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### Stats 16

1. 1. LESSON – 1 STATISTICS FOR MANAGEMENT Session – 3 Duration: 1 hr Technical terms used in formulation frequency distribution a) Class limits: The class limits are the smallest and largest values in the class. Ex: 0 – 10, in this class, the lowest value is zero and highest value is 10. the two boundaries of the class are called upper and lower limits of the class. Class limit is also called as class boundaries. b) Class intervals The difference between upper and lower limit of class is known as class interval. Ex: In the class 0 – 10, the class interval is (10 – 0) = 10. The formula to find class interval is gives on below L−S i= R L = Largest value S = Smallest value R = the no. of classes Ex: If the mark of 60 students in a class varies between 40 and 100 and if we want to form 6 classes, the class interval would be 100 − 40 60 I= (L-S ) / K = = = 10 L = 100 6 6 S = 40 K=6 Therefore, class intervals would be 40 – 50, 50 – 60, 60 – 70, 70 – 80, 80 – 90 and 90 – 100. ♦ Methods of forming class-interval a) Exclusive method (overlapping) In this method, the upper limits of one class-interval are the lower limit of next class. This method makes continuity of data. 1
2. 2. Ex: Marks No. of students 20 – 30 5 30 – 40 15 40 – 50 25 A student whose mark is between 20 to 29.9 will be included in the 20 – 30 class. Better way of expressing is Marks No. of students 20 to les than 30 5 (More than 20 but les than 30) 30 to les than 40 15 40 to les than 50 25 Total Students 50 b) Inclusive method (non-overlaping) Ex: Marks No. of students 20 – 29 5 30 – 39 15 40 – 49 25 A student whose mark is 29 is included in 20 – 29 class interval and a student whose mark in 39 is included in 30 – 39 class interval. ♦ Class Frequency The number of observations falling within class-interval is called its class frequency. Ex: The class frequency 90 – 100 is 5, represents that there are 5 students scored between 90 and 100. If we add all the frequencies of individual classes, the total frequency represents total number of items studied. 2
3. 3. ♦ Magnitude of class interval The magnitude of class interval depends on range and number of classes. The range is the difference between the highest and smallest values is the data series. A class interval is generally in the multiples of 5, 10, 15 and 20. Sturges formula to find number of classes is given below K = 1 + 3.322 log N. K = No. of class log N = Logarithm of total no. of observations Ex: If total number of observations are 100, then number of classes could be K = 1 + 3.322 log 100 K = 1 + 3.322 x 2 K = 1 + 6.644 K = 7.644 = 8 (Rounded off) NOTE: Under this formula number of class can’t be less than 4 and not greater than 20. ♦ Class mid point or class marks The mid value or central value of the class interval is called mid point. (lower limit of class + upper limit of class) Mid point of a class = 2 ♦ Sturges formula to find size of class interval Range Size of class interval (h) = 1 + 3.322 log N Ex: In a 5 group of worker, highest wage is Rs. 250 and lowest wage is 100 per day. Find the size of interval. Range 250 − 100 h= = = 55.57 ≅ 56 1 + 3.322 log N 1 + 3.322 log 50 Constructing a frequency distribution The following guidelines may be considered for the construction of frequency distribution. a) The classes should be clearly defined and each observation must belong to one and to only one class interval. Interval classes must be inclusive and non- overlapping. b) The number of classes should be neither too large nor too small. Too small classes result greater interval width with loss of accuracy. Too many class interval result is complexity. 3
4. 4. c) All intervals should be of the same width. This is preferred for easy computations. Range The width of interval = Number of classes d) Open end classes should be avoided since creates difficulty in analysis and interpretation. e) Intervals would be continuous throughout the distribution. This is important for continuous distribution. f) The lower limits of the class intervals should be simple multiples of the interval. Ex: A simple of 30 persons weight of a particular class students are as follows. Construct a frequency distribution for the given data. 62 58 58 52 48 53 54 63 69 63 57 56 46 48 53 56 57 59 58 53 52 56 57 52 52 53 54 58 61 63 ♦ Steps of construction Step 1 Find the range of data (H) Highest value = 70 (L) Lowest value = 46 Range = H – L = 69 – 46 = 23 Step 2 Find the number of class intervals. Sturges formula K = 1 + 3.322 log N. K = 1 + 3.222 log 30 K = 5.90 Say K = 6 ∴ No. of classes = 6 Step 3 Width of class interval Range 23 Width of class interval = = = 3.883 ≅ 4 Number of classes 6 4
5. 5. Step 4 Conclusions all frequencies belong to each class interval and assign this total frequency to corresponding class intervals as follows. Class interval Tally bars Frequency 46 – 50 ||| 3 50 – 54 |||| ||| 8 54 – 58 |||| ||| 8 58 – 62 |||| | 6 62 – 66 |||| 4 66 – 70 | 1 Cumulative frequency distribution Cumulative frequency distribution indicating directly the number of units that lie above or below the specified values of the class intervals. When the interest of the investigator is on number of cases below the specified value, then the specified value represents the upper limit of the class interval. It is known as ‘less than’ cumulative frequency distribution. When the interest is lies in finding the number of cases above specified value then this value is taken as lower limit of the specified class interval. Then, it is known as ‘more than’ cumulative frequency distribution. The cumulative frequency simply means that summing up the consecutive frequency. Ex: ‘Less than’ Marks No. of students cumulative frequency 0 – 10 5 5 10 – 20 3 8 20 – 30 10 18 30 – 40 20 38 40 – 50 12 50 In the above ‘less than’ cumulative frequency distribution, there are 5 students less than 10, 3 less than 20 and 10 less than 30 and so on. Similarly, following table shows ‘greater than’ cumulative frequency distribution. 5
6. 6. Ex: ‘Less than’ Marks No. of students cumulative frequency 0 – 10 5 50 10 – 20 3 45 20 – 30 10 42 30 – 40 20 32 40 – 50 12 12 In the above ‘greater than’ cumulative frequency distribution, 50 students are scored more than 0, 45 more than 10, 42 more than 20 and so on. Diagrammatic and Graphic Representation The data collected can be presented graphically or pictorially to be easy understanding and for quick interpretation. Diagrams and graphs give visual indications of magnitudes, groupings, trends and patterns in the data. These parameter can be more simply presented in the graphical manner. The diagrams and graphs help for comparison of the variables. Diagrammatic presentation A diagram is a visual form for presentation of statistical data. The diagram refers various types of devices such as bars, circles, maps, pictorials and cartograms etc. Importance of Diagrams 1. They are simple, attractive and easy understandable 2. They give quick information 3. It helps to compare the variables 4. Diagrams are more suitable to illustrate discrete data 5. It will have more stable effect in the reader’s mind. Limitations of diagrams 1. Diagrams shows approximate value 2. Diagrams are not suitable for further analysis 3. Some diagrams are limited to experts (multidimensional) 4. Details cannot be provided fully 5. It is useful only for comparison 6
7. 7. General Rules for drawing the diagrams i) Each diagram should have suitable title indicating the theme with which diagram is intended at the top or bottom. ii) The size of diagram should emphasize the important characteristics of data. iii) Approximate proposition should be maintained for length and breadth of diagram. iv) A proper / suitable scale to be adopted for diagram v) Selection of approximate diagram is important and wrong selection may mislead the reader. vi) Source of data should be mentioned at bottom. vii) Diagram should be simple and attractive viii)Diagram should be effective than complex. Some important types of diagrams a) One dimensional diagrams (line and bar) b) Two-dimensional diagram (rectangle, square, circle) c) Three-dimensional diagram (cube, sphere, cylinder etc.) d) Pictogram e) Cartogram a) One dimensional diagrams (line and bar) In one-dimensional diagrams, the length of the bars or lines is taken into account. Widths of the bars are not considered. Bar diagrams are classified mainly as follows. i) Line diagram ii) Bar diagram - Vertical bar diagram - Horizontal bar diagram - Multiple (compound) bar diagram - Sub-divided (component) bar diagram - Percentage subdivided bar diagram i) Line diagram This is simplest type of one-dimensional diagram. On the basis of size of the figures, heights of the bar / lines are drawn. The distances between bars are kept uniform. The limitation of this diagram are it is not attractive cannot provide more than one information. 7
8. 8. Ex: Draw the line diagram for the following data Year 2001 2002 2003 2004 2005 2006 No. of students passed in first 5 7 12 5 13 15 class with distinction 16 (15) 14 No. of students passed in FCD (13) (12) 12 10 8 (7) 6 (5) (5) 4 2001 2002 2003 2004 2005 2006 Year Indication of diagram: Highest FCD is at 2006 and lowest FCD are at 2001 and 2004. b) Simple bars diagram A simple bar diagram can be drawn using horizontal or vertical bar. In business and economics, it is very a common diagram. Vertical bar diagram The annual expresses of maintaining the car of various types are given below. Draw the vertical bar diagram. The annual expenses of maintaining includes (fuel + maintenance + repair + assistance + insurance). Type of the car Expense in Rs. / Year Maruthi Udyog 47533 Hyundai 59230 Tata Motors 63270 Source: 2005 TNS TCS Study Published at: Vijaya Karnataka, dated: 03.08.2006 8
9. 9. 70000 65000 63270 59230 60000 55000 50000 47533 45000 40000 35000 30000 Maruthi Udyog Hyundai Tata Motors Source: 2005 TNS TCS Study Published at: Vijaya Karnataka, dated: 03.08.2006 Indicating of diagram a) Annual expenses of Maruthi Udyog brand car is comparatively less with other brands depicted b) High annual expenses of Tata motors brand can be seen from diagram. ♦ Horizontal bar diagram World biggest top 10 steel makers are data are given below. Draw horizontal bar diagram. US RIVA Thyssen- Tangshan Steel Arcelo Nippo BAO POSCO JFE Stee NUCOR krupp maker r Mittal n Steel l Prodn. in 110 32 31 30 24 20 18 18 17 16 million tonnes 9
10. 10. Tangshan 16 Thyssen-krupp 17 Top - 10 Steel Makers RIVA 18 NUCOR 18 US Steel 20 BAO Steel 24 JFE 30 POSCO 31 Nippon 32 Arcelor Mittal 110 0 20 40 60 80 100 120 Production of Steel (Million Tonnes) Source: ISSB Published by India Today ♦ Compound bar diagram (Multiple bar diagram) Multiple bar diagrams are used to provide more information than simple bar diagram. Multiple bar diagram provides more than one phenomenon and highly useful for direct comparison. The bars are drawn side-by-side and different columns, shades hatches can be used for indicating each variable used. Ex: Draw the bar diagram for the following data. Resale value of the cars (Rs. 000) is as follows. Year (Model) Santro Zen Wagonr 2003 208 252 248 2004 240 278 274 2005 261 296 302 10
11. 11. 350 296 302 300 278 274 261 252 240 248 250 Value in Rs. 208 200 150 100 50 0 1 2 3 Model of Car Santro Zen Wagnor Source: True value used car purchase data Published by: Vijaya Karnataka, dated: 03.08.2006 Ex: Represent following in suitable diagram Class A B C Male 1000 1500 1500 Female 500 800 1000 Total 1500 2300 2500 2500 2300 2500 Population (in Nos.) 1500 800 1000 2000 1500 500 1000 1500 1500 1000 500 0 1 2 3 Class Male Female 11
12. 12. Ex: Draw the suitable diagram for following data Mode of Investment in 2004 in Rs. Investment in 2005 in Rs. investment Investment %age Investment %age NSC 25000 43.10 30000 45.45 MIS 15000 25.86 10000 15.15 Mutual Fund 15000 25.86 25000 37.87 LIC 3000 5.17 1000 1.52 Total 58000 100 66000 100 110 100 5.17 1.52 90 80 25.86 37.87 % of Investment 70 60 25.86 15.15 50 40 30 43.10 45.45 20 10 0 2004 2005 Year Two-dimensional diagram In two-dimensional diagram both breadth and length of the diagram (i.e. area of the diagram) are considered as area of diagram represents the data. The important two-dimensional diagrams are a) Rectangular diagram b) Square diagram a) Rectangular diagram Rectangular diagrams are used to depict two or more variables. This diagram helps for direct comparison. The area of rectangular are kept in proportion to the values. It may be of two types. i) Percentage sub-divided rectangular diagram ii) Sub-divided rectangular diagram 12
13. 13. In former case, width of the rectangular are proportional to the values, the various components of the values are converted into percentages and rectangles are divided according to them. Later case is used to show some related phenomenon like cost per unit, quality of production etc. Ex: Draw the rectangle diagram for following data Expenditure in Rs. Item Expenditure Family A Family B Provisional stores 1000 2000 Education 250 500 Electricity 300 700 House Rent 1500 2800 Vehicle Fuel 500 1000 Total 3500 7000 Total expenditure will be taken as 100 and the expenditure on individual items are expressed in percentage. The widths of two rectangles are in proportion to the total expenses of the two families i.e. 3500: 7000 or 1: 2. The heights of rectangles are according to percentage of expenses. Monthly expenditure Item Family A (Rs. 3500) Family B(Rs. 7000) Expenditure Rs. %age Rs. %age Provisional stores 1000 28.57 2000 28.57 Education 250 7.14 500 7.14 Electricity 300 8.57 700 10 House Rent 1500 42.85 2800 40 Vehicle Fuel 500 12.85 1000 14.28 Total 3500 100 7000 100 13
14. 14. Provisonal Stores Education Electricity House Rent Vehicle Fuel 100 80 % of Expenditure 60 40 20 0 A B Family b) Square diagram To draw square diagrams, the square root is taken of the values of the various items to be shown. A suitable scale may be used to depict the diagram. Ratios are to be maintained to draw squares. Ex: Draw the square diagram for following data 4900 2500 1600 Solution: Square root for each item in found out as 70, 50 and 40 and is divided by 10; thus we get 7, 5 and 4. 6000 5000 4900 4000 3000 2500 2000 1600 1000 5 7 4 0 1 2 3 14
15. 15. Pie diagram Pie diagram helps us to show the portioning of a total into its component parts. It is used to show classes or groups of data in proportion to whole data set. The entire pie represents all the data, while each slice represents a different class or group within the whole. Following illustration shows construction of pie diagram. Draw the pie diagram for following data Revenue collections for the year 2005-2006 by government in Rs. (crore)s for petroleum products are as follows. Draw the pie diagram. Customs 9600 Excise 49300 Corporate Tax and dividend 18900 States taking 48800 Total 126600 Solution: Item / Source Value in Angle of circle %ge crores 9600 Customs 9600 x 360 = 27.30 o 7.58 126600 49300 Excise 49300 x 360 = 140.20 o 39.00 126600 18900 Corporate Tax and Dividend 18900 x 360 = 53.70 o 14.92 126600 48800 State’s taking 48800 x 360 = 138.80 o 38.50 126600 Total 126600 360o 100 15
16. 16. 7.58 Customs 38.5 Excise 39 Corporate Tax and Dividend State’s taking 14.92 Source: India Today 19 June, 2006 Choice or selection of diagram There are many methods to depict statistical data through diagram. No angle diagram is suited for all purposes. The choice / selection of diagram to suit given set of data requires skill, knowledge and experience. Primarily, the choice depends upon the nature of data and purpose of presentation, to which it is meant. The nature of data will help in taking a decision as to one-dimensional or two-dimensional or three- dimensional diagram. It is also required to know the audience for whom the diagram is depicted. The following points are to be kept in mind for the choice of diagram. 1. To common man, who has less knowledge in statistics cartogram and pictograms are suited. 2. To present the components apart from magnitude of values, sub-divided bar diagram can be used. 3. When a large number of components are to be shows, pie diagram is suitable. Graphic presentation A graphic presentation is a visual form of presentation graphs are drawn on a special type of paper known are graph paper. Common graphic representations are a) Histogram b) Frequency polygon c) Cumulative frequency curve (ogive) 16
17. 17. Advantages of graphic presentation 1. It provides attractive and impressive view 2. Simplifies complexity of data 3. Helps for direct comparison 4. It helps for further statistical analysis 5. It is simplest method of presentation of data 6. It shows trend and pattern of data Difference between graph and diagram Diagram Graph 1. Ordinary paper can be used 1. Graph paper is required 2. It is attractive and easily 2. Needs some effect to understand understandable 3. It is appropriate and effective to 3. It creates problem measure more variable 4. It can’t be used for further analysis 4. Can be used for further analysis 5. It gives comparison 5. It shows relationship between variables 6. Data are represented by bars, 6. Points and lines are used to represent rectangles data Frequency Histogram In this type of representation the given data are plotted in the form of series of rectangles. Class intervals are marked along the x-axis and the frequencies are along the y-axis according to suitable scale. Unlike the bar chart, which is one-dimensional, a histogram is two-dimensional in which the length and width are both important. A histogram is constructed from a frequency distribution of grouped data, where the height of rectangle is proportional to respective frequency and width represents the class interval. Each rectangle is joined with other and the blank space between the rectangles would mean that the category is empty and there are no values in that class interval. Ex: Construct a histogram for following data. Marks obtained (x) No. of students (f) Mid point 15 – 25 5 20 25 – 35 3 30 35 – 45 7 40 45 – 55 5 50 55 – 65 3 60 65 – 75 7 70 Total 30 17
18. 18. For convenience sake, we will present the frequency distribution along with mid-point of each class interval, where the mid-point is simply the average of value of lower and upper boundary of each class interval. 7 Frequency (No. of students) 6 5 4 3 2 1 0 15 25 35 45 55 65 75 Class Interval (Marks) Frequency polygon A frequency polygon is a line chart of frequency distribution in which either the values of discrete variables or the mid-point of class intervals are plotted against the frequency and those plotted points are joined together by straight lines. Since, the frequencies do not start at zero or end at zero, this diagram as such would not touch horizontal axis. However, since the area under entire curve is the same as that of a histogram which is 100%. The curve must be ‘enclosed’, so that starting mid-point is jointed with ‘fictitious’ preceding mid-point whose value is zero. So that the beginning of curve touches the horizontal axis and the last mid-point is joined with a ‘fictitious’ succeeding mid-point, whose value is also zero, so that the curve will end at horizontal axis. This enclosed diagram is known as ‘frequency polygon’. Ex: For following data construct frequency polygon. Marks (CI) No. of frequencies (f) Mid-point 15 – 25 5 20 25 – 35 3 30 35 – 45 7 40 45 – 55 5 50 55 – 65 3 60 65 – 75 7 70 18
19. 19. 10 8 A Frequency polygon 6 Frequency 4 2 0 0 10 20 30 40 50 60 70 80 90 100 Mid point (x) Cumulative frequency curve (ogive) ogives are the graphic representations of a cumulative frequency distribution. These ogives are classified as ‘less than’ and ‘more than ogives’. In case of ‘less than’, cumulative frequencies are plotted against upper boundaries of their respective class intervals. In case of ‘grater than’ cumulative frequencies are plotted against upper boundaries of their respective class intervals. These ogives are used for comparison purposes. Several ogves can be compared on same grid with different colour for easier visualisation and differentiation. Ex: Marks No. of Cum. Freq. Cum. Freq. Mid-point (CI) frequencies (f) Less than More than 15 – 25 5 20 5 30 25 – 35 3 30 8 25 35 – 45 7 40 15 22 45 – 55 5 50 20 15 55 – 65 3 60 23 10 65 – 75 7 70 30 7 19
20. 20. Less than give diagram Less than Cumulative Frequency 30 'Less than' ogive 25 20 15 10 5 20 30 40 50 60 70 Upper Boundary (CI) Less than give diagram 35 30 'More than' ogive More than Ogive 25 20 15 10 10 20 30 40 50 60 70 Lower Boundary (CI) 20