1/1/1 Organizing and Presenting Data Tabulation and Graphs Introduction to Biostatistics Haleema Masud Going back to the definition of Biostatistics The collection, organization, summarization, analysis, presentation and dissemination of DATA, and The drawing of inferences about about POPULATION from the SAMPLE observed.1 Learning Objectives Overall: To give students a basic understanding of best way of organizing and presenting data Specific: Students will be able to Understand how data can be appropriately organized and displayed. Draw Tables Draw Graphs Make Frequency distribution Descriptive statistics involves arranging, summarizing, and presenting a set of data in such a way that useful information is produced. Data Statistics Information Descriptive statistics make use of graphical techniques and numerical techniques (such as averages) to summarize and present the data.. Data Nominal Data Quantitative Qualitative The only allowable calculation on nominal data is to count the frequency of each value of the variable. Discrete Continuous Nominal Ordinal We can organize & summarize the data in a table that presents the categories and their counts called a frequency distribution.. 1
1/1/1 Frequency Distributions A frequency distribution for qualitative data lists all categories and the number of elements that belong to each of the categories. Example A sample of employees from large companies was selected, and these employees were asked how stressful their jobs were. The responses of these employees are recorded very represents very stressful, somewhat means somewhat stressful, and none stands for not stressful at all. 8 Example Solution Some what None Somewhat Very Very None Very Somewhat Somewhat Very Somewhat Somewhat Very Somewhat None Very None Somewhat Somewhat Very Somewhat Somewhat Very None Somewhat Very very somewhat None Somewhat Construct a frequency distribution table for these data. Table Frequency Distribution of Stress on Job Stress on Job Tally Frequency (f) Very Somewhat None 1 Sum = 1 Relative Frequency and Percentage Distributions Relative Frequency and Percentage Distributions cont. A relative frequency distribution lists the categories and the proportion with which each occurs Calculating Percentage Percentage = (Relative frequency) 1 Calculating Relative Frequency of a Category Re lative frequency of a category Frequency of that category Sum of all frequencies 11 1
1/1/1 Determine the relative frequency and percentage for the data in Table Solution Table Frequency Distribution of Stress on Job Stress on Job Tally Frequency (f) Very Somewhat None 1 Sum = Table Relative Frequency and Percentage Distributions of Stress on Job Stress on Job Relative Frequency Percentage Very Somewhat None 1/ =. / =. / =..(1) =..(1) =..(1) =. Sum = 1. Sum = 1 1 Nominal Data (Tabular Summary) - Organization/classification Tabulation The diagrammatic or graphical representation.1 1 Nominal Data (Frequency) Nominal Data (Relative Frequency) Bar Charts are often used to display frequencies Is there a better way to order these? Would Bar Chart Pie Charts show relative frequencies look different if we plotted relative frequency rather than frequency?.1.18
Frequency 1/1/1 Graphical Presentation of Qualitative Data Figure Bar graph for the frequency distribution of Table Definition A graph made of bars whose heights represent the frequencies of respective categories is called a bar graph. It is used to display and compare the number, frequency or other measure (e.g. mean) for different discrete categories or groups. 1 1 1 8 Very Somewhat None Strees on Job 1 Bar charts The heights or lengths of different bars are proportional to the size of the category they represent. Since the x-axis represents the different categories it has no scale. The y-axis does have a scale and this indicates the units of measurement. The bars can be drawn either vertically or horizontally. 1 Graphical Presentation of Qualitative Data cont. Definition A circle divided into portions that represent the relative frequencies or percentages of a population or a sample belonging to different categories is called a pie chart. Pie charts display how the total data are distributed between different categories Table Calculating Angle Sizes for the Pie Chart Figure Pie chart for the percentage distribution of Job Stress Stress on Job Relative Frequency Angle Size Very Somewhat None... (.) = 11.88 (.) = 18.1 (.) =. Sum = 1. Sum = Very,.% None, % Somewhat,.%
Frequency Frequency 1/1/1 Pie chart Civil status of women in a community Widowe Divorce d d8% 11% Free union % Single 8% Civil status of men in a community Free union 1% Widowe d 1% Divorce d 11% Single 1% Married % Married 1% Exercise 1. Prepare a frequency distribution of different characteristics of your class Gender Professional background From where you have got information about this institute (choose as many as applicable) Website Newspaper SMS Bill board Friend Others. Also make suitable graphs 8 Bar chart Grouped bar chart Gastrintestinal infections Gastrointestinal infections 1 Cryptos. E.histolyt. E.coli Giardia Rotavirus Shigella Agents 1 Crypt. E.histolyt. E.coli Giardia Rotavirus Shigella Agents Males Females
Year Percent 1/1/1 Bar Chart 1 Household Ownership of at Least 1 Net or ITN, 8 8 8 Any net LLIN Country 1 Country Country Country Source: Quarterly Country Summaries, 8 Stacked bar chart % Children < with Fever who Took Specific Antimalarial, -8 ACT Amodiaquine Chloroquine Quinine Sulfadoxine-Pyrimethamine Other 8 11 Percent 8 1 ORGANIZING AND GRAPHING QUANTITATIVE DATA
1/1/1 ORGANIZING AND GRAPHING QUANTITATIVE DATA Ordered array Frequency Distributions Constructing Frequency Distribution Tables Relative and Percentage Distributions Graphing Grouped Data Histograms Polygons Stem and leaf plots Organizing & Grouping Data To facilitate the calculation of various descriptive measures such as percentages and averages (Before the days of computers) The main purpose in grouping data now is summarization Summarization is a way of making it easier to understand the information in data 8 Ordered array A first step in organizing data An ordered array is a listing of the values of a collection (either population or sample) in order of magnitude from the smallest value to the largest value. If the number of measurements to be ordered is of any appreciable size, the use of a computer is highly desirable.
1/1/1 Frequency Distributions Frequency Distributions A frequency distribution for quantitative data lists all the classes and the number of values that belong to each class. Data presented in the form of a frequency distribution are called grouped data. Frequency Distributions Variable Third class Table. Weekly Earnings of 1 Employees of a Company Lower limit of the sixth class Weekly Earnings (dollars) 1 to 1 to 8 81 to 1 11 to 1 11 to 1 to 1 Class width Upper limit of the sixth class Number of Employees f 1 Frequency column Frequency of the third class Essential Question : How do we construct a frequency distribution table? Process of Constructing a Frequency Table 8
1/1/1 STEP 1. Determine the tentative number of classes (k) k = 1 +. log N Always round off Process of Constructing a Frequency Table STEP : Determine the range (R). R = Highest Value Lowest Value Note: The number of classes should be between and 1. The actual number of classes may be affected by convenience or other subjective factors STEP. Find the class width by dividing the range by the number of classes. STEP. Write the classes or categories starting with the lowest score. Stop when the class already includes the highest score. Range class width number of classes (Always round off ) c R k Add the class width to the starting point to get the second lower class limit. Add the class width to the second lower class limit to get the third, and so on. List the lower class limits in a vertical column and enter the upper class limits, which can be easily identified at this stage. When constructing frequency tables, the following guidelines should be followed. STEP. Determine the frequency for each class by referring to the tally columns and present the results in a table. The classes must be mutually exclusive. That is, each score must belong to exactly one class. Include all classes, even if the frequency might be zero.
1/1/1 Let s Try!!! All classes should have the same width, although it is sometimes impossible to avoid open ended intervals such as years or older. The number of classes should be between and 1. Time magazine collected information on all people who died from gunfire in the Philippines during one week. Here are the ages of men randomly selected from that population. Construct a frequency distribution table. 1 18 1 1 1 1 18 1 1 1 1 Determine the tentative number of classes (K). K = 1 +. log N = 1 +. log = 1 +. (1.8) =. *Round off the result to the next integer if the decimal part exceeds. K = Find the class width (c). Determine the range. R = Highest Value Lowest Value R = 1 = Range class width number of classes c R k c 8. * Round off the quotient if the decimal part exceeds. 1
1/1/1 Write the classes starting with lowest score. Classes Tally Marks Freq. 8 1 1 1 ///// ///// // /////-// /////-/////-//// /////-/////-/////-// 1 Using Table: What is the lower class limit of the highest class? Upper class limit of the lowest class? Find the class mark of the class 1. What is the frequency of the class 1? Example Classes 8 1 1 1 True Class boundaries. 8... 1... 1..... 1.. ///// ///// Tally Marks Freq. x // /////-// /////-/////-//// /////-/////-/////-// 1 8 Table. gives the total home runs hit by all players of each of the Major League Baseball teams during the season. Construct a frequency distribution table. Table. Home Runs Hit by Major League Baseball Teams During the Season Solution - Team Home Runs Team Home Runs Anaheim Arizona Atlanta Baltimore Boston Chicago Cubs Chicago White Sox Cincinnati Cleveland Colorado Detroit Florida Houston Kansas City Los Angeles 1 1 1 1 1 1 1 1 1 1 Milwaukee Minnesota Montreal New York Mets New York Yankees Oakland Philadelphia Pittsburgh St. Louis San Diego San Francisco Seattle Tampa Bay Texas Toronto 1 1 1 1 1 1 1 18 1 1 18 Approximat e width of each class 1. Now we round this approximate width to a convenient number say,. 11
1/1/1 Solution - Table.1 Frequency Distribution for the Data of Table. The lower limit of the first class can be taken as or any number less than. Suppose we take as the lower limit of the first class. Then our classes will be, 1, 18 18, 1 11, and 1 - Total Home Runs Tally f 1 18 18 1 11 1-1 f = 8 Relative Frequency and Percentage Distributions Example - Relative Frequency and Percentage Distributions Relative frequency of a class Frequency of that class Sum of all frequencies f f Calculate the relative frequencies and percentages for Table.1 Percentage (Relative frequency) 1 Solution - Graphing Grouped Data Table.11 Relative Frequency and Percentage Distributions for Table.1 Total Home Runs 1 18 18 1 11 1 - Class Boundaries 1. to less than.. to less than 1. 1. to less than 18. 18. to less than 11. 11. to less than. Relative Frequency...1.1.1 Percentage.. 1. 1. 1. Sum =. Sum =.% Definition A histogram is a graph in which classes are marked on the horizontal axis and the frequencies, relative frequencies, or percentages are marked on the vertical axis. The frequencies, relative frequencies, or percentages are represented by the heights of the bars. In a histogram, the bars are drawn adjacent to each other. 1 1
Frequency Frequency Relative Frequency Frequency 1/1/1 Figure. Frequency histogram for Table.1. Figure. Relative frequency histogram for Table.1. 1. 1....1 - - 18-1 - 1 18 11 Total home runs 1 - - - 18-1 - 1 18 11 Total home runs 1 - Graphing Grouped Data cont. Figure. Frequency polygon for Table.1. 1 Definition A graph formed by joining the midpoints of the tops of successive bars in a histogram with straight lines is called a polygon. 1 - - 1 18-18 1-11 1 - Figure. Frequency Distribution curve. Example - The following data give the average travel time from home to work (in minutes) for states. The data are based on a sample survey of, households conducted by the Census Bureau (USA TODAY, August, 1). x 8 1
Frequency 1/1/1 Example - Solution -. 1. 1. 1. 1.1 18.. 1..1.. 1.. 1.. 1.8 1. 1. 1... 1. 1.1.8.1... 1. 1.1... 1.. 1. 8. 1... 1....1 1......8 Approximat e width of 1. 1. each class. Construct a frequency distribution table. Calculate the relative frequencies and percentages for all classes. 8 Solution - Example - Table.1 Frequency, Relative Frequency, and Percentage Distributions of Average Travel Time to Work Class Boundaries 1 to less than 18 18 to less than 1 1 to less than to less than to less than to less than f 1 Relative Frequency....18.. Percentage 18 The administration in a large city wanted to know the distribution of vehicles owned by households in that city. A sample of randomly selected households from this city produced the following data on the number of vehicles owned: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Construct a frequency distribution table for these data, and draw a bar graph. Σf = Sum = 1. Sum = 1% 81 8 Solution - Figure. Bar graph for Table.1. Table.1 Frequency Distribution of Vehicles Owned Vehicles Owned Number of Households (f) 18 1 1 18 11 Σf = 1 1 8 No Car 1 Car Cars Cars Cars Cars Vehicles ow ned 8 8
1/1/1 STEM-AND-LEAF DISPLAYS Example -8 Definition In a stem-and-leaf display of quantitative data, each value is divided into two portions a stem and a leaf. The leaves for each stem are shown separately in a display. The following are the scores of college students on a statistics test: 8 8 8 81 1 1 8 8 1 8 8 Construct a stem-and-leaf display. 8 8 8 Solution -8 Solution -8 To construct a stem-and-leaf display for these scores, we split each score into two parts. The first part contains the first digit, which is called the stem. The second part contains the second digit, which is called the leaf. We observe from the data that the stems for all scores are,,, 8, and because all the scores lie in the range to 8 8 88 Figure.1 Stem-and-leaf display. Solution -8 Stems 8 Leaf for Leaf for After we have listed the stems, we read the leaves for all scores and record them next to the corresponding stems on the right side of the vertical line. 8 1
1/1/1 Figure. Stem-and-leaf display of test scores. Figure.1 Ranked stem-and-leaf display of test scores. 8 1 8 1 1 1 8 8 1 8 1 1 1 8 1 Example - Solution - The following data are monthly rents paid by a sample of households selected from a small city. 88 11 111 181 8 1 11 11 1 1 8 11 8 1 1 1 1 1 111 1 1 18 Construct a stem-and-leaf display for these data. Figure.1 Stem-and-leaf display of rents. 8 1 11 1 1 1 8 1 8 81 1 1 1 1 8 Example -1 The following stem-and-leaf display is prepared for the number of hours that students spent working on computers during the last month. Example -1 1 8 1 8 1 8 Prepare a new stem-and-leaf display by grouping the stems. 1
1/1/1 Solution -1 Scatter Diagram Figure.1 Grouped stem-and-leaf display. 8 * 1 * 8 * 1 * 8 * * Example. A real estate agent wanted to know to what extent the selling price of a home is related to its size Collect the data 1) Determine the independent variable (X house size) and the dependent variable (Y selling price) * Relationship between people s weight and height * Relationship between # of calories eaten and weight gain/loss.8 Scatter Diagram It appears that in fact there is a relationship, that is, the greater the house size the greater the selling price Patterns of Scatter Diagrams Linearity and Direction are two concepts we are interested in Positive Linear Relationship Negative Linear Relationship. Weak or Non-Linear Relationship.1 1