will assume that the average chronological age (CA) of these children is exactly ten years, zero months. (CA=10-0) The children are tested in September, at the beginning of the school year.

To qualify as “physically fit,” each child must meet several goals. Push-ups are one measure of upper body strength. Each child must complete as many push-ups as possible in a period of time. Each child’s raw score is the number of push-ups completed. The term raw score is simply another way of describing the number of items correctly answered or performed.

After all of the fifth grade students complete the push-up test, their scores are listed. The results are as follows:

  • Half of the children completed ten push-ups or more.
  • Half of the children completed ten push-ups or less.
  • The average child completed 10 push-ups.
  • The average or mean number of push-ups completed by this class of 100 fifth grade students 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.

As we continue to analyze the children’s scores, we see patterns:

  • One-third of the children scored between 7 and 10 push-ups.
  • One-third of the class completed between 10 and 13 push-ups.
  • Two-thirds of the children scored between 7 to 13 push-ups.
  • Half of the children (50 percent) completed between 8 and 12 push-ups.
  • The lowest scoring child completed 1 push-up.
  • The highest scoring child completed 19 push-ups.

Again, two-thirds of the children in this fifth grade class were able to complete between 7 and 13 push-ups. The remaining third of the children did fewer than 7 or more than 13 push-ups. Nearly all of the children — 98 out of 100 — were able to complete between 4 and 16 push-ups. This information is represented below in a bell curve chart.

Chart showing distribution of number of push-ups completed by children

The test results provide us with a sample of data. As we analyze the data in our sample, we can compare the performance of any individual child with that of the entire group. As we make these comparisons, the data will enable us to recognize any individual child’s strengths and weaknesses when compared with the peer group of similar youngsters.

If we conduct an identical push-up test with children in other grades, we can compare our original group of 100 fifth grade children with other groups of youngsters — children who are older, younger, in different grades, in different schools. If we gather enough information or data from other sources, we can compare our original group of fifth graders — or an individual child within our group — to a national population of children who are being tested for their upper body strength as measured by their ability to do push-ups.

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Measuring Progress: The Bell Curve

In nature, traits and characteristics distribute themselves along theoretical curves. For our purposes, the most important curve is called the normal frequency distribution or bell curve. Because the percentages of areas along the bell curve are well-known and thoroughly researched, they become our frame of reference.

By using the bell curve, we can now develop an actual diagram or graph of the children’s push-up scores. This map — on the bell curve — provides us with additional information. We can see what percentages of children were able to complete specific numbers of push-ups. When we use the bell curve, we can visually demonstrate where any particular child scores, when compared with other children who are the same age or in the same grade. Likewise, with educational test scores, we can visually demonstrate scores and change over time.

If we compare the push-up scores obtained by children who attend different schools, we can determine whether the physical fitness of children, as measured by their ability to do push-ups, varies in different schools, neighborhoods, states, or countries. We can also measure progress over time — with push-ups and with improvement in reading skills. Let’s look at our class of fifth graders again. We want to gather information as to whether the physical fitness class is effective — whether the children’s fitness levels improve. How can we answer this question?

To measure the effectiveness of the fitness class, we will measure the children’s number of push-ups before they take the class and compare this score with their score after they take the class. If the class is effective, we should see individual improvement and group improvement. Some children will have minimal improvement — these children will fall further behind the peer group. Other children who performed below their peers may show significant improvement. Some children will improve so much that they now perform as well or better than the “average” youngster.

We will measure the children’s progress on one or more occasions as they progress through the class. If the fitness class is “working,” that is, if the children’s’ fitness levels are improving, then their ability to perform fitness skills should improve measurably over time. In our example, physical fitness improvement is being assessed using “appropriate objective criteria and evaluation procedures . . .” (34 C.F.R. §300.346)

Because of its enormous usefulness in measuring educational progress, we will return to the subject of the bell curve repeatedly throughout this article.

Continued in Part 2