# 7.9: Relative Frequency Interpretation

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A school lunch break is divided into multiple sessions, the 1st group, consisting of 20 girls and 50 boys, begins lunch at 11:30 am, the 2nd group, with 35 girls and 30 boys begins at 11:40, and the 3rd group, 40 girls and 25 boys, at 11:50.

Assuming that none of the students from an early group leave the cafeteria before everyone goes back to class at 12:10, what is the relative frequency of girls in the cafeteria at 11:35? What about at 11:45? 11:55?

In this lesson we discuss cumulative frequencies, and by the end of the lesson, this sort of question will be no problem at all.

## Cumulative Frequencies

Cumulative frequencies describe a sort of ‘running total’ of frequencies in a frequency distribution. You will find cumulative frequencies in many real-world situations, since we often need to collect data for a larger study by conducting several smaller studies.

To calculate a cumulative frequency, simply create a frequency distribution table, then add the frequency from the first category to that of the second category, then add that total frequency to the third category, and so on. Because cumulative distributions don’t generally show different combinations of categories, it is a good idea to arrange your categories in a suitable order so that the cumulative frequencies are most likely to be of use as the running total increases.

A relative cumulative frequency table shows how the cumulative frequency after each successive interval compares to the total frequency. To create a relative cumulative frequency table, calculate the relative frequency of each interval or category, and then add the relative frequency of each category to all the prior ones.

### Creating a Cumulative Frequency Table

Create a cumulative frequency table showing the number of hours per week that Brian watches television, based on the given information.

Brian’s T.V. Time

flash.pro - www.flickr.com/photos/flashpro/4156535452

• Monday: 1.5 hrs
• Tuesday: 2.25 hrs
• Wednesday: 2 hrs
• Thursday: 1.75 hrs
• Friday: 1.5 hrs
• Saturday: 3.25 hrs
• Sunday: 2.5 hrs

To find the cumulative frequency of hours of T.V. watched, add the frequency of each day to the total frequency of the day before:

 Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday Cumulative Frequency 1.5 hrs 1.5 + 2.25 = 3.75 hrs 3.75 + 2 = 5.75 hrs 5.75 + 1.75 = 7.5 hrs 7.5 + 1.5 = 9 hrs 9 + 3.25 = 12.25 hrs 12.25 + 2.5 = 15 hrs

### Creating a Relative Cumulative Frequency Table

A car dealer is calculating the total sales for the past month, and wants to identify the percentage of monthly sales that have occurred after weeks 1, 2, 3, and 4. Create a relative cumulative frequency table to show the information the dealer wants.

 Week Number Cars Sold 1 21 2 12 3 17 4 24

First total up the sales for the entire month:

21+12+17+24=74 cars

Then find the relative frequencies for each week by dividing the number of cars sold that week by the total:

• The relative frequency for the first week is: 21/74=.284
• The relative frequency for the second week is: 12/74=.162
• The relative frequency for the third week is: 17/74=.23
• The relative frequency for the fourth week is: 24/74=.324

To find the relative cumulative frequencies, start with the frequency for week 1 and for each successive week, total all of the previous frequencies:

 Week Number Cars Sold Relative Frequency Cumulative Frequency 1 21 .284 .284 2 12 .162 .284+.162=.446 3 17 .23 .446+.23=.676 4 24 .324 .676+.324=1.0

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Note that the first relative cumulative frequency is always the same as the first relative frequency, and the last relative cumulative frequency is always equal to 1.

Constructing Tables

Create a cumulative and relative cumulative frequency table for the following data:

Number of customers in store on Black Friday each year for the years 1995 – 2005:

47, 49, 48, 54, 57, 52, 61, 65, 67, 66, 70

Label a table with the years 1995 - 2005 across the top and frequency, cumulative frequency, and relative cumulative frequency down the side. Leave 3 blanks underneath each year, and input the frequency for each year:

 Year: 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Frequency 47 49 48 54 57 52 61 65 67 66 70 Cumulative Frequency Relative Cumulative Frequency

To calculate the cumulative frequencies, sum the total of all previous frequencies under the frequency for each year:

 Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Frequency 47 49 48 54 57 52 61 65 67 66 70 Cumulative Frequency 47 96 144 198 255 307 368 433 500 566 636 Relative Cumulative Frequency

To calculate the relative cumulative frequencies, divide the cumulative frequency for each year by the total cumulative frequency (636 customers, the same as the value for the last year, 2005).

 Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Frequency 47 49 48 54 57 52 61 65 67 66 70 Cumulative Frequency 47 96 144 198 255 307 368 433 500 566 636 Relative Cumulative Frequency 0.074 0.151 0.226 0.311 0.401 0.483 0.579 0.681 0.786 0.89 1

### Earlier Problem Revisited

A school lunch break is divided into multiple sessions, the 1st group, consisting of 20 girls and 50 boys, begins lunch at 11:30 am, the 2nd group, with 35 girls and 30 boys begins at 11:40, and the 3rd group, 40 girls and 25 boys, at 11:50.

Assuming that none of the students from an early group leave the cafeteria before everyone goes back to class at 12:10, what is the relative frequency of girls in the cafeteria at 11:35? What about at 11:45? 11:55?

First create a table showing each session and the number of boys and girls in each session:

 Session 1 - 11:30 Session 2 - 11:40 Session 3 - 11:50 Boys Girls Boys Girls Boys Girls 50 20 30 35 25 40

Next, add a row to the bottom of your table for cumulative relative frequency of girls:

 Session 1 Session 2 Session 3 Boys Girls Boys Girls Boys Girls 50 20 30 35 25 40 Girls: 2070=.286 or 28.6 % Girls: 55135=.407 or 40.7% Girls: 95200=.475 or 47.5%

## Examples

For 25 days, the snow depth at Whiteout Mountain was measured (to the nearest inch) and recorded as follows:

242, 228, 217, 253, 239, 266, 242, 251, 240, 223, 219, 246, 260, 258, 225, 234, 230, 249, 245, 254, 243, 235, 231, 257

1. Determine a reasonable class interval
2. Set up a frequency distribution table
3. Find the frequency for each class interval
4. Calculate the cumulative frequencies progressively from left to right
5. Use the information gathered from the frequency distribution table to plot a cumulative frequency graph.

Solutions:

### Example 1

Determine a reasonable class interval

The values range 217 – 266, a class interval of 10 would break the data up into a reasonable 6 bins.

### Example 2

Set up a frequency distribution table, and find the frequency for each class interval

Create a table with the intervals noted across the top, and the frequency of each below:

 210-220 220-230 230-240 240-250 250-260 260-270 2 3 5 7 5 2

### Example 3

 Interval: 210-220 220-230 230-240 240-250 250-260 260-270 Frequency: 2 3 5 7 5 2 Cumulative Frequency: 224 8.33% 524 20.83% 1024 41.66% 1724 70.83% 2224 91.66% 2424 100%

## Review

1. A weather forecaster highlights the lows over-night for the past 30 days in a small town in Wisconsin. The temperature readings are given in Fahrenheit, and are shown below. Use the data to complete the frequency table.

41,58,61,54,49,46,52,58,67,43,47,60,52,58,48,44,59,66,62,55,44,49,62,61,59,54,57,58,63,60

 Interval Tally Frequency 40 - 44 45- 49 50 - 54 55 - 59 60 - 64 65 - 69

2. The following represents scores that a class received on their most recent Biology test. Complete the frequency table below.

58, 79, 81, 99, 68, 92, 76, 84, 53, 57, 81, 91, 77, 50, 65, 57, 51, 72, 84, 89

 Interval Tally Frequency 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99

3. James received the following scores on his quizzes in US History over the course of 1 year. Complete the frequency table below using the scores:

85, 72, 97, 81, 77, 93, 100, 75, 86, 70, 96, and 80.

 Interval/Grades Tally Frequency 61 - 70 71 - 80 81 - 90 91 - 100

4. Sue competed against 14 others in a time trial for the 400-meter run at the state finals. Their times have been recorded in the table below. What percent of the runners completed the time trial between 50 and 53.9 seconds?

 Interval Tally Frequency 50.0 - 50.9 0 51.0 - 51.9 || 2 52.0 - 52.9 |||| | 6 53.0 - 53.9 ||| 3 54.0 - 54.9 |||| 4

5. The following data is the starting weights of 30 adults, in pounds, who participated in a study on weight loss. Use the data to create a cumulative frequency table. Determine appropriate intervals for the weights given.

195, 206, 100, 98, 150, 210, 195, 106, 195, 168, 180, 212, 104, 195, 100, 216, 195, 209, 112, 99, 206, 116, 195, 100, 142, 100, 135, 98, 160, 155

 Interval Frequency Cumulative Frequency

6. The following table shows the weights in pounds for students attending a “Get Fit” summer program. Use the data to find cumulative frequency.

 Interval Frequency Cumulative Frequency 91-100 6 101-110 3 111-120 0 121-130 3 131-140 0 141-150 2 151-160 2

7. 20 students were asked how many meals their family eats out in one week. The results are listed below. Complete the frequency table using this data.

{6, 5, 4, 5, 0, 7, 1, 5, 4, 4, 3, 2, 2, 3, 2, 4, 3, 4, 0, 7}

 Interval Tally Frequency Cumulative Frequency 0-1 2-3 4-5 6-7

8. The following graph shows the participants in Jane’s aerobics classes. How many total students does she teach on the given day?

CC BY-NC-SA

9. The following table shows a cumulative frequency distribution of the ages of tri-athletes. According to the table, how many tri-athletes are in their forties?

 Age Group Total 20 - 29 8 20 - 39 18 20 - 49 25 20 - 59 31 20 - 69 35

10. The quiz scores for a job placement math assessment for the 10 applicants were: 61, 67, 81, 83, 87, 88, 89, 90, 98, and 100. Which frequency table is accurate for this set of data?

 Interval Freq Interval Freq Interval Freq Interval Freq 61 - 70 2 61 - 70 2 61 - 70 2 61 - 70 2 71 - 80 0 71 - 80 2 71 - 80 0 71 - 80 2 81 - 90 8 81 - 90 8 81 - 90 6 81 - 90 7 91 - 100 10 91 - 100 10 91 - 100 2 91 - 100 10

It is that time of year again. It is Easter, and it is time to hunt eggs. The kids have been divided up into age groups, and their eggs hidden in different areas of a large park. The adult leaders have gathered the data from the three age groups: “Kinders”, “Pee Wee”, and “Big Eggs”, and compiled the results into frequency tables. The problem is, each of the adult leaders reported their information differently.

For questions 11-13, use the data gathered by each adult leader and follow the instructions for interpreting the data given by each.

11. Team “Kinders” coach reported the following results for the number of eggs found by the kiddos in their group, convert them to a cumulative frequency chart.

 Time (s) in minutes Frequency of Eggs Found 0 - 5 20 5 - 10 35 10 - 15 15 15 - 20 22 20 - 25 8 25 - 30 9

12. Team “Pee Wee” also did a great job of finding eggs. Their results were recorded in a cumulative frequency table, convert to a frequency table.

 Time (s) in minutes Cumulative Frequency of Eggs Found 0 - 5 20 5 - 10 27 10 - 15 31 15 - 20 31 20 - 25 37 25 - 30 50

13. The “Big Eggs” were the champions the last three years running. They are known for always collecting the same amount of eggs in each interval. This year they collected a total of 120 eggs.

1. Complete a frequency chart
2. Complete a cumulative frequency chart

## Vocabulary

Term Definition
cumulative frequency Cumulative frequency is used to determine the number of observations that lie above (or below) a particular value in a data set.
cumulative frequency table A cumulative frequency table shows how the cumulative frequency after each successive interval compares to the total frequency.
relative cumulative frequency Relative cumulative frequency is a running total of relative frequencies of all values up to and including the current category.