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ОглавлениеModule 3 Frequency and Percentile Tables
Learning Objectives
Identify the utility of presenting data in a table
Determine how to convert scores to frequency tables
Use a frequency table to create a relative frequency or percentage table
Convert data to a grouped frequency table
Find percentiles and percentile ranks from tabled data
Module Summary
Frequency tables present data in an organized manner that enables specific information to be retrieved efficiently. Through the use of a frequency table, you could easily obtain an estimate of how many participants obtained a specific score, as well as determine how many participants scored above and below that score. All frequency tables follow a similar format. The left displays all possible values and the right displays the frequency, or how many people obtained a specific value. There are three additional columns that you could add to the frequency table to provide you with more information.
The first additional column creates a cumulative frequency table. A cumulative frequency table displays how many scores fall at or below (or possibly above) a specific value. If you were to create a frequency table for test grades, this table would enable you to determine how many students were above your test grade or below your test grade.
A relative frequency or percentage table tells you the proportion (percentage) of the total sample that obtained a specific score. Let’s say you have a data set with 10 numbers ranging from 1 to 5, and three of those numbers are 4s. The relative frequency of 4s would be 30%. This is done by dividing the frequency of a specific score by the amount of scores in the data set and then multiplying the quotient by 100. Because relative frequencies are percentages, they must add up to 100%.
Cumulative relative frequency or cumulative percentage tables provide you with the percentage of scores above or below a specific value. These are created by first finding the relative frequency of each value. Then the relative frequency of the lowest score is added to the relative frequency of the next highest score. Repeat this step until you reach the highest score, which should have a cumulative relative frequency of 100%.
When you have a large number of scores, it can be helpful to group your score into intervals when creating a frequency table (imagine listing all the possible values for the SAT, which has a scale of 0–1,600!). When creating a grouped frequency table, all of the intervals should be equal in size. There are no standard rules for determining when data should be grouped or the size of each interval. The criterion is ease of interpretation.
Cumulative relative frequencies are sometimes also referred to as percentile ranks. Percentile rank indicates the percentage of scores falling at or below a specific score. If you obtain a score of 85 on your next test, which has a cumulative frequency of .94 percentile rank, you can be certain that you did better than 94% of your class.
However, since multiple scores can occur at a specified percentile rank (there may have been 6 students with a score of 85), your percentile rank provides only an estimate of your rank. To determine the precise percentile rank, you need to spread that rank across all the persons with that specific score. This is done by using the UL and LL with the following formula:
After using the above formula, assume that the amount of scores that fall within the real limits of the score (84.5–85.5) are evenly distributed. To find the precise percentile rank, divide the amount of scores at that interval by the proportion (percentage) of scores in that interval and add that amount to the percentage of scores below the interval. If 90% of the scores fell below 84.5 and 4% of the scores fell between 84.5 and 85.5, you can be certain that the true percentile rank of your score was 92.0.
Alternatively, you may be interested in determining the score that corresponds to a specific percentile rank. This can be done using the following formula:XPR = LL + (i /fi)(cum fUL − cum fLL)(0.5)
Computational Exercises
Here are the scores of 15 freshman students rating their confidence they will do well in statistics on a 1 to 10 scale. Use these data for Questions 1 to 8:
5 10 10
7 4 4
3 2 10
2 1 1
2 10 9
1 Arrange the scores into a frequency table in descending order. How many students ranked their confidence as a 5? As a 6? As a 4?
2 Add a column to the table you created for Question 1 to show the cumulative frequency of the scores. How many students ranked their confidence as less than 7? As greater than 4? As less than 10?
3 Add a column to the table you created for Question 1 that shows a relative frequency for each score. What percentage of students ranked their confidence as a 3? As an 8?
4 Add a column to the table that you created for Question 1 that shows the cumulative relative frequency.
5 What is the percentile rank for a person who rated himself or herself at a 5?
6 What is the exact percentile rank for a person who rated himself or herself at a 4? Use the formula.
7 What score falls at the 40th percentile rank?
8 What score falls at the 33rd percentile rank?
Computational Answers
1 Number at 5 = 1; Number at 6 = 0; Number at 4 = 2.
2 Less than 7 = 9; greater than 4 = 7; less than 10 = 11.
3
4
5 60%
6
7 A score of 3.
8 XPR = 1.5 + (1/3)(5 − 2)(0.5). A score of 2.
True/False Questions
1 In creating a frequency table, you need to list scores that have a frequency of zero.
2 Frequency tables are used to organize information more efficiently.
3 A cumulative frequency of 5 means that there are 5 scores below this particular score.
4 Relative frequencies are the proportion of scores at or below a specific score.
5 In a sample containing n = 20 participants, 6 of the participants obtained a score of 12 on a measure. The relative frequency for the score of 12 for this sample is 30%.
6 All the relative frequencies for a sample must sum to 100%.
7 The cumulative relative frequency of the lowest score in a data set is always 0%.
8 When creating a grouped frequency table, you should have a minimum of 5 intervals.
True/False Answers
1 True
2 True
3 False
4 False
5 True
6 True
7 False
8 True
Short-Answer Questions
1 What are the advantages of organizing data in a frequency table as opposed to viewing them as raw scores?
2 You are creating a frequency table for a test anxiety scale that ranges from 1 to 10. After administering the test to 25 people, you notice that no one scored a 5 or an 8. How many intervals should your scale have?
3 If you wanted to determine how many scores were in a data set, which frequency table column would provide this information most efficiently?
4 What does a relative frequency tell you?
5 You are checking the grade of your last English test and notice that your professor provided you with a cumulative relative frequency table as well. You notice that cumulative relative frequency of your score was .34. What does this mean?
6 When should you consider making a grouped frequency table as opposed to a regular frequency table?
7 What is a percentile rank?
Answers
1 The data are organized in an efficient manner such that you can easily determine how many of any particular score are in the data set. When viewing raw scores, especially those out of order, it can be difficult to determine this information.
2 It should still have 10 intervals.
3 The cumulative frequency column.
4 The proportion of the sample that obtained a particular score.
5 34% of the class scored the same or lower than you and 66% of the class scored higher.
6 There is no standard rule for creating a grouped frequency table. However, you should consider using this when the scale for your data set is so large that it would be cumbersome to use a standard frequency table.
7 A percentile rank is the percentage or proportion of cases falling at or below a specific score.
Multiple-Choice Questions
The following scores represent the amount of fear (scored on a scale of 1–15) experienced when going through a haunted house on Halloween. Use the following table for Questions 1 to 6.
1 What is the cumulative frequency (starting from the bottom) at a fear of 5?217293103
2 How many people visited the haunted house this past Halloween?12321315120
3 What is the approximate relative frequency of people that provided a rating of 10?0.070.700.050.50
4 What is the cumulative relative frequency (starting from the bottom) of a rating of 15?0.100.40.981.0
5 If you were to draw a relative frequency curve of these data, how would you describe the shape of this distribution?SymmetricalPositively skewedNegatively skewedBimodal
6 What is the cumulative relative frequency (starting from the bottom) of a score of 11?27.45%88.73%94.31%74.85%
7 In a study of posttraumatic stress, you obtain ratings of the number of flashbacks a person has in a month. If you have a sample of 81 individuals, and 23 have indicated they have had flashbacks twice in the past month, what is the relative frequency of those with two flashbacks in the past month?0.630.400.280.53
Multiple-Choice Answers
1 C
2 B
3 A
4 D
5 B
6 B
7 C
Module Quiz
1 Using the following data, create a frequency table, cumulative frequency table, and relative frequency table.7 8 410 6 45 7 73 5 64 3 9
2 You are investigating a measure of well-being in a geriatric population. The scale has a range of 0 to 50. The cumulative relative frequency for a score of 44 is 34% and the relative frequency is 4% for that interval. Why is the percentile rank of those with a score of 44 equal to 32% and not 34%?
3 You had a percentile rank of 86 in your high school class. What percentage of students were ranked above you?
4 You are trying to arrange the SAT scores of your high school in a frequency table. You realize that you need to use a grouped frequency table because of the large scale (0–1,600), so you decide to create intervals of 400. Was this a good idea?
5 Name a variable that you think would have (a) a negative distribution, (b) a positive distribution, (c) a bimodal distribution, (d) a normal distribution.
Quiz Answers
1
2 Percentile ranks are based on the real limits of a score. The cumulative relative frequency was 34% with a relative frequency of 4%, indicating that 30% of the scores were below this interval and 4% of the scores above this interval. Also, you can assume that the scores were evenly distributed within this interval. Thus, you have to add the amount of scores below (30%) to 1/2 of the amount of scores within it (4%), or 2%. This provides you with a percentile rank of 32%.
3 14%.
4 This was probably not a good idea. You would have 4 groups, and you would expect not many people to fall in the lowest group. Also, you would lose a great deal of precision, as most people would score between 800 and 1,200, which would be just one interval.
5 Answers will vary.
