Frequency Distributions and Graphs PDF

# Frequency Distributions and Graphs ## Chapter 2 ### University of Technology and Applied Sciences **Course**: Managerial Statistics **Course Code**: BSMS2209 **Specialization**: COLLEGE REQUIREMENT **College of Economics and Business Administration** ### Course Objective - This course emphasi...

# Frequency Distributions and Graphs ## Chapter 2 ### University of Technology and Applied Sciences **Course**: Managerial Statistics **Course Code**: BSMS2209 **Specialization**: COLLEGE REQUIREMENT **College of Economics and Business Administration** ### Course Objective - This course emphasizes the development of practical computing skills and problem-solving skills in the field of business, using appropriate statistical tools and graphs, as well as the use of spreadsheets for data management and analysis. ### Learning Outcome - 2 - Create and interpret tables and graphs to organize and present various data sets. ### Chapter 2- OUTLINE - ORGANIZING DATA - FREQUENCY DISTRIBUTION - DISTRIBUTION - TYPES OF FREQUENCY DISTRIBUTIONS - CUMULATIVE FREQUENCY DISTRIBUTION - GRAPHICAL PRESENTATION ### ORGANIZING DATA - A table with 25 rows and 2 columns is displayed. The first column is "Person" and the second column is "Blood type". - The "Person" column includes the numbers 1- 25. - The "Blood type" column includes the blood types A, B, AB, and O. - **By using descriptive statistical methods, we transform raw data into...** - Frequency Distribution (Tables) - Graphs ### 1. Frequency Distributions - A table with 4 rows and 2 columns is displayed. The first column is "Blood type" and the second column is "Number of inductees". - Included in the "Blood type" column are the blood types A, B, O and AB. - Included in the "Number of inductees" column are the numbers 5, 7, 9 and 4. ### 2. Graphs - A pie chart is displayed. The pie chart is divided into 4 sections, each representing a different blood type. - **A** represents 20% - **B** represents 28% - **O** represents 36% - **AB** represents 16% ### FREQUENCY DISTRIBUTION - A frequency distribution is the organization of **raw data** in **table** form, using **classes** and **frequencies**. - When the data are in original form, they are called **raw data** - A **class** is a quantitative or qualitative aspect in which data are accordingly distributed. - A **frequency** of a class is the number of data values placed in this class. ### Examples: - **Raw Data Example** - 49 57 38 73 81 - 74 59 76 65 69 - 54 56 69 68 78 - 65 85 49 69 61 - 48 81 68 37 43 - 78 82 43 64 67 - 52 56 81 77 79 - 85 40 85 59 80 - 60 71 57 61 69 - 61 83 90 87 74 - **Frequency Distribution Table Example** - A table with 9 rows and 3 columns is displayed. The first column is "Class limits", the second column is "Tally" and the third column is "Frequency". - The class limits section includes the following numbers: - 35-41 - 42-48 - 49-55 - 56-62 - 63-69 - 70-76 - 77-83 - 84-90 - The tally column includes marks representing the frequency of each class. - The frequency column lists the numerical representation of data tallied in the tally column. ### DISTRIBUTION - The distribution of a variable refers to the set of all possible values of a variable and the associated frequencies or probabilities with which these values occur. - Sometimes variables are distributed so that all outcomes are equally, or nearly equally likely. Other variables show results that "cluster" around one (or more) particular value. - A heterogeneous distribution is a distribution of values of a variable where all outcomes are nearly equally likely. - A homogeneous distribution is a distribution of values of a variable that cluster around one or more values, while other values are occurring with very low frequencies or probabilities. ### Heterogeneous distribution vs Homogeneous distribution -Example - Example: Suppose a company issues sales reports for two years, 2004 and 2005, as shown in the table and picture below. We can consider this report to consist of two variables (v₂₀₀₄ and v₂₀₀₅, say), each one having 4 values (for North, South, East, and West, respectively). Are the distributions of values hetero- or homogeneous? - A table with 5 rows and 4 columns is displayed. The first column is "Sales Report" and the second column is "2004", the third column is "2005" and the fourth column is "Total". - The "Sales Report" column includes the following categories: - North - South - East - West - Total - A bar graph is displayed, which shows the sales report for both 2004 and 2005. ### TYPES OF FREQUENCY DISTRIBUTIONS - Categorical frequency distribution is used when in the cases of nominal-level or ordinal- level data. - Ungrouped frequency distribution is used in the case of quantitative data with relatively small range or when the data are discrete. - Grouped frequency distribution is used in the case of quantitative data with a large range. ### Categorical frequency distribution - Twenty-five army inductees were given a blood test to determine their blood type. - The data set is: - A, B, B, AB, O, O, B, AB, B, B, B, O, A, O, A, O, O, O, AB, AB, A, O, B, A - This is called the **sample size**. - Construct a frequency distribution for the data: - **Solution- Practice** - A table with 5 rows and 3 columns is displayed. The first column is "Class", the second column is "Tally" and the third column is "Frequency". - The Class Column sections include: - A - B - O - AB - Total - **Solution** - A table with 5 rows and 2 columns is displayed. The first column is "Blood type (Class)" and the second column is "Number of inductees (Frequency)". - The Blood type Column sections include: - A - B - O - AB - Total - The "Number of inductees (Frequency)" column includes the numbers 5, 7, 9 and 4. ### EXERCISE: 1 - Following are the grades scored by the students in Business Mathematics. You are required to construct a frequency distribution. - A table filled with letter grades is displayed, which are: - A, B, A, C, D, F, B, D, A - C, B, A, A, D, D, F, F, F - D, C, C, D, A, A, B, B, A - D, C, F, F, D, A, A, C, D - C, B, A, A, D, D, F, F, F - D, C, C, D, A, A, B, B, A - A, B, A, C, D, F, B, D, A ### Ungrouped frequency distribution - Given below are the marks obtained by 20 students in Accounting out of 25. Construct a frequency distribution for the data. - 21, 23, 19, 17, 12, 15, 15, 17, 17, 19, 23, 23, 21, 23, 25, 25, 21, 19, 19, 19 - **Solution:** - A table with 7 rows and 3 columns is displayed. The first column is "Marks", the second column is "Tally Bars" and the third column is "Frequency". - The "Marks" column includes the following numbers: - 12 - 15 - 17 - 19 - 21 - 23 - 25 - The "Tally Bars" column includes marks representing the frequency of each mark. - The "Frequency" column lists the numerical representation of data tallied in the tally column. ### EXERCISE-1 - From the below information construct an ungrouped frequency distribution table. - A table with 5 rows and 5 columns is displayed. The table is filled with a series of numbers: - 5, 10, 5, 15, 20 - 5, 15, 10, 20, 10 - 15, 20, 10, 15, 5 - 5, 5, 5, 10, 15 - 15, 15, 10, 20, 15 - 5, 15, 10, 15, 10 ### EXERCISE-2 - The data shown here represent the number of miles per gallon (mpg) that 30 selected four-wheel-drive sports utility vehicles obtained in city driving. Construct a frequency distribution. - The following numbers are listed: - 12, 17, 12, 14, 16, 18 - 16, 18, 12, 16, 17, 15 - 15, 16, 12, 15, 16, 16 - 12, 14, 15, 12, 15, 15 - 19, 13, 16, 18, 16, 14 - - **Source:** Model Year Fuel Economy Guide. United States Environmental Protection Agency. ### Groped frequency distribution - In grouped frequency distribution, we need to find classes and frequencies: - **Important terms in Grouped Frequency Distribution:** - 1. Class interval/width - 2. Class limit - 3. Inclusive method - 4. Exclusive method ### Groped frequency distribution - The steps for constructing a grouped frequency distribution are summarized in the following Procedure Table: - ***Procedure Table*** - **Constructing a Grouped Frequency Distribution** - **Step 1**: Determine the classes. - Find the highest and lowest values. - Find the range. - Select the number of classes desired. - Find the width by dividing the range by the number of classes and rounding up. - Select a starting point (usually the lowest value or any convenient number less than the lowest value); add the width to get the lower limits. - Find the upper class limits. - Find the boundaries. - **Step 2**: Tally the data. - **Step 3**: Find the numerical frequencies from the tallies, and find the cumulative frequencies. ### Problem - Weekly wages for the 15 workers in a company are listed below. Construct a frequency distribution with 5 classes. - 21 10 16 32 23 - 23 26 20 29 34 - 18 22 19 27 20 ### Solution - **Step 1**: Determine the classes. - Find the highest and lowest value. H = 34 and L = 10 - Find the range: R = Highest value – Lowest Value - R = 34 – 10 = 24 - Select the number of classes desired (usually between 5 and 20). In this case, 5 classes - Find the class width by dividing the range by the number of classes - **Width** = (Range)/(Number of classes) = 24/5 = 4.8 (round off to 5) ### Solution Contd... - **Step 2**: tally the data. - **Step 3**: Find the numerical frequencies from the tallies - A table with 5 rows and 3 columns is displayed. The first column is "Class Limit", the second column is "Tally" and the third column is "Frequencies". - The "Class Limit" column sections include: - 10-15 - 15-20 - 20-25 - 25-30 - 30-35 - The "Tally" column includes marks representing the frequency of each class limit. - The "Frequencies" column lists the numerical representation of data tallied in the tally column. ### EXERCISE-1 - These data represent the record high temperatures in degrees Fahrenheit (F) for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes. - 112 100 127 120 134 118 105 110 109 112 - 110 118 117 116 118 122 114 114 105 109 - 107 112 114 115 118 117 118 122 106 110 - 116 108 110 121 113 120 119 111 104 111 - 120 113 120 117 105 110 118 112 114 114 - - **Source:** The World Almanac and Book of Facts. ### EXERCISE-2 - The data shown are the number of grams per serving of 30 selected brands of cakes. Construct a frequency distribution using 5 classes. - The following numbers are listed: - 32 47 51 41 46 30 - 46 38 34 34 52 48 - 48 38 43 41 21 24 - 25 29 33 45 51 32 - 32 27 23 23 34 35 - **Source:** The Complete Food Counts. ### CUMULATIVE FREQUENCY DISTRIBUTION - A cumulative frequency distribution is a distribution that shows the number of data values **less than or equal to each upper boundary**. - The values are found by adding the frequencies of the classes less than or equal to the upper class boundary of a specific class. - This gives an ascending cumulative frequency. The last – cumulative frequency must be equal to n(Sample size/ total frequency). ### Cumulative Frequencies and Relative Frequencies (Less than cumulative frequency) - A table is displayed with 7 rows and 4 columns. The first column is "Class Limit", the second column is "Frequencies", the third column is "Less than Cumulative Frequencies" and the fourth column is "Relative Frequencies (%)". - The "Class Limit" column sections include: - 10-15 - 15-20 - 20-25 - 25-30 - 30-35 - Total - The "Less than Cumulative Frequencies" column includes the numbers 1, 4, 10, 13, 15, and 15. - The "Relative Frequencies (%)" column includes the following percentages: - 6.67% - 20% - 33.33% - 40% - 20% - 100% ### Cumulative Frequencies and Relative Frequencies (More than cumulative frequency) - A table is displayed with 7 rows and 4 columns. The first column is "Class Limit", the second column is "Frequencies", the third column is "More than Cumulative Frequencies" and the fourth column is "Relative Frequencies (%)". - The "Class Limit" column sections include: - 10-15 - 15-20 - 20-25 - 25-30 - 30-35 - Total - The "More than Cumulative Frequencies" column includes the numbers 1, 14, 11, 5, 2, and 15. - The "Relative Frequencies (%)" column includes the following percentages: - 6.67% - 20% - 33.33% - 40% - 20% - 100% ### EXERCISE-3 - Following are the marks obtained by 25 students in an examination. Prepare a grouped frequency distribution with 6 classes. Also find out less than cumulative frequencies and relative frequencies. - 25 15 26 65 41 - 55 35 54 78 55 - 65 22 32 62 22 - 45 13 15 16 46 - 62 19 42 33 24 ### EXERCISE-4 - Construct a grouped frequency distribution with 7 classes. Also find out more than cumulative frequencies, less than cumulative frequencies and relative frequencies. - 102 104 140 136 152 132 158 193 128 141 - 130 133 147 148 141 129 133 137 179 147 - 152 114 124 138 129 164 135 128 139 154 - 168 148 152 116 107 136 167 143 139 152 ### GRAPHICAL PRESENTATION - The most commonly used Graphs are: - 1. Bar Chart - 2. Pie Chart - 3. The histogram. - 4. The frequency polygon. - 5. The cumulative frequency graph, or ogive (pronounced o-jive). ### BAR DIAGRAM/CHART - A bar chart presents categorical data/ungrouped frequency distributions, i.e., it is typically used with qualitative data. - A bar graph represents the data by using vertical or horizontal bars whose heights or lengths represent the frequencies of the data. ### EXERCISE-1 - The table shown here displays the number of students joined in different specialization in a college during 2018. Construct a bar chart for the data: - A table is displayed with 4 rows and 2 columns. The first column is "Specializations" and the second column is "Frequency". - The "Specializations" column sections include: - Accounting - Human Resource - Marketing - E-Business - The "Frequency" column includes the numbers 200, 260, 120, and 90. ### EXERCISE-2 - The table shows the average money spent by first-year college students. Draw a horizontal and vertical bar graph for the data. - A table is displayed with 4 rows and 2 columns. The first column is "Specializations" and the second column is "Frequency". - The "Specializations" column sections include: - Electronics - Dorm decor - Clothing - Shoes - The "Frequency" column includes the numbers $728, $344, $141, and $72. ### PIE DIAGRAM/CHART - A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution. - Pie charts represent categorical data. Not suitable for numerical data. - **Remember:** the degree of a class in a pie chart is defined as: - Degrees = $\frac{f}{n} . 360°$ - where f = frequency for each class - n = sum of the frequencies ### EXERCISE-1 - The favorite flavors of ice-cream for the children in a locality are given below. Draw a pie chart to represent the given information. - A table is displayed with 2 rows and 6 columns. The first column is "Flavors", the second column is "Vanilla", the third column is "Strawberry", the fourth column is "Chocolate", the fifth column is "Kesar-pista" and the sixth column is "Mango zap". - The "Flavors" row has "Number of children" in the second row. - The "Vanilla" column includes the number 50. - The "Strawberry" column includes the number 30. - The "Chocolate" column includes the number 20. - The "Kesar-pista" column includes the number 60. - The "Mango zap" column includes the number 40. ### Solution - A table is displayed with 6 rows and 2 columns. The first column is "Flavors" and the second column is "Percentage of Children". - The "Flavors" column includes the following categories: - Vanilla - Strawberry - Chocolate - Kesar-Pista - Mango Zap - The "Percentage of Children" column includes the following percentages: - 25% - 15% - 10% - 30% - 20% - A pie chart is displayed that shows: - Vanilla: 25% - Strawberry: 15% - Chocolate: 10% - Kesar-Pista: 30% - Mango Zap: 20% - **Note:** Pie chart is drawn based on percentages of frequencies ### EXERCISE-2 - This frequency distribution shows the number of pounds of each snack food eaten during the Super Bowl. Construct a pie graph for the data. - A table is displayed with 6 rows and 2 columns. The first column is "Snack" and the second column is "Pounds (frequency)". - The "Snack" column includes the following categories: - Potato chips - Tortilla chips - Pretzels - Popcorn - Snack nuts - The "Pounds (frequency)" column includes the following numbers: - 11.2 million - 8.2 million - 4.3 million - 3.8 million - 2.5 million - Total n = 30.0 million - **Source:** USA TODAY Weekend.