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STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 CORRELATION ANALYSIS Correlation analysis deals with the association between two or more variables. If two or more quantities vary in sympathy so that movements in one tend to be accompanied by corresponding movements in the other then they are said to be correlated. Thus correlation is a statistical device which helps us in analyzing the co variation of two or more variables. The problems of analyzing the relation between different series should be broken into three steps; 1. Determining whether a relation exists and if it does measuring it. 2. Testing whether it is significant. 3. Establishing the cause and effect relation. Significance of correlation • Most of the variables show some kind of relationship like there is relationship between price and supply and income and expenditure. With the help of correlation we can find the degree of relationship between the variables. • When we know the degree of relationship we can know the value of one variable with the help of another variable. • Correlation analysis contributes to the understanding of the economic behavior, aids in locating the critically important variable on which others depend reveal to the economist and suggest to him the paths through which stabilizing forces may become effective. • The effect of correlation is to reduce the range of uncertainty the prediction based on correlation analysis is likely to be more valuable and near to reality. TYPES OF CORRELATION;- Positive or negative correlation;- This depend upon the direction of series. If both variable are increasing or decreasing in same direction than this is positive correlation and if they are varying in opposite directions then they are having negative correlation. If one series is increasing and other is also increasing and if one is decreasing and other series is also decreasing then this is positive correlation and if one series is increasing and other is decreasing and if one is decreasing and other is decreasing they this is negative correlation. Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 Simple partial and multiple correlation; - This depends upon number of variable studied. When only two problems are studied then it is simple correlation. When three or more variables are studied then this is multiple or partial correlation. In multiple correlations three or more variables are studied simultaneously. In partial correlation we recognize three or more variable but make correlation of two variables from the series. Linear and non linear correlation; - It is based upon the constancy of the ratio of change between the variables. If the amount of change in one variables tends to bear constant ratio to the amount of change in other variable then it is called linear correlation or vice versa. METHODS OF CORRELATION;- Scatter diagram method Graphic method Karl Pearson’s coefficient of correlation Rank correlation method Scatter diagram method;- This is the simplest device for ascertaining whether two variables are related is to prepare a dot chart called scatter diagram. The given data are plotted on a graph paper in the form of dots. Merits and demerits of the method;- Merits- It is the simplest form of studying correlation. It is not influenced by the size of extreme it3ems where as most of mathematical methods are influenced by extreme figures. Demerits; - We can get idea of correlation but we can not find out exact degree of correlation. Graphic Method;- The individual values of the two variables are plotted on the graph paper then we obtain two curves one for X variable and another for Y variable by examining the direction and closeness of the two curves so drawn wee can infer they are related or not. Karl Pearson’s coefficient of correlation;- It is most widely used for calculating correlation. The correlation is denoted by r. R = ∑xy / N s.d (X) s.d(y) x= (X-Mean), y = (Y-Mean) s.d(x) = standard deviation of X s.d(y) = standard deviation of Y Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 Direct method of calculating correlation;- N ∑XY – (∑X) (∑Y) ___________________________ R= 2 2 2 2 N ∑X- (∑X) * N ∑Y- (∑Y) When deviations are taken from assumed mean N ∑dx.dy – (∑dx) (∑dy) ___________________________ R= 2 2 2 2 N ∑dx - (∑dx) * N ∑dy - (∑dy) Steps • Take the deviations of X series from an assumed mean and denote these deviations by dx and obtain the total ∑dx • Take the deviations of Y series from an assumed mean and denote these deviations by dy and obtain the total ∑dy. • Square dx and obtain the total ∑dx square • Square dy and obtain the total ∑dy square • Multiply dx and dy and obtain the total ∑dx.dy • Substitute the value of ∑dx.dy, ∑dx, ∑dy, ∑dx square, ∑dy square Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 X Y dx dy dx sq dy sq dx.dy 78 125 9 13 81 169 117 89 137 20 25 400 625 500 99 156 30 44 900 1936 1320 60 112 -9 0 81 0 0 59 107 -10 -5 100 25 50 79 136 10 24 100 576 240 68 123 -1 11 1 121 -11 61 108 -8 -4 64 16 32 ∑Y= ∑dy= ∑X= 593 1004 ∑dx= 41 108 1727 3468 ∑dx.dy=2248 N ∑dx.dy – (∑dx) (∑dy) ___________________________ R= 2 2 2 2 N ∑dx - (∑dx) * N ∑dy - (∑dy) R= (8) (2248) – (41) (108) ________________________________ Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 (8) (1727) – 41*41 (8) (3468) (108*108) = 0.97 The formula of frequency distribution is N ∑fdx.dy – (∑fdx) (∑fdy) ___________________________ R= 2 2 2 2 N ∑fdx - (∑fdx) * N ∑fdy - (∑fdy) The limitations of this method are great care must be expected for calculating correlation. Conditions; • When r is +1 it means there is perfect positive relationship between the variables, • When r is -1 it means there is perfect negative relationship between the variables • When r is 0 it means there is no relationship between the variables Rank correlation method This method was developed by British psychologist Charles Edward Spearman in 1904. The ranking is done of the variables whether in ascending or descending order. Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 2 R= 6∑D 1- _____________________ 2 N (N - 1) R denotes rank coefficient of correlation and D refers to the difference of rank between paired items in two series. X Rx Y Ry D square 97.8 3 73.2 1 4 99.2 7 85.8 6 1 98.8 6 78.9 4 4 98.3 4 75.8 2 4 98.4 5 77.2 3 4 96.7 1 87.2 7 36 97.1 2 83.8 5 9 2 R= 1- 6∑D _____________________ 2 N (N - 1) R= 1- 6* 62 ________________ 7 (7*7 – 1) 1- 1.07 = - 0.107 Equal ranks- When there are two equal variables then it will be very difficult to provide them ranks. If two individuals are ranked equal at fifth place they are each given the rank Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 5+6 /2= 5.5 and if three are ranked equal at fifth place they are given the rank 5+6+7 /2 = 6. 2 3 3 R= 6∑D + 1/12 (m –m) + 1/12 (m –m) 1- ________________________________ 2 N (N - 1) Concurrent deviation method This is the simplest method of all the methods. The only thing is required under this method is to find out the direction of change of X variable and Y variable. The formula is Rc= ± (2c-n) / n n= number of pairs of observation compared. Steps • Find out the direction of change of X variable as compared with the first value whether the second value is increasing or decreasing or is constant. If it is increasing put a plus sign and if it is decreasing then minus sign and if it is constant then zero sign will be there. And denote them by dx. • In the same manner find out the direction of change of y variables and denote the column by Dy. • Multiply dx with dy and determine the value of c the number of positive signs. And apply the above formula. Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 X dx Y dy dxdy 60 65 55 - 40 - + 50 - 35 - + 56 + 75 + + 30 - 63 - + 70 + 80 + + 40 - 35 - + 35 - 20 - + 80 + 80 + + 80 0 60 - 0 75 - 60 0 0 C=8 Concurrent deviation method- coefficient of correlation ± (2c-n) / n ± (2*8 – 10) / 10 0.774 answer Regression analysis Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 The analysis of coefficient of correlation finds the closeness of the variables but the regression analysis helps us to find out one variable as we know the other variable. The given variable is independent variable and other which has to find out is called dependent variable. Definitions- “regression is the measure of the average relationship between two or more variable if terms of the original unit of the data” “One of the most frequently used techniques in economics and business research to bind a relation between two or more variables that are related casually is regression analysis. Difference between correlation and regression- Whereas coefficient is a measure of degree pf co variability between x and y the objective of regression is to find out the nature of relationship between the variables. Correlation is merely a tool of ascertaining the degree of relationship between two variables and we can not say that one variable is the cause another effect. There may be nonsense correlation between two variables which is purely due to chance and has no practical relevance there is nothing like nonsense regression, Correlation coefficient is independent of change of scale and origin regression coefficients are independent of change of scale but not of origin. Regression lines- There may be two regression lines one is x on y and another is y on x. x on y represents x as dependent variable on y and vice versa. Least square method Regression equation of y on x Y= a+ bx To determine the value Making summation of the equation ∑Y= Na + b∑X 2 ∑XY= a∑X +bX Regression equation of x on y X= a+ bY Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 To determine the value Making summation of the equation ∑X= Na + b∑Y 2 ∑XY= a∑Y +bY Deviations taken from arithmetic means of X and Y Regression equation of X on Y: X- mean = rσx/σy (y-mean) Rσx/σy= regression coefficient of x on y Deviations taken from assumed means X on Y : X- mean = rσx/σy (Y-mean) Rσx/σy = N ∑dx.dy – (∑dx) (∑dy) __________________ 2 2 N ∑dy - (∑dy) Dx= (X-A) Dy= (Y-A) Yon X; (Y-mean)= rσy/σx (X- mean) Rσy/σx= N ∑dx.dy – (∑dx) (∑dy) __________________ 2 2 N ∑dx - (∑dx) In the case of frequency distribution Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 Rσx/σy = N ∑fdx.dy – (∑fdx) (∑fdy) __________________ *ix/iy 2 2 N ∑fdy - (∑fdy) Rσy/σx= N ∑fdx.dy – (∑fdx) (∑fdy) __________________ *iy/ix 2 2 N ∑fdx - (∑fdx) Limitations of regression analysis- In making estimate from a regression it is important to remember that the assumption is being made that relationship has not changed since the regression equation was computed. Time Series Analysis “A time series is a set of statistical observations arranged in chronological order.” “A time series consists of statistical data which are collected recorded and observed over successive increments of time.” It is clear from the definitions that if we arrange the data according to time then it is called time series. UTILITY OF THE TIME SERIES ANALYSIS • It helps in understanding past behavior- by observing data over a period of time one can easily understand what changes have been taken place in the past. This analysis will be extremely helpful in predicting the future behavior. • It helps in planning future operations- plans for the future can not be made without forecasting events and relationship they will have. Statistical techniques like time series helps to make decision for future. • It helps in evaluating current accomplishments.- The actual performance can be compared with the expected performance and the cause of variation analyzed • It facilitates comparison- Different time series are compared and important decisions are concluded. Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 COMPONENTS OF TIME SERIES • Secular trend • Seasonal variations • Cyclical variations • Irregular variations Y = T* S*C *I Y denotes the result of the four element and T stands for trend, S for seasonal, C for cyclical and I for irregular variations. Another approach is to treat each observation of a time series as the sum of these four components. Y= T+S+C+I SECULAR TREND- The term trend is very commonly used in day to day parlance. For example we talk of rising trend of population; prices etc. are called secular trend or long term trend. The concept of secular trend indicates only for long term data. SEASONAL VARIATIONS Seasonal variations are those periodic movements in business activity which occur regularly every year and have their origin in the nature of year itself. The factors that cause seasonal variations are- 1. Climate and weather conditions- The most important factor causing seasonal variations is the climate. Changes in the climate and weather and weather conditions such as rainfall, humidity, heat act on different products and industries differently. 2. Customs and traditions and habits- Though nature is primarily responsible for seasonal variations in the times series, customs traditions and habits also have their impact. For example on certain occasions like deepawali, dusserha Christmas there is big demand for sweets and also there is large demand for cash before the festivals because they need money for shopping and gifts. CYCLICAL VARIATIONS The term cycle refers to the recurrent variations in time series that usually last longer than a year and are regular neither in amplitude nor in length. Cyclical Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 fluctuations are long term movements that represent consistently recurring rises and declines in activity, a business cycle consists of the recurrence of the up and downs movements of business activity from some sort of statistical trend. IRREGULAR VARIATIONS- Irregular variations are called erratic accidental, random refer to such variations inn business activity which do not repeat in a definite pattern. There are two reasons for recognizing irregular movements. 1. To suggest that on occasions it may be possible to explain certain movements in the data due to specific causes and to simplify further analysis. 2. To emphasize the fact that predictions of economic conditions are always subject to degree of error owing to the unpredictable erratic influences which may enter? MEASUREMENT OF TREND 1. Free hand method 2. Semi- average method 3. Moving average 4. Method of least square • FREE HAND METHOD Plot the time series on a graph paper then examine carefully the direction of the trend based on the plotted information. Then draw a straight line which will best fit to the data according to personal judgment and this line will show the direction of the trend Semi- average method The given data is divided into two parts. Preferably with the same number of years Illustration- Year sale of firm A 1997 102 1998 105 1999 114 2000 110 2001 108 2002 116 Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 2003 112 Since seven years are given the middle year shall be left out and an average of first three years and last three years shall be appointed. The average of first three years is 102+105+114 /3 = 107 and average of last three years is 108+116+112 /3 112 thus we get two points and by joining points we shall obtain the re4quired trend line it can be used for prediction or for determining intermediate value. Method of moving average. This method is selected of period of three years, five years and eight years The three years moving average shall be computed as follows , a+b+c /3, b+c+d /3, c+d+e/3, d+e+f/3 and for five years moving average is a+b+c+d+e/5, b+c+d+e+f /5, c+d+e+f+g/5 Three years moving average year production 3 year total moving average 1989 15 - - 1990 21 66 22 1991 30 87 29 1992 36 108 36 1993 42 124 41.33 1994 46 138 46 1995 50 152 50.67 1996 56 169 56.33 1997 63 189 63 1998 70 207 69 1999 74 226 75.33 2000 82 246 82 2001 90 267 89 2002 95 287 95.67 2003 102 - - Five years moving average year no of students 5 year total moving average 1994 332 - - Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 1995 317 - - 1996 357 1800 360 1997 392 1873 374.6 1998 402 1966 393.2 1999 405 2036 407.2 2000 410 2049 409.8 2001 427 2085 417 2002 405 - - 2003 438 - - Method of least square- Equation of y on x Y= a+ bX To determine the value Making summation of the equation ∑Y= Na + b∑X 2 ∑XY= a∑X +bX Equation of x on y X= a+ by To determine the value Making summation of the equation ∑X= Na + b∑Y 2 ∑XY= a∑Y +bY Mathematics of management Business mathematics consists of a set of mathematical and statistical tools that can be used for the fulfillment of one or more objective of a businesslike the Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 maximization of out or sales, maximization of profits, minimization of cost etc. These tools are often known as quantitative techniques. Main features of quantitative techniques 1. Every management problem can be represented by one or more equations with the help of certain symbols. These symbols denote relevant variables and constants of the problems. 2. The solution of the model is obtained by the application of one or more techniques from the set of quantitative techniques. 3. The quantitative techniques take care of the polcies and capacities of different departments and hence avoid the occurrence of any contradiction between them. 4. This is interdisciplinary approach to problem solving. 5. These techniques attempt to analyze the business problem in actual working environment which often differ from the ideal conditions assumed in mathematics, economics and other disciplines. IMPORTANCE OF QUANTITATIVE TECHNIQUES. 1. Basis for scientific analysis- With the increase in complexities of modern business it is not possible to rely on the unscientific decisions based on the intuitions. This provides the scientific methods for tackling various problems for modern business. 2. Tools for scientific analysis- Quantitative techniques provide the managers with a variety of tools from mathematics, statistics, economics and operational research. These tools help the manager to provide a more precise description and solution of the problem. The solutions obtained by using quantitative thechniques are often free from the bias of the manager or the owner of the business. 3. Solution for various business problems. Quantitative techniques provide solutions to almost every area of a business. These can be used in production, marketing, inventory, finance and other areas to find answers to various question like (a) how the resources should be used in production so that profits are maximized. (b) How should the production be matched to demand so as to minimize the cost of inventory? 4. Optimum allocation of resources- An allocation of resources is said to be optional if either a given level of output is being produced at minimum cost or maximum output is being produced at a given cost. A quantitative technique enables a manager to optimally allocate the resources of a business or industry. 5. Selection of an optimal strategy- Using quantitative techniques it is possible to determine the optimal strategy of a business or firm that is Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 facing competition from its rivals. The techniques for determining the optimal strategy is dependent upon game theory. 6. Optimal deployment of resources- Using quantitative technique It is possible to find out the earliest and latest time for successful completion of project and this is called program evaluation and review technique. 7. Facilitate the process of decision making- quantitative techniques provide a method of decision making in the face of uncertainty. These techniques are based upon decision theory. SCOPE OF QUANTITATIVE TECHNIQUES Production management- quantitative techniques are useful to the production management in (a) selecting the location site for a plant, scheduling and controlling its development and designing of plant layout. (b) Locating within the plant and controlling the movements of required production material and finished goods inventories and (c) scheduling and sequencing production by adequate preventive maintenance with optimum product mix. Personnel management- quantitative techniques are useful to personnel management to find out (a) optimum manpower planning, (b) the number of employees to be maintained on the permanent or full time roll, (c) the number of persons to be kept in a work pool intended for meeting the absenteeism, (d) in studying personnel recruiting procedures, accidents rates, labor turnover. Marketing management- Quantitative techniques equally help n marketing management to determine (a) warehouse distribution point and where warehousing should be located, their size quantity to be stocked and the choice of customers, (b) The optimum allocation of sales budget to direct selling and promotional expenses, (c) The choice of different media of advertising and bidding strategies and (d) The customer preferences relating to size, color, packaging et for various products as well as to outbid and outwit customers. Financial management - Quantitative techniques are also very useful to the financial management in (a) finding long range capital requirements as well as how to generate these requirements, (b) Determining optimum replacement policies (c) working out a profit plan for the firm (d) developing capital investment plan, (e) estimating credit and investment risk ARITHMATIC PROGRESSION Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 A series in which each successive term is obtained by adding a constant quantity (known as common difference) to its proceeding term is called Arithmetic Progression The general term of an A.P with first term equal to a and common difference equal to d is written as a, a+d, a+2d, a+3d -----------, a+(n-1)d Where n denotes the number of terms l = a+ (n-1) d is the last term of A.P. with n terms. Sum of n terms of an A.P= N/2 [2a + (n-1) d] On substituting l = a+ (n-1) d the formula can be written as Sum = n/2{a+l} Example- find the sum of first 15 terms of the following series- 10, 15, 20, 25, -------- Solution- here a = 10 d = 5, n=15 Sum = 15/2 {2*10 (15-1) 5} 675 ans Example-The fourth term of an A.P is 14 and the eighth term is 26 find the sum of first ten terms. Solution- Let a be the first term and d be the common difference Then it is given that a+3d = 14 and a+7d = 26 eliminating a from these equations 4d= 12 so d=3 A+3*3 = 14 so a = 5 And sum = 10/2 {2*5+9*3} = 185 GEOMETRIC PROGRESSION A series in which each successive term is obtained by multiplying the proceeding term by a constant quantity (known as common ratio) is called geometric progression The general term of G.P with first term equal to a and common ratio equal to R, is written as 2 3 (n-1) A, aR, aR. -------------aR Sum of n terms of G.P Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 The sum of n terms of a G.P with the first term equal to a and common 2ratio equal to R, is written as2 n Sum = a (1-R ) _______________ 1-R Sum of an infinite G.P When R < 1 and n becomes infinite the formula for the sum of G.P is given by sum = a / (1-R) Example- The first term of a G.P is 8 and the common ratio is 3 . Find the sum of first 10 terms. Solution- a= 8, R=3 and n = 10 10 Sum= 8(1-3) _________ 1-3 236192 answer QUANTITATIVE METHODS CONTENTS:- 1. BASIC MATHEMATICS 2. ARITHMETIC PROGRESSION 3. GEOMETRIC PROGRESSION 4. MEASUREMENT OF CENTRAL TENDENCY 5. MEASUREMENT OF DISPERSION Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 6. SKEWNESS, MOMENTS, KURTOSIS 7. CORRELATION ANALYSIS 8. REGRESSION ANALYSIS 9. ANALYSIS OF TIME SERIES 10. PROBABILITYAND EXPECTED VALUE 11. THEORETICAL DISTRIBUTION PROBABILITY AND EXPECTED VALUE In day to day conversation we normally use the terms chance etc. and generally people have a vague idea about its meaning. For example we come across statements like probably it may rain tomorrow it is likely that Mr. A may not coming for taking the class today, these all vague ideas are probability, Definition of probability- The probability of a given event is an expression of likelihood or chance of occurrence of an event. A probability is a number which ranges from 0 to 1 zero for an event which can not occur and 1 for an event certain to occur. Calculation of probability- 1. Experiments and events- The term experiments refer to describe an act which can be repeated under some given conditions. Random experiments are those experiments whose results depend on chance such as tossing of a coin, throwing a dice. The result of a random experiments are called outcomes 2. Mutually exclusive events- Two events are said to be mutually exclusive or incompatible when both cannot happen simultaneously in a single trial or the occurrence of any one of them precludes the occurrence of the other. For example if a single coin is tossed either head can be up or tail can be up. Both cannot be up at the same time. These events are called mutually exclusive events. if both cases can be happened then these events are called not mutually exclusive events. 3. Independent and dependent events- Two or more events are said to be independent when the outcome of one does not affect and is not affected by other. For example if a coin is tossed twice the result of the second throw would in no way be affected by the result of the first throw. Similarly the results obtained by throwing a dice are independent of the results obtained by drawing an ace from a pack of cards. 4. Equally likely events- Events are said to be equally likely when one does not occur more often than the others. For example if an unbiased coin or dice is thrown each face may be expected to be observed approximately the same number of times in the long run, similarly the cards of a pack of playing cards are so closely alike that we expect each card to appear equally often when a large number of drawings are made with replacement. However if the coin or dice is Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 biased we should not expect each face to appear exactly the same number of times. 5. Simple and compounds events- In case of simple events we consider the probability of the happening or not happening of single events. For example we might be interested in finding out the probability of drawing a red ball from al bag containing 10 white and 6 red balls. On the other hand in case of compound events we consider the joint occurrence of two or more events. THEOREMS OF PROBABILITY- The additional theorem The multiplication theorem The additional theorem- The additional theorem states that if two events A and B are mutually exclusive the probability of the occurrence of either A or B is the sum of the individual probability of A and B. P (Aor B) = P (A) + P (B) Example- One head is drawn from a standard pack of 52. What is the probability that it is either a king or a queen? Solution- There is 4 kings and 4 queens in a pack of 52 cards. The probability that the card drawn is a king = 4/52 And the probability that the card drawn is a queen = 4/52 Since the events are mutually exclusive events the probability that the card drawn is either king or queen 4/52 +4/52 = 8/52 = 2/13 answer When events are not mutually exclusive events P (Aor B) = P (A) + P (B) – P (A and B) In the example taken the probability of drawing a king or a heart shall be- P (Heart or king) = P (Heart) +P (King) – P (Heart and king) 4/52 +4/52 -1/52 = 4/13 answer Multiplication theorem- This theorem states that if two events A and B are independent the probability that they both will occur is equal to the product of their individual probability. If A and B are independent then P (A and B) = P (A) * P (B) Conditional probability Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 Two events A and B are said to be dependent when B Can occur only when A is known to have occurred only the probability attached to such an event is called the conditional probability. Example- Find the probability of drawing a queen, a king and a knave in that order from a pack of cards in three consecutive draws the cards drawn not being replaced. Solution- The probability of drawing a queen = 4/52 The probability of drawing a king after a queen has been drawn – 4/51 The probability of drawing a knave after queen and king have been drawn 4/50 Since they are dependent event the required probability of the compound events us 4/52 *4/51 *4/50 = 64/132600= 0.00048 THEORETICAL DISTRIBUTION Probability distributions are used in discrete and continues series. The distributions are 1. Binomial distribution, poison distribution and normal distribution. BINOMIAL DISTRIBUTION- Binomial distribution is known as Bernoulli distribution is associated with the name of a Swiss mathematician James Bernoulli. Binomial distribution is a probability distribution expressing the probability of one set of dichotomous alternatives .i.e. success or failure. The assumptions are • An experiment is performed under the same conditions for a fixed number of trials say n. • In each trial there are two possible outcomes of the experiment. For lack of a better nomenclature they are called success or failure. • The probability of a success denoted by p remains constant from trial to trial. The probability of a failure denoted by q is equal to (1-p) if the probability of success is not the same n each trial we will not have binomial distribution. • The trials are statistically independent. The binomial distribution- P(r) = (n-r) r Nc q p r p = probability of success in a single trial q = (1-p) n = number of trials r= number of success Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 POISSON DISTRIBUTION- Poisson distribution is a discrete distribution and is used in statistical work. This distribution is used to describe the behavior of rare events. This is expected in cases where the chance of any individual event being a very small success such as no of accidents on road, printing mistakes on a paper. The poison distribution -m r P(r) = e m __________________ R! Normal distribution- normal distribution is used in continuous series in this distribution value of z is find out and Z= X – mean / S.D. Properties of normal distribution- 1. Normal distribution is bell shaped. 2. It is perfectly symmetrical about mean. 3. This is unimodal means it has one modal. 4. Mean median and mode are equal in normal distribution. MEASUREMENT OF DISPERSION Dispersion;- Dispersion is the measure of the variation of the items. The concept of dispersion is related to the extent of variability in observations. The variability in an observation is often measured as its deviation from central value. A suitable average of all such deviations is called measure of dispersion. Measure of dispersion enables a comparison to be made of two or more series with regard to their variability, Significance of Dispersion:- Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 1. To determine the reliability of an average;- The dispersion is used to know how much the average is reliable. A low variation or dispersion shows more reliability and consistency in data. 2. To serve as a basis for the control of variability: - dispersion acts as a basis of variability. Many measurements can be done by the dispersion and if the variations are high then controlling tools can be used. 3. Useful in quality control;- due to variation methods it can be known that the product are up to grade or not if they are not means if variation are high then they can rectify the technique of productions. 4. To facilitate the use of other statistical tools;- Many powerful analytical tools in statistics such as correlation analysis, the testing of hypothesis, analysis of variance regression analysis are based of measurement of variation Properties of Good Measure of Variation 1. It should be simple to understand. 2. It should be easy to compute. 3. It should be rigidly defined. 4. It should be based on each and every item of the distribution. 5. It should have sampling stability. 6. It should not be affected by the extreme items. Methods of measuring dispersion;- 1. Range 2. Quartile deviation 3. Mean deviation 4. standard deviation Range;- Range is simplest method of studying dispersion. It is the difference between the value of the smallest item and the value of the largest item include in distribution Range = L- S L = largest value S= Smallest value Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 Coefficient of range= L- S _____________ L+S Quartile deviation It represents the difference between third quartile and first quartile. Q.D. = Q3 –Q1 / 2 Coefficient of Quartile deviation;- = Q3 –Q1/ Q3+ Q1 Q3 = 3N/4 Q3= L+ 3N/4 - c.f. ______________ * H F Q1 = N/4 Q1= L + N/4 - c.f. ______________ * H F L = Least value C.F. = cumulative frequency F = Frequency H = class interval Mean Deviation M.D. = ∑ │D│ Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 ___________ N ∑ │D│ = summation of deviation from actual mean . Mean deviation in discrete series. M.D. = ∑ F│D│ ___________ N F = frequencies Coefficient of mean deviation; - M.D. / Median Standard Deviation;- Standard deviation measures absolute dispersion or variability of the series. 2 ∑ │D│ _ ∑ │D│ 2 S.D = *i N N Standard deviation in continues series;- 2 ∑ F│D│ ∑F│D│ 2 - *i N N Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 Standard deviation is denoted by 2 Variance = {S.D} Coefficient of standard deviation = S.D / Mean * 100 Skewness, moments and kurtosis Skew ness;- When a series is not symmetrical it is said to be skewed Absolute measure of skew ness; - Mean – Mode Relative measure of skew ness;- 1. Karl Pearson’s coefficient of skew ness;- Mean - Mode SKp = _________________ Std. deviation 2. Bow ley’s coefficient of skew ness;- SKb = Q3 – Q1 – 2 median ____________________________ Q3 + Q1 3. Kelly’s coefficient of skew ness;- SKk = P10 + P90 – 2median _____________________________ P90 - P10 = D1 + D9 – 2median Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 _________________________ D9 - D1 P10 = 10N/100 10N/100 - c.f. L+ ______________ *H F P90 = 90N/100 D1 = 1N/100 D9 = 9N/100 All the methods are just like median. Moments Moments are the sum of the deviations. It is the sum of the deviation is also known as the first moment of dispersion. It is the sum of the deviations of the items of a series from mean of the series, divided by the total number of items in the distribution. In other words, it is the average deviation of the items from the mean. The arithmetic means of the various powers of the deviations of the items in a distribution are called the moments of the distribution. Moments are denoted by u (mu) U1 = ∑ (X – mean) / n 2 U2 = ∑ (X – mean) / n 3 U3 = ∑ (X – mean) / n For frequency distribution;- U1 = ∑ F (X – mean) / n 2 Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 U2 = ∑ F (X – mean) / n 3 U3 = ∑ F (X – mean) / n First moment about origin is mean Second moments about mean are variance. Third moments about mean is skew ness Fourth moments about mean are kurtosis. Kurtosis Kurtosis is the degree of peaked ness of a distribution, usually taken relative to a normal distribution. If a curve is peaked then a normal curve it is called leptokurtic. If a curve is flat topped than the normal curve it is called platykurtic. The normal curve is called mesokurtic, 2 β1 = u4 / u2 β= Kurtosis u4, u2 fourth moment and second moment Visit us at www.sgiithisar.com Contact at 94163-59920 STATISTICAL ANALYSIS NOTES FOR YEAR 2008-2009 Important Questions other than above notes 1. Define the various methods of data collection. Explain with example. 2. Explain the term probability and its theorems. 3. Explain the following a. Action space b. Bayesian rule c. Games theory d. Expected pay off table e. Null hypothesis and alternate hypothesis f. One tail and two tail test 4. What is a sampling distribution? What purpose does it serve? 5. How does the size of population and the kind of random sampling determine the shape of a sampling distribution? 6. Ch No 10 7. explain the following with the help of example a. Z- test b. T-test c. χ2 Test 8. What is chi- square test of goodness of fit? what precautions are necessary in using this test? 9. Briefly discuss the advantages and disadvantages of non-parametric methods as compared with parametric methods in statistics. 10. What are index numbers? What purpose do they serve? Discuss the various problems faced in the construction of index numbers. 11. Explain the test used in index number? Visit us at www.sgiithisar.com Contact at 94163-59920

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Notes for MBA 2nd Sem of GJU University Hissar Haryana In India

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