- 3. 1.Md.Sajal Ahmed c-08031 2.Moazzem hossen foyez c-08013 3.Ashikur rahman torik c-08002 4.Nishita nimu c-08022 5.Jubayer hossen c-08001 6.Arif c-08032 7.Pranto c-08021 8.Syful hossen c-08006 9.Rukiya parvin ruku c-08012
- 4. Presenting Tropics Measure of dispersion. Importance and necessity of measure of dispersion The classification of measure of dispersion. Merits & Demerits of Standard Deviation Merits & Demerits of Mean Deviation Merits & Demerits of Quartile Deviation Advantages & Disadvantages of Range
- 5. Presenting Tropics Difference between Absolute and Relative Measure of Dispersion Good qualities of a good measure of dispersion Best measure of dispersion Difference between a) SD and CV b) MD and SD c) Variance and CV Standard deviation is the best measure of dispersion Coefficient of correlation (Karl Pearson’s),scatter diagram, rank correlation(spearman’s)
- 6. Presenting Tropics Perfect correlation and curvilinear correlation with example. Characteristics and limitation of correlation. Endogenous and exogenous variable. Least square method. Difference between correlation analysis and regression analysis. Degree of correlation. Difference between correlation and Regression analysis.
- 7. Presenting Tropics Regression & it’s important Simple and multiple regression & regression coefficient Probability Distributions Continuous and discrete probability distributions Continuous distribution Example of the distribution of weights Distribution plot of the weight of adult males Discrete distribution Example of the number of customer complaints
- 8. Presenting Tropics The different types of discrete and continuous distribution Condition of binomial distribution Properties of binomial distribution Relation of Poisson and Normal distribution
- 9. Dispersion The measure dispersion are called average of second order. A measure of dispersion describes the degree of scatter shown by observation and is usually measure as an average deviation about some central value. Importance and necessity of measure of dispersion To realize the reliability of the measures of central tendency To compare the variability of two or more sets of data. To suggest various methods for controlling the variations in a set of observations To facilitate as a basis for further statistical analysis
- 10. Dispersion There are two type of measure of dispersion: 1.The absolute measure of dispersion 2.The relative measure of dispersion The absolute measure of dispersion When dispersion is measure in original units then it is known as absolute measure of dispersion. There are four type of absolute measure of dispersion 1.Range 2.Mean deviation 3.Standard deviation 4.Quartile deviation
- 11. Dispersion The relative measure of dispersion A relative measure of dispersion is independent of original units. Generally, relative measure of dispersion are expressed interns ratio, percentage etc. The relative measure of dispersion are as follows:- 1.Coefficient of range 2.Coefficient of mean deviation 3.Coefficient of standard deviation 4.Coefficient of quartile deviation
- 12. Standard Deviation The standard deviation is the positive square root of the mean of the square from their mean of a set of values. It is generally denoted by (sigma) and expressed is Merits of Standard Deviation It is rigidly defined and used in many general use It is less affected by sampling fluctuations It is useful for calculating the skewness, Kurtosis, Coefficient of correlation and so forth It measures the consistency of data. It is less erratic. Dispersion
- 13. Dispersion Demerits of Standard Deviation It is not so easy to compute It is affected by extreme values. Merits of Mean Deviation It is easy to compute and understand. It is based on all the value of set. It is useful measure of dispersion. It is not greatly affected by extreme value.
- 14. Dispersion Demerits of Mean Deviation This measure is not sates factory unless the data is symmetrical. This is not suitable for further mathematical treatment. Use of Mean Deviation It is used in certain economic and anthropological studies. It is often sufficient when an informal measure of dispersion required.
- 15. Dispersion Merits of Quartile Deviation It is easy to compute and simple to understand. It is not affected by extreme value. This measure of dispersion is superior to range. It is useful also in measuring variation in case of open ended distribution.
- 16. Dispersion Demerits of Quartile Deviation It is not based on all the observations in the series. It is highly affected by sometimes fluctuations. It is not suitable for further algebraic or mathematical treatment. Advantages of Range It is the simplest measure of dispersion. It is easy to compute the range and easy to understand.
- 17. Dispersion It is based on extreme observations only and no detail information is required. The chief merit of range is that it gives us a idea of the variability of a set of data. It does not depend on the measures of central tendency. Disadvantages of Range The range is not precise measure. It is influenced completely by the extreme values. It cannot be computed for open ended distribution. It is not suitable for further mathematical treatment.
- 18. Dispersion Difference between absolute and relative measure of dispersion
- 19. Difference between absolute and relative measure of dispersion
- 20. Dispersion The good measure of dispersion It should be rigidly defined. It should be easy to understand and easy to calculate. It should be based on all the values of the given data. It should be useful for further mathematical treatment. It should be less affected by sampling fluctuation. It should be unduly influenced by extreme values of all the observations.
- 21. Dispersion Range The range is easy to compute and rigidly defined but it is not based on all the observations. It is influenced by extreme values and not useful for further mathematical treatment. It is not use for open class interval and also affected by sampling fluctuation. This measure is employed in quality control whether forecast etc.
- 22. Mean deviation Mean deviation rigidly defined and easy to understand. It is based on all observations. It is capable for algebraic manipulation. It is less affected by sampling function and extreme values. It can’t be determine for open class interval. It is freely used in stoical analysis of economics, business etc. It is less suitable than standard deviation.
- 23. Dispersion Quartile deviation Quartile deviation rigidly defined to easy to understand and easy to calculate. It is not based on all the values. It is not capable to further mathematical treatment and affected by extreme values. It is used to measure variation in open-end distribution. It is not influenced by extreme values. It is suitable to study the social phenomenon.
- 24. Standard deviation Standard deviation rigidly defined. It is based on all the values. It is less affected by sampling function. It is capable for further algebraic manipulation. It is not easy to understand and not easy to calculated. It is not suitable for open-end distribution. It is affected by extreme values. Standard deviation is extensively used in the theory of sampling, regression, correlation, analysis of variance etc.
- 25. Standard deviation From the above comparison of dispersion, standard deviation is the most popular and used method for measuring dispersion in a series. It has great practical utility in sampling and statical inference. It is the most important measure of variation which is one of the pillars of statistics.
- 27. Differentiate between: SD and CV
- 28. Differentiate between: MD and SD
- 29. Differentiate between: Variance and CV
- 30. Standard deviation is the best measure of dispersion Standard deviation is the most popular and used method for measuring dispersion in a series. It has great practical utility in sampling and statically inference. It is the most important measure of variation which is one of the pillars of statistics. Standard deviation is extensively used in the theory of sampling, regression, correlation, analysis of variance etc.
- 31. Coefficient of correlation Karl Pearson’s Karl Pearson’s Coefficient of Correlation is widely used mathematical method where in the numerical expression is used to calculate the degree and direction of the relationship between linear related variables. Pearson’s method, popularly known as a Pearson an Coefficient of Correlation, is the most extensively used quantitative methods in practice. The coefficient of correlation is denoted by “r”.
- 32. Coefficient of correlation If the relationship between two variables X and Y is to be ascertained, then the following formula is used: Where, mean of X variable mean of Y variable Scatter diagram A graphic representation of vicariate data as a set of points in the plane that have Cartesian coordinates equal to corresponding values of the two varieties.
- 33. Coefficient of correlation Rank correlation (Spearman’s) This method of finding out co-variability or the lack of it between two variables was developed by the British psychologist Charles, Edward Spearman in 1904. This measure is especially useful when quantitative measure of certain factor (such as in the evaluation of leadership ability or the judgment of female beauty) cannot be fixed, but the individuals in the group can be arranged in order thereby obtaining for each individual a number indicating his rank in the group.
- 35. Coefficient of correlation Significant of the coefficient of correlation indicator. It makes understand the value of r. The closer r to +1 or -1, the closer the relationship between the variables and the closer r is to 0, the less closely the relationship. Beyond this is not safe to go. The fill interpretation of r depends upon circumstances, one of which is the size of simple. All that can really be said that when estimating the value of one variable from the value of another; the higher the value of r, the better the estimate. So, r should be the value, it has more significant.
- 36. Correlation Perfect correlation When a change in the values of one variable associated whit a corresponding and proportional change in the other then the correlation is called perfect correlation. Curvilinear correlation When the change in the values of one variable tends to be a constant ratio with the corresponding change in the other variable, then the correlation is said to be linear.
- 37. Example:
- 39. Correlation Characteristics of correlation The major characteristics of Pearson’s coefficient of correlation, r is as follows: (a). The value of r lies between (-1, +1) When, r= + 1, then there exists perfect positive correlation. When, r= - 1, then there exists perfect negative correlation. (b). When r=0, then there is no linear relationship between two variables but there may be curvilinear or non-linear relationship. When, r= 0, then the two variables may also be independent
- 40. Correlation (c). It gives the degree of concomitant movements or variation between two variables. (d). ‘r’ is independent of origin and scale. (e). The coefficient of correlation is the geometric mean of two regression coefficients. Symbolically,
- 41. Correlation Limitation of correlation To determine the coefficient of correlation (r) we have to assume that there is a liner relationship and non- linear relationship. It is valid when we have a random sample from a vicariate normal distribution. If the sample size is small then it doesn’t give us a better result to determine the relation
- 42. Variable Endogenous variable Endogenous variables are used in econometrics and sometimes in linear regression. They are similar to(but not exactly the same as) dependent variables. According to Daniel Little, University of Michigan- Dearborn, an endogenous Variable is defined in the following way: A variable is said to be endogenous within the causal model M if its value is determined or influenced by one or more of the independent variables X(excluding itself). Exogenous Variables An exogenous variable is a variable that is not affected by other variables in the system.
- 44. Variable Least square method For finding out the correlation of coefficient by least square method we have to calculate the values of two regression coefficient: 1. The regression coefficient of x on y. 2. The regression coefficient of y on x. Then the correlation coefficient is the square root of the product term of the regression coefficients symbolically
- 45. Correlation Degree of correlation Perfect correlation: If the value of r is near 1 it said to be perfect correlation; as one variable increase (or decrease), the other variable tend to also increase (or decrease). Higher degree of correlation: If the coefficient of r lies between 50 and bellow 1 then it is said to high degree on strong correlation. Moderate degree: If the value lies between 0.30 and 0.49 then it is said to be a medium/moderate degree of correlation. Low degree: When the value lies below 0.29 it is said to be low degree of correlation. No correlation: When the value of r = 0 then there is no correlation.
- 46. Correlation and Regression analysis
- 48. Regression Regression is a statistical measure used in finance, investing and disciplines that attempts to determine the strength of the relationship between one dependent variable Importance of Regression The regression equation provides a coincide and meaningful summery of the relationship between the dependent variable y and independent variable x. The relation can be used for predictive purpose. If the form of the relationship between x and by x is known then the parameters of interest can be estimated from relevant data. In situations, where the variable of interest(y) depends on a number of factors, then it is possible to assess and entangle the contribution of the factors individually with the help of regression analysis.
- 49. Regression Simple Regression When the dependency of the dependent variable is estimated by only one independent variable then it is said simple regression. Multiple Regression When there are more than one independent variables then it is said multiple regression. Equation for multiple regression:
- 50. Regression Regression coefficient When the regression line (y=+) the regression coefficient is the constant () that represents the rate of change of one variable(y) as a function of changes in the other (x); it is the slope of the regression line. And alternative, Here, regression coefficient of y on x and = regression coefficient of x on y.
- 51. Probability Distributions A listing of all the values the random variable can assume with their corresponding probabilities make a probability distribution. Probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. A note about random variables. A random variable does not mean that the values can be anything (a random number). Random variables have a well defined set of outcomes and well defined probabilities for the occurrence of each outcome. The random refers to the fact that the outcomes happen by chance – that is, you don’t know which outcome with occur next.
- 52. Probability Distributions Here’s an example probability distribution that results from the rolling of a single fair die. Probability distributions are either continuous probability distributions or discrete probability distributions, depending on whether they define probabilities for continuous or discrete variables. Continuous and discrete probability distributions
- 54. Probability Distributions Continuous distribution A continuous distribution describes the probabilities of the possible values of a continuous random variable. A continuous random variable is a random variable with a set of possible values (known as the range) that is infinite and uncountable. Probabilities of continuous random variables (X) are defined as the area under the curve of its PDF. Thus, only ranges of values can have a nonzero probability. The probability that a continuous random variable equals some value is always zero.
- 55. Probability Distributions Distribution of weights The continuous normal distribution can describe the distribution of weight of adult males. For example, you can calculate the probability that a man weighs between 160 and 170 pounds.
- 56. Probability Distributions Distribution plot of the weight The shaded region under the curve in this example represents the range from 160 and 170 pounds. The area of this range is 0.136; therefore, the probability that a randomly selected man weighs between 160 and 170 pounds is 13.6%. The entire area under the curve equals 1.0. However, the probability that X is equal to some value is zero because the area under the curve at a single point, which has no width, is zero. For example, the probability that a man weighs exactly 190 pounds to infinite precision is zero. You could calculate a nonzero probability that a man weighs more than 190 pounds, or less than 190 pounds, or between 189.9 and 190.1 pounds, but the probability that he weighs exactly 190 pounds is zero.
- 58. Probability Distributions Discrete distribution A discrete distribution describes the probability of occurrence of each value of a discrete random variable. A discrete random variable is a random variable that has countable values, such as a list of non- negative integers. With a discrete probability distribution, each possible value of the discrete random variable can be associated with a non-zero probability. Thus, a discrete probability distribution is often presented in tabular form.
- 59. Probability Distributions Number of customer complaints With a discrete distribution, unlike with a continuous distribution, you can calculate the probability that X is exactly equal to some value. For example, you can use the discrete Poisson distribution to describe the number of customer complaints within a day. Suppose the average number of complaints per day is 10 and you want to know the probability of receiving 5, 10 and 15 customer complaints in a day. You can also view a discrete distribution on a distribution plot to see the probabilities between ranges.
- 62. Probability Distributions Discrete distribution Binomial and Poisson are discrete probability distribution and normal distribution is continuous probability distribution. Binomial distribution A random variable X is said to follow binomial distribution if it assume only non negative values and its probability mass function is given by P(X=x) x=0,1,2,3,…….n P+q=1,q=1-p, 0 P P= success, q=failure, n= number of Bernoulli trials When n=1, it is called Bernoulli distribution
- 63. Probability Distributions Condition of binomial distribution The experiment consist of n repeated trials Each trail result in just two possible outcomes, we call one of these outcomes a success and other, a failure The probability of success, denoted by p, is the same on every trail The trails are independent; that is the outcomes on trail does not affect the outcome other trails
- 64. Probability Distributions Properties of binomial distribution It is discrete probability distribution Mean and variance of binomial distribution are np and npq respectively. Parameters of binomial distribution are n and p When n is large enough i.e, n and p then binomial distribution tends to normal distribution When n is large enough i.e, n then binomial distribution tends to Poisson distribution.
- 65. Probability Distributions Normal of Poisson distribution A random x is said to have a normal distribution if and only if for >0 and - , the density function of x is y = f(x)= , - y = the computed height of an ordinate at a distance of x from the mean = Standard deviation of the given normal distribution = 3.1416 e = the constant = 207183 = mean or average of the given normal distribution.
- 66. Probability Distributions The graph of y = f(x) is a famous “bell shaped” curve
- 67. Probability Distributions A random variable with any mean and standard deviation can be transformed to a standardized normal variable by subtracting the mean and dividing by standard deviation. For a normal distribution with mean and standard deviation , the standardized variable z obtained as z =
- 68. Probability Distributions Relation of Binomial, Poisson and Normal distribution When n is large and the probability p of occurrence of an event is close to zero so that np remains a finite constant, then the binomial distribution tends to Poisson distribution. Similarly, there is a relation between binomial and distributions. Normal distribution is limiting from of binomial distribution under the following conditions: n, the number of trials is very large, i.e n Neither p nor q is very small.
- 69. Probability Distributions In fact, it can be proved that the binominal distribution approaches a normal distribution with standardized normal variable, I.e., z = or, z = will follow the normal distribution with mean zero and variance one. Similarly, Poisson distribution also approaches a normal distribution with standardized normal variable, i.e. , z = In other words, z = will follow the normal distribution with mean zero and variance one.