Jubail University College MATH 117 Statistical Methods for Management I Chapter Two There are a wide variety of ways to summarize, organize, and present data: I. Tables 1. Distribution Table (Categorical or Numeric discrete) 2. Grouped Distribution Table (for Interval or Ratio data) 3. Joint Table for Two Variables (Qualitative or Quantitative) II. Graphs 1. Bar Chart (Vertical, Horizontal, Clustered, Stacked) 2. Pie Chart 3. Histogram 4. Pareto Charts 5. Polygon 6. Ogive 7. Stem & leaf 8. Line chart (Time Series) 9. Scatter plots (Scatter Diagrams) 10. Pictograph Definitions: Raw Data: The Data that have not been processed or treated. Data Array: Data that have been arranged in numerical order. : The number of times a certain value or class of values occurs. (How often something occurs). Distribution Table: Lists in one column the categories or classes and in another column the corresponding frequencies. Categorical Distribution: A frequency distribution in which the data is only nominal or ordinal such as colors, blood type, Marital Status, and gender. Ungrouped Distribution: A frequency distribution of numerical data. The raw data is not grouped such as the number of absent students, number of papers sold. Grouped Data Distribution: The variable of interest is continuous such as height, weight, temperature, stock price, the data will be grouped into continuous classes such as (10-<20, 20--<30) or if the values are rounded to the nearest integer we use discrete classes such as (10-19, 20-29). For discrete variables such as total credit hours taken by students, if the ungrouped list of possible values is two long we use discrete classes. Ms. Ghaida Barghouthi, JUC, Semester 341 Page 1
Class Limits: Separate one class in a grouped frequency distribution from another. The limits could actually appear in the data and have gaps between the upper limit of one class and the lower limit of the next. For the class 10-19, the lower class limit is 10 and the upper is 19. Class Exact Limits: Separate one class in a grouped frequency distribution from another. The exact limits have one more decimal place than the raw data and therefore do not appear in the data. There is no gap between the upper exact limit of one class and the lower exact limit of next class. The exact lower class limit is found by subtracting 0.5 unit from lower class limit and exact upper class limit is found by adding 0.5 units to upper class limit. Example for the class 10-19 the exact limits are 9.5-10.5. If the raw data values are 14.5, 9.2, 8.6 and the class limits are 10.5-18.5, the exact class limits are found by adding and subtracting 0.05. The exact limit for the class 10.5-18.5 will be 10.45-18.55 Class Width: The difference between the upper and lower exact limits of any class. The class width is also the difference between lower limits of two consecutive classes or the upper limits of two consecutive classes. It is not the difference between the upper and lower limit of the same class. Class Mark (Mid-Point): The number in the middle of class. It is found by adding the upper and lower limits and dividing by two. It can also be found by adding the upper and lower boundaries and dividing by two. The midpoint for the class 10-19 is (10+19)/2=14.5 Cumulative : The number of values less than the upper class boundary for the current class is known as cumulative frequency. This is a running total of frequencies. Relative : The divided by total frequency. This gives the percentage of values falling in that class. Cumulative Relative (Relative Cumulative ): The running total of the cumulative frequency divided by the total frequency. That gives the percent of the values which are less than the upper class boundary. Joint Distribution Table: A set of data consisting of paired responses for two variables. The number of rows in the table represents the number of categories of one variable and the number of columns represents the number of categories of the second variable. Joint Relative Distribution Table: Divide each cell frequency by the total number of paired observations. Classes in a Grouped Table must be: a. Mutually exclusive: No element can belong to more than one class no overlapping b. Equal width: Include each and every class. c. All- Inclusive: Contains all possible data values. Ms. Ghaida Barghouthi, JUC, Semester 341 Page 2
Other Guidelines for Constructing Classes: a. Avoid open ended classes if possible such as 75 and over, 10 or less. b. Target between 5 to 20 classes depending on the range and number of data points. c. Keep class limits as simple as possible (multiple of class width). If class width 5 make lower limits 5, 10, 15, and 20. d. Make the width odd number so the midpoint will be a whole number. For example 10-15 the width W=6 which is even number and the midpoint is 14.5, but for the class 10-14, the class width W=5 and the mid-point is 12. Graphs: Bar Chart: A graphical representation of categorical data sets in which rectangle or bars represent category or class. The height of each bar represents the frequencies, relative frequency or percentage relative frequency of observations. Bars may be vertical or horizontal one color or different colors. Cluster Bar Chart: A clustered bar graph is used to represent discrete values for more than one item that share the same category. A clustered bar chart helps summarize your data for groups of cases. There is one cluster of bars for each value of the variable you specified under Rows. The variable that defines the bars within each cluster is the variable you specified under Columns. There is one set of differently colored or patterned bars for each value of this variable. If you specify more than one variable under Columns or Rows, a clustered bar chart is produced for each combination of two variables. Stacked Bar Chart: A bar graph has the bar divided into subparts that represent the discrete value for items that represent a portion of a whole group Pie Charts: Graphical depiction of data as slices of a pie. The frequency determines the size of the slice. The number of degrees in any slice is the relative frequency time s 360 degrees. (F/N)(360 ) Histogram: A graph which display the data by using vertical bars with (no gaps between the bars) of various heights to represent frequencies, relative frequency or percentage relative frequency. The horizontal axis represents classes of continuous data, the class marks or class exact limits. Pareto Charts: A bar graph for qualitative data with the bars arranged in descending order of frequency from left to right. Polygon: A Line graph where the frequency is placed along the vertical axis and the class midpoints are placed along the horizontal axis. The points are connected with lines. Ogive (oh-jive): A frequency polygon of the cumulative relative frequency. The vertical axis is the cumulative frequency. The horizontal axis is the class boundaries. The graph always start at zero at lowest class boundary and will end up to total frequency for cumulative frequency. Steam & Leaf Plot: A data plot which uses part of the data value as the stem and the rest of the data value (the leaf) to form groups or classes. This is very useful for sorting data quickly. Ms. Ghaida Barghouthi, JUC, Semester 341 Page 3
Line chart (Time Series): A two dimensional chart showing time on the horizontal axis and the variable of interest on the vertical axis, such as profit, price, number of cars sold Scatter Plot or Scatter Diagram: A two dimensional graphical display of two quantitative variables. The independent variable is placed on the x-axis and the dependent variable is placed on the y-axis. Pictograph: A graph that uses pictures of an object such as coins, airplanes to represent data. Examples of Tables and Graphs: Tables/ Relative / Percentage Relative Categorical data Grades Relative Percentage Relative A 1 0.05 5% B 3 0.15 15% C 9 0.45 45% D 5 0.25 25% F 2 0.10 10% Total 20 1.00 100% Column Bar Chart Pie Chart Numeric Discrete data Cars Sold Relative Percentage Relative 10 5 0.25 25% 11 4 0.20 20% 12 3 0.15 15% 13 6 0.30 30% 14 2 0.10 10% Total 20 1.00 100% Ms. Ghaida Barghouthi, JUC, Semester 341 Page 4
Column Bar Chart Pie Chart Relative (3D) Bar Chart Percentage (3D) Bar Chart Ms. Ghaida Barghouthi, JUC, Semester 341 Page 5
Grouped Tables of a data set that represent daily wages rounded to the nearest dollar/ Relative / Percentage Relative/Cumulative / Cumulative Relative Discrete classes Classes Exact Class Limits Relative Number of classes is 5. Class width is 20-10=10. Class midpoints are: 14.5, 24.5, 34.5, 44.5 and 54.5. Percentage Relative Histogram showing the distribution of Ages with x-axis is the class limits Cumulative (Less than) Cumulative Relative (Less than ) 10-19 9.5-19.5 1 0.05 5% 1 0.05 20-29 19.5-29.5 3 0.15 15% 4 0.20 30-39 29.5-39.5 9 0.50 45% 13 0.65 40-49 39.5-49.5 5 0.25 25% 18 0.90 50-59 49.5-59.5 2 0.10 10% 20 1.00 Total 20 1.00 100% Histogram showing the distribution of Ages with x-axis is the exact class limits Note: The y-axis can be frequency, relative frequency or percentage relative frequency. Grouped Tables of a data set the represent weight Relative / Percentage Relative/Cumulative/ Cumulative Relative Continuous Classes Ms. Ghaida Barghouthi, JUC, Semester 341 Page 6
Classes Bins Relative Histogram showing the distribution of Weight Percentage Relative Cumulative (Less than) Cumulative Relative (Less than) 10--<20 19.9 1 0.05 5% 1 0.05 20--<30 29.9 3 0.15 15% 4 0.20 30--<40 39.9 9 0.45 45% 13 0.65 40--<50 49.9 5 0.25 25% 18 0.90 50--<60 59.9 2 0.10 10% 20 1.00 Total 20 1.00 100% Distribution of Weight 10 8 6 4 2 0 19.9 29.9 39.9 49.9 59.9 Weight Stem and leaf constructed using Excel Add-in PHStat2 for the following data: 83, 130, 90, 178, 92, 116, 181, 138, 79, 85, 76, 146, 134, 110, 145, 156, 68, 73, 88, 162, 105, 147, 156, 93, 119, 103, 71, 74 Stem unit: 10 6 8 7 1 3 4 6 9 8 3 5 8 9 0 2 3 10 3 5 11 0 6 9 12 13 0 4 8 14 5 6 7 15 6 6 16 2 17 8 18 1 Ms. Ghaida Barghouthi, JUC, Semester 341 Page 7
Cumulative Relative Grades A 1 B 3 C 9 D 5 F 2 Total 20 Classes Midpoint 10-19 14.5 1 20-29 24.5 3 30-39 34.5 9 40-49 45.5 5 50-59 55.5 2 Exact Limit Cumulative Cumulative Relative 9.5 0 0.00 19.5 1 0.05 29.5 4 0.20 39.5 13 0.65 49.5 18 0.90 59.5 20 1.00 1.20 1.00 0.80 0.60 0.40 0.20 0.00 Ogive of Wages 9.5 19.5 29.5 39.5 49.5 59.5 Wages Line Chart of the Number of Passengers over the Years 2005-2010 Ms. Ghaida Barghouthi, JUC, Semester 341 Page 8
Year Passengers 2005 3,217 2006 2,740 2007 3,326 2008 4,040 2009 4,935 2010 5,512 Scattered Diagram of Sales versus Number of Ads Ads Sales 8 56 5 45 10 55 6 45 8 48 12 66 4 48 9 62 7 54 3 40 The chart shows a positive relation between the sales and advertising Joint Table Joint Relative Table Ms. Ghaida Barghouthi, JUC, Semester 341 Page 9
Number of Students Grades BA MIS Total A 1 0 1 B 1 2 3 C 6 3 9 D 3 2 5 F 1 1 2 Total 12 8 20 Grades BA MIS Total A 1/20=0.05 0/20=0.00 1/20=0.05 B 1/20=0.05 2/20=0.10 3/20=0.15 C 6/20=0.30 3/20=0.20 9/20=0.45 D 3/20=0.15 2/20=0.10 5/20=0.25 F 1/20=0.05 1/20=0.05 2/20=0.10 Total 12/20=0.60 8/20=0.40 20/20=1.00 Clustered (3D) Bar Chart Stacked Bar Chart Clustered (3D) Bar Chart Distribution of Grades by Major 6 5 4 3 2 1 0 BA Majors MIS A B C D F Different Pictographs Ms. Ghaida Barghouthi, JUC, Semester 341 Page 10
Ms. Ghaida Barghouthi, JUC, Semester 341 Page 11