Reporting and Interpreting
the Results
of Data Collection Processes
The purpose of Chapter 7 is to help you meet the objectives listed below:
1. Identify examples of nominal, ordinal, and interval data.
(Review questions 1-4)(Textbook emphasis pp. 156-158)
2. Define the mode, median, and mean, and identify situations in which each is used.
(Review questions 5-9)(Textbook emphasis pp. 158-160)
3. Describe and interpret examples of the standard deviation.
(Review questions 10-13)(Textbook emphasis pp. 160-161)
4. Distinguish between norm-referenced and criterion-referenced measurement techniques.
(Review questions 14-16)(Textbook emphasis pp. 162-163)
5. Interpret examples of percentiles and derived scores.
(Review questions 17-20)(Textbook emphasis pp. 162-164)
1. Dale shot 14 straight free throws successfully. This put him first in the class. What scale of measurement is 14?
a. Nominal.
b. Ordinal.
c. Interval/ratio.
d. It's impossible to tell.
2. Belinda missed the third and eighteenth questions on the 20-item exam. What scale of measurement is eighteenth?
a. Nominal.
b. Ordinal.
c. Interval/ratio.
d. It's impossible to tell.
3. Grace missed item 24 on the test. What scale of measurement is 24?
a. Nominal.
b. Ordinal.
c. Interval/ratio.
d. It's impossible to tell.
4. A recent survey shows that 95% of Americans have at least one telephone. This places Americans first in the world. What scale of measurement is 95%?
a. Nominal.
b. Ordinal.
c. Interval/ratio.
d. It's impossible to tell.
5. Mary scored at the 5th percentile. Bob scored at the 10th percentile. Bob claims he did twice as well as Mary on the test. Is Bob right?
a. Yes. He is correct.
b. No. He definitely did not do twice as well as Mary.
c. We would need more information to make a judgment.
6. Of the 60 persons who ran in the mini-marathon, 18 didn't finish the race. The fastest time was 34.50 minutes. The average time was 45.0 minutes. What measure of central tendency was probably used to calculate this "average"?
a. Mode.
b. Median.
c. Mean.
7. The average American above the age of 50 has never used a microcomputer. What measure of central tendency was probably used to calculate this "average"?
a. Mode.
b. Median.
c. Mean.
8. The teacher kept track of how often Wally was out of his seat. During the two-week observation period, he was out of his seat an average of 41.6% of the time. What measure of central tendency was probably used to calculate this "average"?
a. Mode.
b. Median.
c. Mean.
9. The teacher kept track of how long the students took to complete the test. Four of the twenty were still not finished when the testing session was over. However, the average amount of time was 34 minutes, which was well within the 45-minute class period. What measure of central tendency was probably used to calculate this "average"?
a. Mode.
b. Median.
c. Mean.
Use this information for questions 10 to 13.
Test A had a mean of 78.9 and a standard deviation of 5.6.
Test B had a mean of 23.6 and a standard deviation of 7.0.
Test C had a mean of 35.5 and a standard deviation of 10.5.
Test D had a mean of 45.6 and a standard deviation of 10.0.
10. On which test were the scores probably the most widely spread out?
a. Test A.
b. Test B.
c. Test C.
d. Test D.
11. On which test were the scores probably the most closely clustered together?
a. Test A.
b. Test B.
c. Test C.
d. Test D.
12. Assuming the scores were normally distributed, which test probably had the lowest score?
a. Test A.
b. Test B.
c. Test C.
d. Test D.
13. Assuming the scores were normally distributed, which test probably had the highest score?
a. Test A.
b. Test B.
c. Test C.
d. Test D.
14. Kristin was on the committee to select the homecoming queen. She wanted to find out which of the many candidates had the sharpest wit. Does she need a criterion-referenced or a norm-referenced test?
a. Criterion-referenced.
b. Norm-referenced.
c. It wouldn't matter.
15. Toby wants to find out what percentage of college graduates can spell at least 90% of the words on her standard spelling list. Does she need a criterion-referenced or a norm-referenced test?
a. Criterion-referenced.
b. Norm-referenced.
c. It wouldn't matter.
16. Mr. Boswell wants to find out what percentage of the American public are familiar with the titles of all of Mr. Johnson's major works. Does he need a criterion-referenced or a norm-referenced test?
a. Criterion-referenced.
b. Norm-referenced.
c. It wouldn't matter.
17. Glen scored at the 75th percentile. Wanda did better than 40% of the students in the class. Who did better?
a. Glen.
b. Wanda.
c. It's impossible to tell.
18. Matt got a score of 115 on a test with a mean of 100 and a standard deviation of 15. June got 50% of the items right on the same test. Who did better?
a. Matt.
b. June.
c. It's impossible to tell.
19. Sandy got a score of 700 on a standardized test with a mean of 500 and a standard deviation of 100. What was the percentile equivalent of this score?
a. About 2nd percentile.
b. About 34th percentile.
c. About 84th percentile.
d. About 98th percentile.
e. There's no way to tell.
20. Raleigh scored 85 on a test with a mean of 100 and a standard deviation of 15. Sam scored 70 on the same test. What percentage of the standardization group scored between Sam's and Raleigh's scores?
a. about 10%.
b. About 14%.
c. About 50%.
d. About 34%.
e. There's no way to tell.
POSSIBLE PROBLEMS AND SOLUTIONS
1. Confusion among nominal, ordinal, and interval data. The easiest way to deal with this problem is to focus on ordinal data as being "rank order." The word order is practically in the word ordinal. Many students are already familiar with the concept of ordinal (first, second, etc.) vs. cardinal numbers. Then nominal is not even as specific as ordinal, and interval is more specific than ordinal. Most students can master these concepts, if they are willing to focus their attention and spend the time needed.
2. Confusion over the word average. Many students equate the word average with the mean. This leads to confusion, because the English language permits the word average to mean typical. For example, it makes perfect sense to say, "The average Christmas tree is green." This is clearly a mode, but the word average is legitimate. The best way to deal with this seems to be to tell the students that you expect them to be more precise in their use of terms describing central tendency. The only three acceptable terms are mode, median, and mean.
3. Confusion among measures of central tendency. These terms are clearly treated in both the textbook and the workbook. The major problem seems to be that some students think these concepts are so easy that they don't bother to study the information carefully.
4. Confusion over the use of the mean or median when both are possible. This is a legitimately difficult question, and often the choice is arbitrary. Instructors should be willing to accept either measure of central tendency when it's a close call. The basic principle is to use the mean as often as possible, except when there's a clear need for the median. (The reason is because the mean is more easily combined with other useful statistics, such as the standard deviation, correlation coefficient, and t-test.) There are two related situations that would call for the median: (1) When one or two extreme scores would distort the calculation of the mean so that it would not really be representative. For example, if the boss of a company makes $100,000,000 a year and the rest of the employees make about $30,000, it would be better to use the median to describe the typical income. (2) When there is a distribution of scores that have unusual clusters at come point - usually the top or bottom of the distribution. In some cases, the distribution makes it actually impossible to calculate the mean. For example, if 30 runners start a race, and ten drop out, it would be impossible to calculate a mean time for the racers: the median would make more sense. In other cases, the scores are just tightly clustered: for example, if out of 30 students 10 received scores of 100% and the rest ranged somewhere between 60 and 90, it would be better to report the median. The best way to deal with this is to have students deal with numerous examples until they can make the appropriate distinctions.
4. Confusion about standard deviations. Some students get just plain panicky about the standard deviation, because it sounds like mathematics. The mathematics in this chapter is actually almost non-existent. The best strategy seems to be to go over examples until the student grasps the concept. It is extremely easy (and useful) for students themselves to generate examples like Questions 10 through 13 of this workbook.
SEMIPROGRAMMED UNIT A: SCALES OF MEASUREMENT
1. Numbers can serve several purposes. At their simplest level, numbers merely attach a label to what they "measure." If a number merely attaches a label and carries no additional meaning, it's conveying nominal information. (If you're interested in roots of words, you can see the idea "naming" in the word "nominal." That's what a nominal number does. It names something, or labels it.)
2. Which of the following numbers express merely nominal information?
a. Joe wore the number 14 on his jersey. (Go to 3.)b. Bill hit 20 home runs. (Go to 4.)
3. Right. The number on Joe's jersey probably is just a label. However, it's conceivable that the number 14 might be more than nominal information. For example, the coach might have 15 players on the team and might assign the number 1 to the best player and the number 15 to the worst player, with all the other numbers falling in between. If this were the case, then 14 would do more than label Joe. The number would also indicate that he ranked low in the team's hierarchy. (Go to 5.)
4. Wrong. The number does more than label Bill. It gives a precise count of the number of runs he hit. Return to 1 and start over.
5. Here is a sample item from a hypothetical questionnaire.
A) What is your sex?1. Male.2. Female.
What level of measurement is the number "2" in this questionnaire item?
a. Nominal. (Go to 7.)b. More than nominal. (Go to 6.)
6. Wrong. The number "2" is merely a substitute for the word "female." There's no apparent assumption that females are better or worse than males or that one has a higher value than the other. The numbers could be reversed with no loss of meaning, as long as the data collector remembered what the labels stood for. Even if you got this wrong, go ahead to 7. Perhaps when you understand the other levels, nominal numbers will become easier to understand.
7. The second level of measurement is the ordinal scale. Ordinal numbers do more than attach a label, they also put what they measure in a rank-order. For example, a person receiving one number is classified as being either higher or lower in some respect than a person receiving another number. (This is easy to remember, since the word "order" is obviously contained in the word "ordinal.")
8. Listed below are two sample items from a hypothetical questionnaire.
A) What is your academic major?1. English
2. Foreign Languages
3. History
4. Psychology
5. PhilosophyB) What is your annual family income?
1. Less than $15000
2. $15001 to $30,000
3. $30,001 to $50,000
4. $50,001 to $75,000
5. Over $75,000If a respondent answered "4" to question A, this would be nominal information. This is because the "4" is just a label. There's no apparent logic behind the idea that psychology is higher than history and lower than philosophy. We could, in fact, arbitrarily rearrange the labels, so that the question would read as follows:
A) What is your academic major?1. Foreign Languages.
2. Philosophy.
3. History.
4. English.
5. Psychology.
The rearranged labels would make no difference, as long as we remember what the numbers stand for.
9. If a respondent answered "4" to question 8B, however, this would be ordinal information. This is because the "4" is more than a label. There is logic behind the idea that a person labeled "4" comes from a family with a higher annual income than one labeled "3" and lower than one labeled "5." There is a rank-ordering of information. We would not sensibly rearrange the question to read as follows:
B) What is your annual family income?1. $50,001 to $75,000
2. Less than $15,000
3. Over $75,000
4. $15,001 to $30,000
5. $30,001 to $50,000
The labels in this revised version would still be accurate (if the respondent took the extra trouble to locate his or her correct category), but the numbers would have lost the rank-ordered meaning of the original version.
10. Which of the following hypothetical questionnaire items will generate a nominal response, and which will generate an ordinal response?
A) What is your race?1. Caucasian
2. African American
3. Hispanic
4. Asian American
5. Other
B) What was your undergraduate grade-point average?
1. Below 1.50
2. 1.50 to 1.99
3. 2.00 to 2.49
4. 2.50 to 2.99
5. 3.00 to 3.49
6. Above 3.50
Which is nominal?________________Which is ordinal?________________
Answers: A is nominal data.
B is ordinal data.
11. The final level of measurement is referred to as interval. (Actually, there are two separate levels, "interval" and "ratio," but the distinction is not important for educational purposes. Therefore, they are treated together in this text and workbook.) Interval data not only attaches a label and describes a rank order, but it also provides meaningful intervals. It attempts to attach a precise measurement to a variable. The difference can be seen if we examine both the ordinal and the interval (or ratio) description of the performance of runners in a track meet. Listed below are the order of finish and the running times for runners in a race.
Runner Order of Finish Time Smith
1st
4:01
Jones
2nd
4:02
Brown
3rd
4:03
Black
4th
4:15
Wilson
5th
4:20
Jablonski
6th
4:30
In this example, the order of finish is given in terms of ordinal measurement (rank-ordering). The ordinal measurement gives us the order - we know that Smith finished ahead of Jones, for example; but we don't know how far ahead. On the other hand, the time is given in terms of interval/ratio data. The sizes of the intervals between runners are meaningful. We know that Brown, for example, finished slightly behind Jones, but well ahead of Black. Likewise, the ordinal measurement gives the impression that the relationship between Brown and Smith (3rd and 1st) was similar to that between Jablonski and Black (6th and 4th). However, the interval data gives us more precise information; and we can see that Brown was only 2 seconds behind Smith, whereas Jablonski was 15 seconds behind Black. The interval data provides much more precise information.
12. Although it is entirely possible (and sometimes useful) to have subtle discussions about the levels of measurement, the basic idea is quite simple.
- Nominal data provides mere labels.
- Ordinal data provides an ordering of data, but not meaningful intervals.
- Interval data provides meaningful intervals.
13. Listed below are several items from a hypothetical questionnaire. Examine the questionnaire items, and then classify the levels of measurement as requested afterwards.
A) Major Code:______ (Choose from list provided.)B) Number of credit hours taken last semester:_______
C) Grade-point average last semester:______
D) Do you plan to take Advanced Pyrotechnics next semester?
1. Yes.
2. No.
3. Uncertain.
4. I've already taken that course.E) Examine the following list of new course titles. Write "1" next o the course you would most like to take, "2" next to your second choice, and continue until you have written "5" after the course you would least like to take.
______ The New American Novel.
______ Contemporary Paleontology.
______ Vocabulary Development.
______ Irish and Russian Poetry.
______ Introductory Etiquette.
What level of measurement would be provided by the answer to each of the questions?A._________________________________
B._________________________________
C._________________________________
D._________________________________
E._________________________________
Answers:
A. Nominal. All this code does is attach a label.B. Interval. This provides a precise count.
C. Interval. This provides a precise count. Asking for "class rank," on the other hand, would result in ordinal measurement.
D. Nominal. There is no logical order in the possible responses.
E. Ordinal. The respondent is putting these responses in a rank- order. It's not interval measurement because there's no reason to assume that the difference in preference between, say, first and second choices is the same as the degree of difference between fourth and fifth choices.
14. Here's one more hypothetical questionnaire. Examine it, and then try to classify the levels of measurement which would result from each question.
A) How did you like the book?
- I hated it.
- I was indifferent towards it.
- I liked it.
- I loved it.
- I liked it better than any book I've ever read before.
- It was undoubtedly the best book ever written.
B) How did you obtain the book?
- I purchased it in hardback.
- I purchased it in paperback.
- I shared the purchase with a friend.
- I borrowed it from someone.
- I borrowed it from the library.
- Other (Specify:_________________________ )
C) Approximately how many hours did it take you to read the book?________
D) How many other books can you name by the same author?________
E) Why did Jason not return from Kimberly's house at the end of the book?
- Because he knew that Victoria would reject him.
- Because Kimberly made exquisite vegetable soup.
- Because Kristin might try to shoot him again.
- Because he had found the true meaning of life at Kimberly's house.
- Jason did return from Kimberly's house.
What level of measurement would be provided by the answer to each of these questions?A._________________________________
B._________________________________
C._________________________________
D._________________________________
E._________________________________
Answers:
A. Ordinal. The responses suggest an increasing order or preference for the book. However, it's not obvious that the intervals are equal, and therefore it's not interval measurement.B. Nominal. There is no apparent order within this list of responses.
C. Interval. It gives a precise count.
D. Interval. Another precise count.
E. Nominal. There is no apparent order within this list of responses.
SEMIPROGRAMMED UNIT B: MEASURES OF CENTRAL TENDENCY
1. Instead of stating a whole list of numbers, it's often useful to state a single number that is a good representative of the whole list of numbers. For example, if a class of 30 pupils took a test and the principal asked the teacher how the students did, the teacher could recite the entire list of scores. On the other hand, she could simply tell him that the "average" score on the test was 85%. The principal would probably feel less overwhelmed with the latter piece of information. "Average" or "typical" scores that can be stated to summarize a larger list of scores are referred to as measures of central tendency.
2. Which of the following statements contain a measure of central tendency?
a. Bob hit 18 home runs and James hit 15 home runs this season.b. The average player on the team hit 12 home runs.
c. The typical player on the team uses a Louisville Slugger bat.
d. Bob struck out his first two times at bat.
e. Carl strikes out every 13.5 times at bat. He hits a home run every 35.2 times at bat.
f. There were 3400 people at yesterday's game.
g. Season attendance averaged about 5500 this year.
Answers: b, c, e, and g are the correct answers. If you got this wrong, you may find it useful to review Frame 1 before you continue.
3. The mode is one measure of central tendency. In any set of numbers, the mode is the most frequently occurring number. Look at a few statements containing a reference to the mode.
- The mode on the test was 16. (This means that more students got a score of 16 than any other score.)
- The test had two modes, 16 and 24. (This means that an equal number of students got scores of 16 and 24, and there was no score obtained by a higher number of people.)
- Sometimes the word "average" is used instead of "mode." In such cases, it is necessary to tell from the logic of the situation that a mode was used to compute the "average."
- The average American has brown eyes. (This means that more Americans have brown eyes than another color.)
- The average American is a white female between the ages of 25 and 35. (This statement, if it is true, means that there are more persons who fall into this category than in any of the other categories in the analysis. To make complete sense of the statement, it would be useful to know what the other categories were.)
4. When measurement consists of nominal data, the only "average" that you can compute is the mode. In other situations, you can use either the mode or another type of "average." You should decide which type of "average" expressed your ideas most accurately, and then use that measure of central tendency.
The average player on the team hit 12 home runs. (If this is a mode, it means that more players hit 12 home runs than any other number. On the other hand, if the "average" is a median or a mean, the meaning is different. These other meanings will be discussed below.)
5. The median refers to the middle score in a set of scores. To compute the median, you simply arrange the scores in ascending or descending order, and then identify the score in the middle. (In cases of tied scores and when there is no precise score that falls in the exact middle, the computation is slightly more complex. Consult a statistics text for specifics.) Here are a few examples of medians:
- The median income in the United States that year was $5000. (This means that half the citizens earned over $5000 and half earned less. $5000 was the midpoint.)
- The median score on the test was 16. (This means that half the students scored above 16 on the test and half scored lower.)
Sometimes the word "average" is used instead of
"median." In such cases, it is necessary to tell from the logic of
the situation or from the statement of methodology that the median
was used to compute the "average."
- The average income that year was $5000.
- The average score on the test was 16.
These statements are interpreted in the same way as the others. However, further information would be necessary to know that these "averages" are medians.
6. The median is used in preference to the mode in situations when it is valuable to take into consideration the rank-ordering of the scores. It is possible to use the median whenever you have ordinal data. However, with ordinal data, it is also possible to use the mode. With interval data the median can be used, but the mean (Frame 9) is often preferred.
The average player on the team hit 12 home runs. (If this is a median, it means that half the players hit more than 12 home runs and half hit fewer than 12.)
7. In situations where one end of a distribution is artificially truncated, it is often advisable to use a median instead of a mean. For example, if you want to measure performance in a race, and 10 of the 33 cars don't even finish, it would be difficult to compute the mean time it took to complete the race. On the other hand, it would be easy to identify the time below and above which half the times fell (the median).
8. Likewise, in situations where one score (or a few scores) would produce a mean which is not really "typical" of all the scores, then the median is the statistic to use. For example, if the top brass in a company make huge salaries and the peons all make much less, the use of the mean as a measure of central tendency would produce an "average" which would be typical of no one. On the other hand, the median income would provide a good estimate of the "typical" income.
9. The mean is the average attained by adding all the scores together and dividing by the number of scores used in this calculation. It is what we learned as the "average" in our arithmetic classes. If there are 10 players on a baseball team and the team hit 120 home runs, then 12 would be the mean number of home runs. (Compare this last sentence to the statements given as examples in 4 and 6.)
10. To compute a mean, you need interval or ratio data. This means that if you have only a ranking of scores you have to use a median (or perhaps a mode) instead of a mean. Likewise, if you have nominal data, you have to use the mode instead of the mean. On the other hand, if you have interval data, you have a choice. You can use the mean, but you might wish to choose another statistic. The mean takes into consideration all the scores in the set of scores available for analysis. In most cases, this is what you want, and so the mean will be the statistic, of choice. In a few cases (such as the salary example in 8 and the race example in 7) a few scores will introduce inaccuracy or confusion. If this is the case, then you want a statistic which does not take into consideration all the scores; and for this reason a mode or median may be more useful.
11. Examine the following list of statements and decide what measure of central tendency was used.
a. A very large number of students couldn't even translate the instructions to start the exam. However, the average score on the exam was 70%.b. The average American drives a Japanese car.
c. There were over 1100 requests to be absent during the first semester. However, that comes down to an average of only 2.3 per teacher.
d. Herb can type an average of about 5 pages an hour.
e. The teacher talked at the rate of about 150 words per minute.
f. The average pupil in the school system used to be white. Now the average student is black.
g. The study showed that the average American uses Brand X toothpaste.
Answers:
a. Median. Since this distribution of scores is artificially truncated (a lot of students scored zero because they could not translate the instructions), the median would be the best way to describe the "average" performance.b. Mode. This is simply a count of what type of car is driven by more Americans than any other type of car.
c. Mean. This "average" appears to have been obtained by dividing the number of requests by the number of teachers.
d. Mean. This average would be obtained by counting the number of pages and dividing by the number of hours it took to type those pages.
e. Mean. This "average" would be obtained by counting the total number of words and dividing by the number of minutes it took the teacher to say those words.
f. Mode. This is a simple count of how many students there were of each race and comparing the totals.
g. Mode. This is a simple count of which toothpaste was used by more persons than any other toothpaste in the study.
SEMIPROGRAMMED UNIT C: THE STANDARD DEVIATION
1. The standard deviation is a way to express how much spread there is in a set of scores. A large standard deviation means that the scores are widely dispersed. A small standard deviation means that the scores are more tightly clustered together.
2. A standard deviation is meaningless unless it accompanies a statement of the mean. By knowing the mean and standard deviation of a set of scores, we can actually have a pretty good summary of what the whole list of scores would look like.
3. Examine the following two statements. Then answer the questions in the following frames.
a. The students averaged 53.5 on the algebra test. The standard deviation was 10.6.b. The students averaged 53.5 on the geometry test. The standard deviation was 3.4.
4. On which test in Frame 3 did the students on the average do better?
a. Algebra test (Go to 5.)b. Geometry test (Go to 5.)
c. Both the same (Go to 6.)
5. Wrong. Look again. The score was 53.5 on each test. They were both the same. (Go to 6.)
6. The average performance on both tests was the same. Now, on which test were the scores more widely distributed?
a. Algebra test (Go to 7.)b. Geometry test (Go to 8.)
c. Both the same (Go to 9.)
7. Right. The standard deviation on 10.6 is much higher than the standard deviation of 3.4 on the geometry test. This means that the spread of scores was much greater on the algebra test than on the geometry test. (Go to 10.)
8. Wrong. The standard deviation of 3.4 on the geometry test is much smaller than the standard deviation of 10.6 on the algebra test. (Try 6 again.)
9. Wrong. The scores would be equally spread out only if the standard deviations were equal. The means are equal, but the standard deviation of 10.6 is much greater than 3.4. (Try 6 again.)
10. On which of the tests in Frame 3 was the higher score obtained by the student who did best on that test.
a. Algebra test (Go to 11.)b. Geometry test (Go to 12.)
c. There's no way to tell. (Go to 13.)
11. Right. The scores are more spread out on the algebra test. Since this is the case, it's likely that the higher score would occur in this group. (In other words, the student with the highest score on the algebra test is likely to be quite far from the mean, whereas the student with the highest score on the geometry test is likely to be close to the mean, since the geometry scores are more tightly clustered.) (Go to 14.)
12. Wrong. The scores on the geometry test are tightly clustered together. Therefore, the highest scoring student is likely to be closer to the mean on that test than on the algebra test, where the scores are much more widely dispersed. (Try 10 again.)
13. Wrong. You know that one set of scores is tightly clustered together, whereas the other set of scores is more widely dispersed. Use this information to formulate an intelligent guess about which one would have the higher top score. (Try 10 again.)
14. On which of the tests in Frame 3 did the student who did worst on that test score lower?
a. Algebra test (Go to 15.)b. Geometry test (Go to 16.)
c. There's no way to tell. (Go to 17.)
15. Right. The logic here is the same as in Frame 10. The scores are more widely distributed, and therefore the person with the lowest score on the algebra test is probably quite far below the mean. (Go to 18.)
16. Wrong. The scores on the geometry test are quite closely clustered. Therefore, the person with the lowest score on the geometry test is probably not as far below the mean as the person with the lowest score on the algebra test, where the scores are more widely dispersed. (Try 14 again.)
17. Wrong. You know that one set of scores is tightly clustered together, whereas the other set of scores is more widely dispersed. Use this information to formulate an intelligent guess about which one would have the lower bottom score. (Try 14 again.)
18. Here is some information about four sets of scores. Examine this information and answer the questions in the following frames.
Test I had a mean of 16.4 and a standard deviation of 3.1.Test II had a mean of 24.1 and a standard deviation of 5.0.
Test III had a mean of 18.0 and a standard deviation of 0.5.
Test IV had a mean of 20.0 and a standard deviation of 6.0.
a. Which test had the most tightly clustered scores?________b. Which test had the most widely dispersed scores?________
c. Assuming that all the scores fell within three standard deviations above and below the mean, which test had the highest score?________
d. Assuming that all scores fell within three standard deviations above and below the mean, which test had the lowest score?________
Answers:
a. IIIb. IV
c. II
d. I
19. Standard deviations can be computed only with interval or ratio data. Likewise, they have to be interpreted in conjunction with the mean. This is one good reason to use the mean (rather than the median or mode) whenever the mean is appropriate.
20. The idea of incorporating the standard deviation into the interpretation of the normal distribution will be discussed in Programmed Unit D of this chapter.
21. Computation of the standard deviation is relatively easy. However, it is not discussed here. Consult a statistics text.
SEMIPROGRAMMED UNIT D: WAYS TO EXPRESS SCORES
1. Scores can be expressed in certain ways which make comparisons easier. For example, a percentile score indicates how a person ranks with regard to other persons who took the same test. The percentile score indicates how many persons fell below the score of the person whose score is being reported. For example, a score at the 84th percentile means that the person with that score did better on the test than 84 percent of the others who took the same test. (These "others" could take the test simultaneously with the present respondent, or they could be a standardization group - a group of respondents who took the test for the specific purpose of setting up percentiles.)
2. A percentile score is not the same as a "percentage-correct" score. Knowing a person's percentile score tells us nothing at all about how many answers the respondent got correct. All it tells us is how the respondent's performance compares to that of others. Percentiles, therefore, are extremely useful for norm-referenced reporting. They have no relevance whatsoever for criterion-referenced reporting. (The difference between norm-referenced and criterion-referenced tests is discussed in the textbook.)
3. John scored at the 57th percentile. What does this mean?
a. He got 57% of the questions right. (Go to 4.)b. He scored better than 57% of those who took the test. (Go to 5.)
c. Only 57% of those who took the test scored as well as he did. (Go to 6.)
4. Wrong. Reread 2 and then try 3 again.
5. Right. This is exactly what the statement means. (Go to 7.)
6. Wrong. Reread 1 and try 3 again.
7. Many scores happen to fall into a pattern called the normal curve. The normal curve is described and diagrammed in the textbook. When scores fall into this pattern (or come close to doing so), it is possible to state them in terms of "derived scores." Such derived scores use a common mean and standard deviation. By doing this, they make it easier to compare the performance of the same persons on different tests. They also make it possible to compare the performance of different persons on the same test or set of tests.
8. The normal curve and many of the most common derived scores are diagrammed in the textbook. The logic of the curve and these scores is best understood by examining the information in the textbook. At this point, I will merely provide a few exercises on the use of Figure 5.3 to interpret derived scores.
9. If Willa got a score of 130 on a test with a mean of 100 and a standard deviation of 15, what was the percentile equivalent of her performance?
Try to work this out by referring to Figure 5.3 in the textbook before looking at the answer, which comes next.Answer: 98th percentile. To arrive at this answer, you would figure out how many standard deviations Willa scored above the mean. (She was 2 standard deviations above the mean.) Looking at the diagram, we can see that only 2.2% of the respondents (2.1% + 0.1% = 2.2%) scored higher than 2 standard deviations above the mean. Therefore, Willa scored at approximately the 98th percentile (100% - 2% = 98th percentile).
10. Andy scored 600 on the Graduate Record Exam (GRE). Sandy scored 400 on the same test. What percentage of the respondents scored between Sandy and Andy?
Try to work this out before looking at the answer, which follows next.Answer: 68%. You can see that Andy scored 1 standard deviation above the mean. The diagram tells us that 34% of the respondents would score between the mean and one standard deviation above the mean. Likewise, Sandy scored one standard deviation below the mean. The diagram shows us that 34% fall between the mean and one standard deviation below the mean. This means that 68% fell between Sandy and Andy (34% + 34% = 68%).
11. The approach described in frames 9 and 10 can be applied to any derived scores. By using Figure 5.3 or a more refined version of a similar table or set of tables, you can draw useful conclusions about derived scores.
12. Derived scores are useful only for norm-referenced tests. They have no use whatsoever for criterion-referenced tests. Because of their extreme usefulness in interpreting norm-referenced tests, people who design tests often go to great efforts to design tests whose results will almost certainly fall into the pattern of the normal curve - thereby making it appropriate to use derived scores. If your job requires you to make widespread use of such standardized tests, it would be useful for you to become familiar with the logic of Figure 5-3.
CROSS-REFERENCES TO OTHER CHAPTERS
Chapter 7 makes reference to the following concepts that are defined and discussed in other chapters. These are listed in the order in which they occurred in Chapter 7.Questionnaires (discussed on page 157) are discussed in Chapter 6 on page 124.
Statistical analysis (which is mentioned on page 158) is discussed in detail in Chapters 13 and 14.
Graphing data (introduced on page 161) is discussed in greater detail in Chapter 19 on page 439.
Standardized tests (which are mentioned on page 162 and 163) are discussed in Chapter 6 on page 145.
Norms (which are mentioned on page 163) are discussed in Chapter 6 on page 146.
Operational definitions (which are an important consideration in the discussion of validity, beginning on page 101) were covered in Chapter 4.
Unobtrusive measurement (mentioned on page 105) is further discussed on page 142.
EXAMPLES OF IMPORTANT CONCEPTS IN THIS CHAPTER
Sometimes readers want to go directly to examples of topics. Anecdotes or examples of each of these concepts can be found on the following pages of the textbook:
Nominal data - pp. 156-157Ordinal data - p. 157
Interval/ratio data - p. 157-158
Mode - p. 159
Median - p. 159, 160
Mean - p. 159
Standard deviation - p. 163
Normal distribution - p. 163
Statistical estimates of reliability - pp. 97-98, 111
Validity - pp. 103, 104, 106, 111
Operational definitions - pp. 103, 104, 111
The following matching exercise focuses on the key terms in this chapter. Instead of using it as a matching exercise, you may find it effective to try to define each of the terms. The correct answers can be found by checking the answers to the matching exercise.
MATCHING EXERCISE - KEY TERMS
Listed below are several key terms from this chapter. Match each of these terms with one of the definitions.
a. Nominal datab. Ordinal data
c. Interval/ratio data
d. Mean
e. Median
f. Mode
g. Standard deviation
h. Absolute score
i. Criterion-referenced score
j. Norm-referenced score
k. Percentile
l. Derived score
m. Normal distribution
Review Quiz
Matching Exercise
1. c2. b
3. j
4. e
5. m
6. d
7. h
8. a
9. f
10. i
11. l
12. k
13. g