2.1: Frequency and Group Frequency Distribution
- Page ID
- 43575
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible.
Answers and Rounding Off
A simple way to round off answers is to carry your final answer one more decimal place than was present in the original data. Round off only the final answer. Do not round off any intermediate results, if possible. If it becomes necessary to round off intermediate results, carry them to at least twice as many decimal places as the final answer. For example, the average of the three quiz scores four, six, and nine is 6.3, rounded off to the nearest tenth, because the data are whole numbers. Most answers will be rounded off in this manner.
It is not necessary to reduce most fractions in this course. Especially in Probability Topics, the chapter on probability, it is more helpful to leave an answer as an unreduced fraction.
Frequency
Frequency distribution shows how data are partitioned among several categories by listing the categories along with the number of data values in each of them.
According to Table Table \(\PageIndex{1}\), there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:
5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3.
Table lists the different data values in ascending order and their frequencies.
| DATA VALUE (Hours worked) | FREQUENCY (number students) |
|---|---|
| 2 | 3 |
| 3 | 5 |
| 4 | 3 |
| 5 | 6 |
| 6 | 2 |
| 7 | 1 |
A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.
| DATA VALUE (Hours worked) | FREQUENCY | RELATIVE FREQUENCY |
|---|---|---|
| 2 | 3 | \(\frac{3}{20}\) or 0.15 |
| 3 | 5 | \(\frac{5}{20}\) or 0.25 |
| 4 | 3 | \(\frac{3}{20}\) or 0.15 |
| 5 | 6 | \(\frac{6}{20}\) or 0.30 |
| 6 | 2 | \(\frac{2}{20}\) or 0.10 |
| 7 | 1 | \(\frac{1}{20}\) or 0.05 |
The sum of the values in the relative frequency column of Table \(\PageIndex{2}\) is \(\frac{20}{20}\), or 1.
Cumulative frequency is the accumulation of the previous frequencies. To find the cumulative frequencies, add all the previous frequencies to the frequency for the current row, as shown in Table \(\PageIndex{3}\).
Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table \(\PageIndex{3}\).
| DATA VALUE (Hours worked) | FREQUENCY | CUMULATIVE FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|---|
| 2 | 3 | 3 | \(\frac{3}{20}\) or 0.15 | 0.15 |
| 3 | 5 | 3+5 = 8 | \(\frac{5}{20}\) or 0.25 | 0.15+0.25 = 0.40 |
| 4 | 3 | 8 + 3 = 11 | \(\frac{3}{20}\) or 0.15 | 0.40 + 0.15 = 0.55 |
| 5 | 6 | 11 + 6 = 17 | \(\frac{6}{20}\) or 0.30 | 0.55 + 0.30 = 0.85 |
| 6 | 2 | 17 + 2 = 19 | \(\frac{2}{20}\) or 0.10 | 0.85 + 0.10 = 0.95 |
| 7 | 1 | 19 + 1= 20 | \(\frac{1}{20}\) or 0.05 | 0.95 + 0.05= 1.00 |
The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.
Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.
Table \(\PageIndex{1}\) represents the amount of student hours worked. What fraction of surveyed students worked between 3 and 4 hours.
- Answer
-
\(\frac{14}{20}\)
Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows: 2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10. Table \(\PageIndex{4}\) was produced:
| DATA (miles) | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|
| 3 | 3 | \(\frac{3}{19}\) | 0.1579 |
| 4 | 1 | \(\frac{1}{19}\) | 0.2105 |
| 5 | 3 | \(\frac{3}{19}\) | 0.1579 |
| 7 | 2 | \(\frac{2}{19}\) | 0.2632 |
| 10 | 3 | \(\frac{3}{19}\) | 0.4737 |
| 12 | 2 | \(\frac{2}{19}\) | 0.7895 |
| 13 | 1 | \(\frac{1}{19}\) | 0.8421 |
| 15 | 1 | \(\frac{1}{19}\) | 0.8948 |
| 18 | 1 | \(\frac{1}{19}\) | 0.9474 |
| 20 | 1 | \(\frac{1}{19}\) | 1.0000 |
- Is the table correct? If it is not correct, what is wrong?
- True or False: Three percent of the people surveyed commute three miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections.
- What fraction of the people surveyed commute five or seven miles?
- What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)?
Answer
- No. The frequency column sums to 18, not 19. Not all cumulative relative frequencies are correct.
- False. The frequency for three miles should be one; for two miles (left out), two. The cumulative relative frequency column should read: 0.1052, 0.1579, 0.2105, 0.3684, 0.4737, 0.6316, 0.7368, 0.7895, 0.8421, 0.9474, 1.0000.
- \(\frac{5}{19}\)
- \(\frac{7}{19}\), \(\frac{12}{19}\), \(\frac{7}{19}\)
Table \(\PageIndex{7}\) contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012.
| Year | Total Number of Deaths |
|---|---|
| 2000 | 231 |
| 2001 | 21,357 |
| 2002 | 11,685 |
| 2003 | 33,819 |
| 2004 | 228,802 |
| 2005 | 88,003 |
| 2006 | 6,605 |
| 2007 | 712 |
| 2008 | 88,011 |
| 2009 | 1,790 |
| 2010 | 320,120 |
| 2011 | 21,953 |
| 2012 | 768 |
| Total | 823,356 |
Answer the following questions.
- What is the frequency of deaths measured from 2006 through 2009?
- What percentage of deaths occurred after 2009?
- What is the relative frequency of deaths that occurred in 2003 or earlier?
- What is the percentage of deaths that occurred in 2004?
- What kind of data are the numbers of deaths?
- The Richter scale is used to quantify the energy produced by an earthquake. Examples of Richter scale numbers are 2.3, 4.0, 6.1, and 7.0. What kind of data are these numbers?
Answer
- 97,118 (11.8%)
- 41.6%
- 67,092/823,356 or 0.081 or 8.1 %
- 27.8%
- Quantitative discrete
- Quantitative continuous
Table \(\PageIndex{6}\) contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 2011.
| Year | Total Number of Crashes | Year | Total Number of Crashes |
|---|---|---|---|
| 1994 | 36,254 | 2004 | 38,444 |
| 1995 | 37,241 | 2005 | 39,252 |
| 1996 | 37,494 | 2006 | 38,648 |
| 1997 | 37,324 | 2007 | 37,435 |
| 1998 | 37,107 | 2008 | 34,172 |
| 1999 | 37,140 | 2009 | 30,862 |
| 2000 | 37,526 | 2010 | 30,296 |
| 2001 | 37,862 | 2011 | 29,757 |
| 2002 | 38,491 | Total | 653,782 |
| 2003 | 38,477 |
Answer the following questions.
- What is the frequency of deaths measured from 2000 through 2004?
- What percentage of deaths occurred after 2006?
- What is the relative frequency of deaths that occurred in 2000 or before?
- What is the percentage of deaths that occurred in 2011?
- What is the cumulative relative frequency for 2006? Explain what this number tells you about the data.
- Answer
-
- 190,800 (29.2%)
- 24.9%
- 260,086/653,782 or 39.8%
- 4.6%
- 75.1% of all fatal traffic crashes for the period from 1994 to 2011 happened from 1994 to 2006.
Table \(\PageIndex{6}\) contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 2011.
| Year | Total Number of Crashes | Year | Total Number of Crashes |
|---|---|---|---|
| 1994 | 36,254 | 2004 | 38,444 |
| 1995 | 37,241 | 2005 | 39,252 |
| 1996 | 37,494 | 2006 | 38,648 |
| 1997 | 37,324 | 2007 | 37,435 |
| 1998 | 37,107 | 2008 | 34,172 |
| 1999 | 37,140 | 2009 | 30,862 |
| 2000 | 37,526 | 2010 | 30,296 |
| 2001 | 37,862 | 2011 | 29,757 |
| 2002 | 38,491 | Total | 653,782 |
| 2003 | 38,477 |
Answer the following questions.
- What is the frequency of deaths measured from 2000 through 2004?
- What percentage of deaths occurred after 2006?
- What is the relative frequency of deaths that occurred in 2000 or before?
- What is the percentage of deaths that occurred in 2011?
- What is the cumulative relative frequency for 2006? Explain what this number tells you about the data.
- Answer
- 190,800 (29.2%)
- 24.9%
- 260,086/653,782 or 39.8%
- 4.6%
- 75.1% of all fatal traffic crashes for the period from 1994 to 2011 happened from 1994 to 2006.
An insurance company determines vehicle insurance premiums based on known risk factors. If a person is considered a higher risk, their premiums will be higher. One potential factor is the color of your car. The insurance company believes that people with some colors of cars are more likely to ve involved in accidents. To research this, the insurance company examines police reports for recent total-loss collisions. The data are as follows: blue, green, grey, white, black, white, grey, black, grey, red, black, red, green, red, white, black, white, red, green, blue, green, red, green, green, and blue.
Solution
| Color | Frequency |
|---|---|
| Blue | 3 |
| Green | 6 |
| Red | 5 |
| Grey | 3 |
| White | 4 |
| Black | 4 |
| Total | 25 |
Group Frequency
Table \(\PageIndex{8}\) represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.
| HEIGHTS (INCHES) | FREQUENCY | RELATIVE FREQUENCY | CUMULATIVE RELATIVE FREQUENCY |
|---|---|---|---|
| 59.95–61.95 | 5 | \(\frac{5}{100} = 0.05\) | \(0.05\) |
| 61.95–63.95 | 3 | \(\frac{3}{100} = 0.03\) | \(0.05 + 0.03 = 0.08\) |
| 63.95–65.95 | 15 | \(\frac{15}{100} = 0.15\) | \(0.08 + 0.15 = 0.23\) |
| 65.95–67.95 | 40 | \(\frac{40}{100} = 0.40\) | \(0.23 + 0.40 = 0.63\) |
| 67.95–69.95 | 17 | \(\frac{17}{100} = 0.17\) | \(0.63 + 0.17 = 0.80\) |
| 69.95–71.95 | 12 | \(\frac{12}{100} = 0.12\) | \(0.80 + 0.12 = 0.92\) |
| 71.95–73.95 | 7 | \(\frac{7}{100} = 0.07\) | \(0.92 + 0.07 = 0.99\) |
| 73.95–75.95 | 1 | \(\frac{1}{100} = 0.01\) | \(0.99 + 0.01 = 1.00\) |
| Total = 100 | Total = 1.00 |
The data in this table have been grouped into the following intervals:
- 61.95 to 63.95 inches
- 63.95 to 65.95 inches
- 65.95 to 67.95 inches
- 67.95 to 69.95 inches
- 69.95 to 71.95 inches
- 71.95 to 73.95 inches
- 73.95 to 75.95 inches
In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints.
The grouped frequency distribution of soccer players heights can be expressed using the following terminology:
Lower class limits are the smallest numbers that can belong to each of the different classes.
Upper class limits are the largest numbers that can belong to each of the different classes.
Class boundaries are the numbers used to separate the classes, but without the gaps created by class limits.
Class midpoints are the values in the middle of the classes.
midpoint = \(\frac{upper limit + lower limit}{2}\)
| Class Limits | Class Boundaries | Class Midpoint | FREQUENCY |
|---|---|---|---|
| 60-61 | 59.95–61.95 | \(\frac{61+60}{2}=60.5\) | 5 |
| 62-63 | 61.95–63.95 | \(\frac{63+62}{2}= 62.5\) | 3 |
| 64-65 | 63.95–65.95 | \(\frac{65+64}{2}= 64.5\) | 15 |
| 66-67 | 65.95–67.95 | \(\frac{67+66}{2}=66.5\) | 40 |
| 68-69 | 67.95–69.95 | \(\frac{69+68}{2}=68.5 \) | 17 |
| 70-71 | 69.95–71.95 | \(\frac{71+70}{2}= 70.5\) | 12 |
| 72-73 | 71.95–73.95 | \(\frac{73+72}{2}= 72.5\) | 7 |
| 74-75 | 73.95–75.95 | \(\frac{75+74}{2}= 74.5\) | 1 |
| Total = 100 |
Additional, the grouped frequency distribution of soccer players heights has a class width of two since the difference between the first class lower limit, 60 and second class lower limit, 62 is 2.
Class width is the difference between two consecutive lower class limits in a frequency distribution.
- From the Table \(\PageIndex{8}\), find the percentage of heights that are less than 65.95 inches.
- Find the percentage of heights that fall between 61.95 and 65.95 inches.
- Answer
-
- If you look at the first, second, and third rows, the heights are all less than 65.95 inches. There are \(5 + 3 + 15 = 23\) players whose heights are less than 65.95 inches. The percentage of heights less than 65.95 inches is then \(\frac{23}{100}\) or 23%.
- This percentage is the cumulative relative frequency entry in the third row. Add the relative frequencies in the second and third rows: \(0.03 + 0.15 = 0.18\) or 18%.
Use the heights of the 100 male semiprofessional soccer players in Table \(\PageIndex{8}\). Fill in the blanks and check your answers.
- The percentage of heights that are from 67.95 to 71.95 inches is: ____.
- The percentage of heights that are from 67.95 to 73.95 inches is: ____.
- The percentage of heights that are more than 65.95 inches is: ____.
- The number of players in the sample who are between 61.95 and 71.95 inches tall is: ____.
- What kind of data are the heights?
- Describe how you could gather this data (the heights) so that the data are characteristic of all male semiprofessional soccer players.
Remember, you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.
- Answer
-
- 29%
- 36%
- 77%
- 87
- quantitative continuous
- Get rosters from each team and choose a simple random sample from each
Table \(\PageIndex{9}\) shows the amount, in inches, of annual rainfall in a sample of towns.
- Find the percentage of rainfall that is less than 9.01 inches.
- Find the percentage of rainfall that is between 6.99 and 13.05 inches.
| Rainfall (Inches) | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 2.95–4.97 | 6 | \(\frac{6}{50} = 0.12\) | \(0.12\) |
| 4.97–6.99 | 7 | \(\frac{7}{50} = 0.14\) | \(0.12 + 0.14 = 0.26\) |
| 6.99–9.01 | 15 | \(\frac{15}{50} = 0.30\) | \(0.26 + 0.30 = 0.56\) |
| 9.01–11.03 | 8 | \(\frac{8}{50} = 0.16\) | \(0.56 + 0.16 = 0.72\) |
| 11.03–13.05 | 9 | \(\frac{9}{50} = 0.18\) | \(0.72 + 0.18 = 0.90\) |
| 13.05–15.07 | 5 | \(\frac{5}{50} = 0.10\) | \(0.90 + 0.10 = 1.00\) |
| Total = 50 | Total = 1.00 |
- Answer
-
- \(0.56\) or \(56%\)
- \(0.30 + 0.16 + 0.18 = 0.64\) or \(64%\)
From Table \(\PageIndex{9}\), find the number of towns that have rainfall between 2.95 and 9.01 inches.
- Answer
-
\(6 + 7 + 15 = 28\) towns
In your class, have someone conduct a survey of the number of siblings (brothers and sisters) each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions:
- What percentage of the students in your class have no siblings?
- What percentage of the students have from one to three siblings?
- What percentage of the students have fewer than three siblings?
References
- “State & County QuickFacts,” U.S. Census Bureau. quickfacts.census.gov/qfd/download_data.html (accessed May 1, 2013).
- “State & County QuickFacts: Quick, easy access to facts about people, business, and geography,” U.S. Census Bureau. quickfacts.census.gov/qfd/index.html (accessed May 1, 2013).
- “Table 5: Direct hits by mainland United States Hurricanes (1851-2004),” National Hurricane Center, National Hurricane Table [www.nhc.noaa.gov] [www.nhc.noaa.gov] (accessed May 1, 2013).
- “Levels of Measurement,” infinity.cos.edu/faculty/wood...ata_Levels.htm (accessed May 1, 2013).
- Courtney Taylor, “Levels of Measurement,” about.com, Levels of Measurement in Statistics [statistics.about.com] (accessed May 1, 2013).
- David Lane. “Levels of Measurement,” Connexions, Levels of Measurement in Statistics [cnx.org] (accessed May 1, 2013).
Review
Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round off your final answer to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth. When organizing data, it is important to know how many times a value appears. How many statistics students study five hours or more for an exam? What percent of families on our block own two pets?
Glossary
- Frequency
- The number of times a value of the data occurs.
- Relative Frequency
- The ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes to the total number of outcomes.
- Cumulative Relative Frequency
- The term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value.