DATA PRESENTATION At the end of the chapter, you will learn to: Present data in textual form Construct different types of table and graphs Identify the characteristics of a good table and graph Identify the proper uses and possible misuses of graphs 1
Presenting Data with Text - data or highlights of the data are incorporated to a paragraph or in a textual presentation. Advantages: a) This presentation gives emphasis to significant figures and comparisons. b) It is simplest and most appropriate approach when there are only a few numbers to be presented such as numerical measures that summarize the data. 2
Disadvantages: a) When a large mass of quantitative data is included in a text or paragraph, the presentation becomes almost incomprehensible. b) Paragraphs can be tiresome to read especially if the same words are repeated so many times. For example: In a statistics class with 50 students, only 42 passed the midterm examinations. Of the 42 students who passed the examinations, 30 are females and 12 are males. Of the 8 students who failed, 2 are females and 6 are males. 3
Presenting Data with Tables - the systematic organization of data in rows and columns Advantages: a) more concise than textual presentation b) easier to understand c) facilitates comparison and analysis of relationship among different categories d) presents data in greater detail than a graph 4
Some Guidelines in Constructing Tables 1. The title should be concise, written in telegraphic style, not in complete sentence. 2. Column labels should be precise. Stress differences rather than similarities between adjacent columns. 3. Categories should not overlap. 4. The units of measurements must be clearly stated. 5
5. Show any relevant total, subtotals, percentages, etc. 6. Indicate if data were taken from another publication by including a source note. 7. Tables should be self-explanatory, although they may be accompanied by a paragraph that will provide an interpretation or direct attention to important figures. 6
Frequency Distribution Table tabular arrangement of data by grouping the values into mutually exclusive classes and showing the number of observations falling in each class. Single-Value Grouping Frequency Distribution Table form of frequency distribution where the distinct values are used as classes; commonly used for qualitative types of data. No. of accidents in a certain intersection No. Of Cars Frequency Relative Frequency 0 2 0.14 1 5 0.36 2 4 0.29 3 1 0.07 7 4 2 0.14
STEPS to construct a Single-Value Grouping Frequency Distribution Table Step 1. First column, list all the distinct groupings. Step 2. second column, make a tally of the responses for the frequency. Step 3. Third column, compute for the relative frequency, rf = f n where rf = relative frequency f = frequency n = total number of observations. 8
Two-Way Contingency - a statistical table that shows the observed frequencies or proportions of data elements classified according to two variables, with the rows indicating one variable and the columns indicating the other variable. The variables are either qualitative or quantitative. Number of Students Who Play Basketball by Gender 9
Grouping by Class Intervals - form of frequency distribution where mutually exclusive classes are in the form of intervals; used for quantitative types of data. Class Interval Class Boundaries Frequency F Relative frequency Class Mark x 7-9 6.5-9.5 2 0.04 8 10-12 9.5-12.5 8 0.16 11 13-15 12.5-15.5 14 0.28 14 16-18 15.5-18.5 19 0.38 17 19-21 18.5-21.5 7 0.14 20 10
Steps in constructing a frequency distribution table that uses class intervals as groupings. 1. Calculate the range R = highest value lowest value. 2. Determine the number of classes d = n, where n is the total number of observations. d is round off to the nearest whole number. In some instances your number of classes will be one more than d. 3. Compute the class size C = R d The class size is determined by rounding up c to the nearest value whose precision is the same as those of the given data. For example, if the given data are integers and c is computed as 7.2, then the class size should be 8. If the given data are rounded up to the nearest tenths, and c is computed as 7.21, then the class size should be 7.3. 11
Define the class intervals by first determining the lower and upper limits of each class. a. The lower limit of the first class interval is the lowest observed value in the data set. The lower limits of the succeeding class intervals are obtained by adding the class size to the lower limit of the preceding class. b. b. The upper limit of the class interval is obtained by subtracting one unit of measure from the lower limit of the next class. 5. Tally the observations into the classes to obtain the frequency of each class interval. 12
6. Class boundaries of a certain class are numbers that split classes without forming gaps counted by the upper and lower limits. Class Boundaries carry out one more decimal place than the class interval. Lower Class Boundaries = Lower Limit 0.5(one unit of measure) Upper Class Boundaries = Upper Limit + 0.5(one unit of measure) 7. The class mark is the average of the lower and upper limits of a given class interval. CM = lower limt + upper limit 2 13
8. Relative Frequency (RF). Relative frequency is obtained by dividing the frequency of a given class by the total number of observations. 9. Relative Percentage. Relative percentage is obtained by multiplying the relative frequency by 100%. 10.Cumulative Frequency (CF). Cumulative frequency is the accumulated frequency of a class. I a. Less than CF (< CF). This is the total number of observations within a class whose values is less than or equal to the upper limit of the class. b. Greater than CF (> CF). This is the total number of observations within a class whose values are greater than or equal than the lower limit of the class. 14
Construct a frequency distribution table for the given data 2.2 4.1 3.5 4.5 3.2 3.7 3.0 2.6 3.4 1.6 3.1 3.3 3.8 3.1 4.7 3.7 2.5 4.3 3.4 3.6 2.9 3.3 3.9 3.1 3.3 3.1 3.7 4.4 3.2 4.1 1.9 3.4 4.7 3.8 3.2 2.6 3.9 3.0 4.2 3.5 15
Presenting Data with Graphs A graph is a device for showing numerical values or relationships in pictorial form. Advantages a) main features and implications of a body of data can be grasped at a glance b) can attract attention and hold the reader s interest c) simplifies concepts that would otherwise have been expressed in so many words d) can readily clarify data, frequently bring out hidden facts and relationships 16
Qualities of a Good Graph a) Accurate should not be deceptive, distorted or misleading b) Simple should be straightforward, not loaded with irrelevant or trivial symbols and ornamentation c) Clear should be easily read and understood d) Appearance to attract and hold attention 17
Types Of Graph a) Line Graph useful for showing trends over a period of time 18 Age at First Marriage
b) Bar Graph consists of a series of rectangular bars where the length of the bar represents the magnitude to be demonstrated 19
c) Pie or Circle Graph a circle is divided into sectors in such a way that the area of each sector is proportional to the size of the quantity represented by that sector - useful in showing how a total quantity is distributed among a group of categories 20
d) Pictogram pictures or symbols are used to represent certain quantity or volume 21
e) Histogram a bar graph of a frequency distribution table - class boundaries are represented by the width of the bars and the frequencies that fall within the classes are represented by the height of the bars 20 18 16 14 12 10 8 6 6.5-9.5 9.5-12.5 12.5-15.5 15.5-18.5 18.5-21.5 4 2 0 1 22
f) Stem and Leaf Plots Steps to construct a stem and leaf plot 1.Divide each measurement into two parts: the stem and the leaf. 2.List the stems in a column, with a vertical line to their right. 3.For each of the measurement, record the leaf portion in the same row as its corresponding stem. 4.Order the leaves from lowest to highest in each stem. 5.Provide the key to your stem and leaf coding so that the reader can recreate the actual measurements if necessary. 90 70 70 70 75 70 65 68 60 74 70 95 75 70 68 65 40 65 4 0 5 Leaf unit=1 6 0 5 5 5 8 8 7 0 0 0 0 0 0 0 4 5 5 8 9 0 5 23
Construct a stem and leaf plot for the following data. 1. 123 111 119 139 126 132 127 116 145 110 141 149 134 138 124 2. 123 235 178 261 378 213 539 606 675 321 207 176 543 145 354 322 24
g) Box plot A boxplot is a concise graph showing the five point summary. Multiple boxplots can be drawn side by side to compare more than one data set. 25
h) Map chart A map chart displays data by shading sections of a map, and must include a key. A total data number should be included. 26
MISUSES of Graphs 27
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Pictograph A pictograph uses an icon to represent a quantity of data values in order to decrease the size of the graph. A key must be used to explain the icon. Advantages Easy to read Visually appealing Handles large data sets easily using keyed icons Disadvantages Hard to quantify partial icons Icons must be of consistent size Best for only 2-6 categories Very simplistic Map chart A map chart displays data by shading sections of a map, and must include a key. A total data number should be included. Advantages Good visual appeal Overall trends show well Disadvantages Needs limited categories No exact numerical values Color key can skew visual interpretation Pie chart A pie chart displays data as a percentage of the whole. Each pie section should have a label and percentage. A total data number should be included. Advantages Visually appealing Shows percent of total for each category Disadvantages No exact numerical data Hard to compare 2 data sets "Other" category can be a problem Total unknown unless specified Best for 3 to 7 categories Use only with discrete data 29
Histogram A histogram displays continuous data in ordered columns. Categories are of continuous measure such as time, inches, temperature, etc. Histogram Explorer Advantages Visually strong Can compare to normal curve Usually vertical axis is a frequency count of items falling into each category Disadvantages Cannot read exact values because data is grouped into categories More difficult to compare two data sets Use only with continuous data Bar graph A bar graph displays discrete data in separate columns. A double bar graph can be used to compare two data sets. Categories are considered unordered and can be rearranged alphabetically, by size, etc. Advantages Visually strong Can easily compare two or three data sets Disadvantages Graph categories can be reordered to emphasize certain effects Use only with discrete data Line graph A line graph plots continuous data as points and then joins them with a line. Multiple data sets can be graphed together, but a key must be used. Advantages Can compare multiple continuous data sets easily Interim data can be inferred from graph line Disadvantages Use only with continuous data 30
Stem and Leaf Plot Stem and leaf plots record data values in rows, and can easily be made into a histogram. Large data sets can be accomodated by splitting stems. Advantages Concise representation of data Shows range, minimum & maximum, gaps & clusters, and outliers easily Can handle extremely large data sets Disadvantages Not visually appealing Does not easily indicate measures of centrality for large data sets Box plot A boxplot is a concise graph showing the five point summary. Multiple boxplots can be drawn side by side to compare more than one data set. Advantages Shows 5-point summary and outliers Easily compares two or more data sets Handles extremely large data sets easily Disadvantages Not as visually appealing as other graphs Exact values not retained 31