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
