Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write. H.G. Wells

Transcription

1 Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write. H.G. Wells 1

2 Statistics is a science which deals with collection, tabulation, presentation, analysis, and interpretation of the data. 2

3 DATA VALUES ARE THE VALUES OF THE CHARACTERISTICS UNDER STUDY. Types of data Qualitative data (attribute) Quantitative data (variable) Discrete variable Continuous variable 3

4 SOURCES OF DATA 1.Primary source - Direct investigation 2. Secondary source Recorded data (data collected by some other person) 4

5 COLLECTION OF PRIMARY DATA 1. Census survey or population survey 2. Sample survey 5

6 SAMPLING TECHNIQUES 1. Simple random sampling (SRS) 2. Stratified random sampling 3. Systematic sampling 4. Cluster sampling 6

7 SIMPLE RANDOM SAMPLING 1. Lottery method 2. Random number table method With replacement sampling Without replacement sampling 7

8 Parameter Statistic A measure based on population units Eg. Population mean, population proportion A measure based on sample units Eg. Sample mean, sample proportion 8

9 DATA SCALE Nominal : Sex, colour, class. Ordinal : rank, class obtained.. Interval : marks, height, weight, Ratio : no of leaves per branch, no of students in a class, 9

10 TABULATION The table should be precise, easy to understand and self explanatory. One-way table Two-way table Three-way table(complex table) 10

11 The main objectives of tabulation are: It simplifies complex data and the data presented are easily understood. It facilitates comparison of related facts. It facilitates computation of various statistical measures. It presents facts in minimum possible space. Moreover, the needed information can be easily located. Tabulated data are good for references and to present the information 11

12 Components of Table An ideal table should consist of the following main parts: Table number For easy reference assign number for the table Title of the table : Each table must have title. It must be brief and self explanatory. 12

13 Captions or column headings Stubs or row heading Body of the table It is most important part of the table It contains numerical information including row and column totals. Footnotes normally written at the bottom of the table. Sources of data: It mentions the source of the information used in the table. 13

14 ONE WAY TABLE No. of students in the college for the year F. Y. B. Sc S. Y. B. Sc. T. Y. B. Sc. TOTAL 14

15 TWO WAY TABLE No. of students in the college for the year class-> sex Male F.Y.B.Sc S.Y.B.Sc T.Y.B.Sc Total Female Total 15

16 THREE WAY TABLE NO. OF STUDENTS IN THE COLLEGE FOR THE YEAR Class -> Sex \ category F.Y.B.Sc. S.Y.B.Sc. T.Y.B.Sc. Total male Total Female Total Total Total open Reserved open Reserved open Reserved 16

17 THEORY OF ATTRIBUTES TWO ATTRIBUTES A α Total B (AB) (α B) (B) β (A β ) (α β) (β ) Total (A) (α) N 17

18 RELATIONS BETWEEN CLASS FREQUENCIES N = ( A ) + ( α) = (B) + (β ) (A)= (AB) + (A β ) (α) = (α B) + ( β α ) (B) = (AB) + (α B) (β ) = (A β) + (α β) 18

19 THREE ATTRIBUTES A α Total B C (ABC) (α BC) (BC) γ (AB γ ) (α B γ ) (Bγ) Total (AB) (α B) (B) β C (A β C) (α β C) (β C) γ (A β γ) (α β γ) (β γ) Total (A β ) (α β ) (β ) Total C (A C ) (α C ) (C ) γ (A γ ) (α γ) (γ ) Total (A) (α) N 19

20 CONSISTENCY OF THE DATA ALL CELL ENTRIES SHOULD BE POSITIVE. In case of two attributes with two classes the table can be represented as Conditions: (AB) 0 (αb) 0 (Aβ) 0 (αβ) 0 A α Total B (AB) (αb) (B) β (Aβ) (αβ) (β) Total (A) (α) N 20

21 ASSOCIATION BETWEEN TWO ATTRIBUTES Positively associated Negatively associated Independent (AB)/(B)> (Aβ)/(β) Or (AB) >(A)*(B)/N Or (AB)*(αβ)> (Aβ)(αB) (AB)/(B)< (Aβ)/(β) Or (AB) <(A)*(B)/N Or (AB)*(αβ)< (Aβ)(αB) (AB)/(B)= (Aβ)/(β) Or (AB) =(A)*(B)/N Or (AB)*(αβ)= (Aβ)(α B) 21

22 MEASURES OF ASSOCIATION Yule s coefficient of association Q ( AB)( ) ( AB)( ) ( A )( B) ( A )( B) Yule s coefficient of colligation Y ( AB)( ( AB)( Relationship between Q & Y Q ) ) 2Y 1 Y 2 ( A ( A )( )( B) B) 22

23 ATTRIBUTES A & B ARE Positively associated if Q>0 or Y>0 Negatively associated if Q<0 or Y>0 Independent if Q= 0 or Y=0 23

24 DIAGRAMS 1. One dimensional diagrams -Bar diagrams Simple bar Segmented percentage Multiple 2. Two dimensional diagrams Squares Rectangles Pie diagrams 24

25 SIMPLE BAR DIAGRAM Crop Rice Wheat Tur Cotton Sugarcane Gram Production (in 000 tons)

26 production SIMPLE BAR DIAGRAM Principle crops in Maharastra rice wheat crop tur cotton sugarcane gram production (in '000 tons) 26

27 SEGMENTED BAR DIAGRAM RESULT OF DEGREE COLLEGE FOR Result F.Y.B.Sc. S.Y.B.Sc. T.Y.B.Sc. I class II III Fail

28 No. of students SEGMENTED BAR DIAGRAM RESULT FAIL PASS II I 0 F.Y.B.Sc S.Y.B.Sc T.Y.B.Sc CLASS 28

29 precentage SEGMENTED BAR DIAGRAM FOR PERCENTAGES FOR COMPARISON result 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% FAIL PASS II I 0% F.Y.B.Sc S.Y.B.Sc T.Y.B.Sc class 29

30 MULTIPLE BAR DIAGRAM SALE AND PROFIT OF XYZ COMPANY. (ALL FIGURES ARE IN 000 RS.) Year Total sales Profit

31 in '000 Rs. MULTIPLE BAR DIAGRAM Sale & profit of XYZ CO Total sales profit year 31

32 Two dimensional diagrams : In two dimensional diagrams area is proportional to the value of the variable. Commonly used two dimensional diagrams are squares rectangles and Pie diagram. 32

33 Square: A square is used to compare the values of the characteristic which differ widely from one another. The area of the square is proportional to the value of the variable. Hence side of the square is proportional to the square root of the value of the variable. 33

34 Rectangles: When we have the data on two characteristics relating to different items and the product of these values have a significance. Then we use rectangle. One characteristic is taken along height and other along width. Eg data is with respect to the selling price and number of units sold of a certain commodity, then the product represent revenue generated from that commodity. 34

35 Pie diagram : When the number of components of a variable are large, the segmented bar diagram fails to give proper visual representation. In such case we use pie diagram. 35

36 A pie diagram consists of a circle divided into number of sectors representing different components of a variable. The areas of these sectors are proportional to the values of the components. Since area of the sector is proportional to its angle. We consider value of the component proportional to angle. We determine angle for each sector by using formula Angle of the sector = 360*(value of the component / total of all components). 36

37 PIE DIAGRAM Item Food Clothing Rent Fuel & lights Education Miscellaneous Expenditure (in Rs.)

38 PIE DIAGRAM Item Expenditure (in Rs.) Angle =360*value/total Food Clothing Rent Fuel & lights Education Miscellaneous Total

39 PIE DIAGRAM monthly expenditure food clothing rent fuel education misc. 39

40 Merits of Diagrams The diagrams are visuals and hence easy to understand the data. They provide the information instantly. The diagrams are much more attractive than the numerical data. Diagrams can be easily remembered than the numerical figures. Comparisons can be done more easily using diagrams. 40

41 Demerits The diagrams cannot represent exact values. The diagrams cannot give us detailed information. Diagrams are supplement to tabular representation but not alternative to it. For further statistical analysis diagrams are not useful 41

42 CLASSIFICATION THE BASIC OBJECTIVES FOR CLASSIFICATION ARE to condense the huge data into few classes so that similarities and dissimilarities in the data are easily recognized. To help comparison. To highlight important features and to pinpoint most significant features. To do away with unimportant features. To present data in a form from which further statistical analysis is feasible. 42

43 TYPES OF CLASSIFICATION Geographical classification (Area-wise classification) Chronological classification (time series data.) Quantitative data (frequency distribution.) 43

44 GEOGRAPHICAL CLASSIFICATION (AREA- WISE CLASSIFICATION) CROP YIELD OF WHEAT IN THE YEAR 2009 State Gujrat Madhya Pradesh Maharastra Punjab Haryana Yield (in lakhs of tons)

45 CHRONOLOGICAL CLASSIFICATION (TIME SERIES DATA.) RESULT OF T.Y.B.SC. FOR YEARS 2000 TO 2004 Year I Class II Class Pass Class Fail Total

46 FREQUENCY DISTRIBUTION IF WE HAVE N OBSERVATIONS AND THESE ARE TO BE GROUPED IN TO CLASSES OF EQUAL WIDTH THEN WE HAVE TO FOLLOW FOLLOWING STEPS Number of classes=k= log N K should be integer. Locate maximum and minimum of the data then width of class is given by Width of the class = d =( max-min)/k Write K classes with width d in such a way that minimum value get included in the first class and maximum value in the last class. Class limits should be round figures, and class intervals should be nonoverlapping and must include all observations. 46

47 EX 1. CONSIDER THE FOLLOWING DATA WHICH GIVES US THE WEIGHTS (IN KGS.) 52.5, 59.5, 49.5, 52.9, 57.4, 52.9, 64.7, 51.8, 61.3, 71.4, 50.7, 73.5, 58.7, 61.8, 62.8, 56.6, 69.0, 56.4, 62.8, 47.8, 55.4, 69.9, 48.1, 51.2, 62.5, 57.1, 64.3, 45.6, 64.8, 60.9, 57.2, 56.8, 50.5, 63.4, 49.2, 61.2, 56.6, 67.6, 61.7, Number of observations =N=40 Maximum=73.5 minimum=45.1 Number of classes=k= log N=6.3 = 6 Width of the class=( max-min)/k =( )/6=4.7=5 (appr) 47

48 FREQUENCY DISTRIBUTION Classes (wts in kgs) Frequency

49 A FREQUENCY DISTRIBUTION CAN BE REPRESENTED GRAPHICALLY BY 1. Histogram 2. Frequency curve 3. Frequency polygon 49

50 HISTOGRAM 50

51 frequency FREQUENCY CURVE frequency curve class mark 51

52 frequency FREQUENCY POLYGON Frequency polygon classmark 52

53 Negatively skewed curve 53

54 TYPES OF CURVES Positively skewed curve 54

55 Symmetric Curve 55

56 OGIVES Ex: Classes : Frequency: Class boundary Cum freq< type Cum freq > type

57 cumulative freq. OGIVES ogives class boundary < type type 57

58 HISTOGRAM ( CLASSES ARE NOT OF EQUAL WIDTH) Wages (in,00 Rs.) No. of workers freq Class width d Freq. density =( f/d)* freq *10 d

59 HISTOGRAM 59

60 FREQUENCY DISTRIBUTION Classes with equal width and continuous classes Eg Further analysis is easy Classes not of equal width Eg , analysis becomes difficult Classes with equal width and discontinuous classes Eg , make the classes continuous before analysing 60

61 STEM AND LEAF DISPLAY STEM AND LEAF DISPLAY IS USED TO REPRESENT UNGROUPED DATA Data: 241, 252, 259, 261, 274, 245, 258, 265, 272, 264, 268, 267, 254, 269, 274. stem leaf 24 1, 5, 25 2, 4, 8, , 4, 5, 7, 8, , 4, 4 61

62 Thank you 62