Math 154-1 Quantitative Methods: Data Presentation PDF

MATH154-1 QUANTITATIVE METHODS COURSE OUTCOME 1 DATA PRESENTATION Mapua University – Department of Mathematics  OBJECTIVES At the end of the lesson, the students are expected to Identify and learn various ways of presenting data; Describe data through tables...

MATH154-1 QUANTITATIVE METHODS COURSE OUTCOME 1 DATA PRESENTATION Mapua University – Department of Mathematics  OBJECTIVES At the end of the lesson, the students are expected to Identify and learn various ways of presenting data; Describe data through tables, graphs, and charts; Describe and interpret data presented in various charts; and Practice different ways or presenting data. Mapua University – Department of Mathematics  TYPES OF DATA PRESENTATION Textual Form - Data presentation using sentences and paragraphs in describing data Tabular Form - Data presentation that uses tables arranged in rows and columns for various parameters Graphical Form - Pictorial representation of data Mapua University – Department of Mathematics  DATA PRESENTATION Ungrouped Data - Data points are treated individually. Grouped Data - Data points are treated and grouped according to categories. Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Numerous data can be analyzed by grouping the data into different classes with equal class intervals and determining the number of observations that fall within each class. This procedure is done to lessen work done in treating each data individually by treating the data by group. Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Class limits - The smallest and the largest values that fall within the class interval (class) - Taken with equal number of significant figures as the given data. Class boundaries (true class limits) - More precise expression of the class interval - It is usually one significant digit more than the class limit. - Acquired as the midpoint of the upper limit of the lower class and the lower limit of the upper class Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Frequency - The number of observations falling within a particular class. - Counting and tallying Class width (class size) - Numerical difference between the upper and lower class boundaries of a class interval. Class mark (class midpoint) - Middle element of the class - It represents the entire class and it is usually symbolized by x. Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Cumulative Frequency Distribution - can be derived from the frequency distribution and can be also obtained by simply adding the class frequencies - Partial sums Types of Cumulative Frequency Distribution - Less than cumulative frequency (cf) refers to the distribution whose frequencies are greater than or above the lower class boundary the correspond to. Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Relative Frequency - Percentage frequency of the class with respect to the total population - For presenting pie charts Relative Frequency (%rf) Distribution - The proportion in percent the frequency of each class to the total frequency - Obtained by dividing the class frequency by the total frequency, and multiplying the answer by 100 Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Class Frequency x LCB UCB cf %rf Interval Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Class Frequency x LCB UCB cf %rf Interval 5-10 6 7.5 4.5 10.5 6 20 6/20=30 11-16 7 13.5 10.5 16.5 13 14 7/20=35 17-22 3 19.5 16.5 22.5 16 7 3/20=15 23-28 0 25.5 22.5 28.8 16 4 0/20=0 29-34 4 31.5 28.5 34.5 20 4 4/20=20 N=20 100% Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Steps in Constructing a Frequency Distribution Table (FDT) 1. Get the lowest and the highest value in the distribution. We shall mark the highest and lowest value in the distribution. 2. Get the value of the range. The range denoted by R, refers to the difference between the highest and the lowest value in the distribution. Thus, R = H ─ L. Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Steps in Constructing a Frequency Distribution Table (FDT) 3. Determine the number of classes. In the determination of the number of classes, it should be noted that there is no standard method to follow. Generally, the number of classes must not be less than 5 and should not be more than 15. In some instances, however, the number of classes can be approximated by using the relation 𝒌 = 𝟏 + 𝟑. 𝟑𝟐𝟐 𝒍𝒐𝒈 𝒏 (Sturges’ Formula), where k = number of classes and n = sample size. is the ceiling operator (meaning take the closest integer above the calculated value). Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Steps in Constructing a Frequency Distribution Table (FDT) 4. Determine the size of the class interval. The value of C can be obtained by dividing the range by the desired number of classes. Hence, 𝐶 = 𝑅Τ𝑘. 5. Construct the classes. In constructing the classes, we first determine the lower limit of the distribution. The value of this lower limit can be chosen arbitrarily as long as the lowest value shall be on the first interval and the highest value to the last interval Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Steps in Constructing a Frequency Distribution Table (FDT) 6. Determine the frequency of each class. The determination of the number of frequencies is done by counting the number of items that shall fall in each interval. Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table 88 62 63 88 65 85 83 76 72 63 1. Using the steps discussed, 60 46 85 71 67 construct the frequency 75 78 87 70 43 distribution of the following results 63 90 63 60 73 of a test in statistics of 50 students 55 62 62 83 79 given. 78 43 51 56 80 90 47 48 54 77 86 55 76 52 76 43 52 72 43 60 Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table 88 62 63 88 65 Class Interval Frequency 85 83 76 72 63 𝑅𝑎𝑛𝑔𝑒 43-49 7 60 46 85 71 67 𝐶= 𝑘 75 78 87 70 43 𝐻−𝐿 50-56 7 = 63 90 63 60 73 1 + 3.322𝑙𝑜𝑔𝑁 57-63 10 90 − 43 55 62 62 83 79 = 64-70 3 78 43 51 56 80 1 + 3.322log(50) = 7.07 71-77 9 90 47 48 54 77 𝐶=7 78-84 6 86 55 76 52 76 43 52 72 43 60 85-91 8 Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Class Frequency x LCB UCB cf %rf Interval 43-49 7 46 42.5 49.5 7 50 7/50=14 50-56 7 53 49.5 56.5 14 43 7/50=14 57-63 10 60 56.5 63.5 24 36 10/50=20 64-70 3 67 63.5 70.5 27 26 3/50=6 71-77 9 74 70.5 77.5 36 23 9/50=18 78-84 6 81 77.5 84.5 42 14 6/50=12 85-91 8 88 84.5 91.5 50 8 8/50=16 N=50 100% Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table 2. The following are the 22 31 55 76 48 49 50 85 17 38 scores of 40 students in 92 62 94 88 72 65 63 25 88 88 a Math quiz. Prepare a 86 75 37 41 76 64 66 58 66 76 frequency distribution 52 40 42 76 29 72 59 42 54 62 for these scores using a class size of 10. Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Class Interval Frequency 17-26 3 22 31 55 76 48 49 50 85 17 38 27-36 2 92 62 94 88 72 65 63 25 88 88 37-46 6 86 75 37 41 76 64 66 58 66 76 47-56 6 52 40 42 76 29 72 59 42 54 62 57-66 9 67-76 7 77-86 2 87-96 5 Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table 3. The thickness of a particular metal of an optical instrument was measured on 121 successive items as they came off a production line under what was believed to be normal conditions. The results are shown in Table Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Mapua University – Department of Mathematics  DATA PRESENTATION Graphical Form of Frequency Distribution Frequency Polygon - Line graph - The points are plotted at the midpoint of the classes. Histogram (Frequency Histogram or Relative Frequency Histogram) - Bar graph - Plotted at the exact lower limits of the classes Mapua University – Department of Mathematics  DATA PRESENTATION Graphical Form of Frequency Distribution Ogive - Line graph - Graphical representation of the cumulative frequency distribution - The < ogive represents the ogive represents the >cf. Mapua University – Department of Mathematics  DATA PRESENTATION Frequency Distribution Table Class Interval Frequency 25-29 1 30-34 1 35-39 5 4. Construct a 40-44 8 frequency polygon, 45-49 15 histogram, and ogives 50-54 4 of the given distribution. 55-59 4 60-64 3 65-69 4 70-74 3 75-79 2 Mapua University – Department of Mathematics  DATA PRESENTATION Graphical Form of Frequency Distribution Frequency Polygon In the preparation of a 16 15 polygon, the frequency values 14 13 are always plotted on the y- 12 11 axis (vertical) while the classes 10 Frequency (f) 9 are plotted on the x-axis 8 7 (horizontal). Here we use the 6 5 class midpoints. 4 3 2 1 0 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 Class Midpoint (x) Mapua University – Department of Mathematics  DATA PRESENTATION Graphical Form of Frequency Distribution Frequency Histogram The histogram is plotted using 16 15 14 the frequencies against the 13 12 exact limit of the classes. 11 10 Frequency (f) 9 8 7 6 5 4 3 2 1 0 19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5 59.5 64.5 69.5 74.5 79.5 Exact Class Limit Mapua University – Department of Mathematics  DATA PRESENTATION Graphical Form of Frequency Distribution Frequency Histogram 16 The histogram is plotted using 15 14 the frequencies against the 13 12 class midpoint. 11 10 Frequency (f) 9 8 7 6 5 4 3 2 1 0 22 27 32 37 42 47 52 57 62 67 72 77 82 Class Midpoint (x) Mapua University – Department of Mathematics  DATA PRESENTATION Graphical Form of Frequency Distribution Ogives Ogive Graph 55 50 45 < ogive > ogive 40 Cumulative Frequency (CF) 35 30 25 20 15 10 5 0 19.5 24.5 29.5 34.5 39.5 44.5 49.5 54.5 59.5 64.5 69.5 74.5 79.5 84.5 Class Boundary (CB) Mapua University – Department of Mathematics  DATA PRESENTATION Graphical Form of Frequency Distribution Class Interval Frequency EXERCISE 5. Construct a 43-49 7 frequency polygon, 50-56 7 histogram, and ogives 57-63 10 of the frequency 64-70 3 distribution from problem #1. 71-77 9 78-84 6 85-91 8 Mapua University – Department of Mathematics  SUMMARY Stem-and-leaf diagram is one way of data presentation tabular form. Frequency distribution can be depicted in two ways: tabular and graphical (frequency polygon, histogram, and ogives) forms Mapua University – Department of Mathematics  REFERENCES Linear Programming and Resource Allocation Modeling by Wiley Quantitative Methods / Mathematics for Business, Mark Cleary, 2019 Introduction to Quantitative Methods in Business, Bharat Kolluri, 2016 Mapua University – Department of Mathematics