Published on September 23, 2008
Applied Statistics in Business & Economics Applied Statistics in Business & Economics Discrete Data List of observation eg 14,20, 23, 25, 28 Grouped Data List of observations & frequency e.g. Weekly Wages & No Of Labours Continuous Series In series from 1 value to another eg. Marks 0-10 10-20 20-30 30-40 40-50 # Of Student 6 11 14 20 15 Measures of Central Tendencies Arithmetic Mean Discrete Data Mean = X N Grouped Data Mean = fX f Continuous Series Mean = fX * X here is the Mid Value of the data in series eg. If Marks is 0-10, 10-20 then the f Mid Values will be 5, 15 respectively. Median Median is the middle most observation if the data is sorted. If the number of observations (N) is odd then the centre most value is the observation. Eg. In Observations: 14, 20, 23, 25, 28 Median is 23 If the number of observations (N) is even then the N/2-1th and N/2+1th observations are summed and divided by 2. Eg. In Observations: 14, 20, 23, 25, 28, 32 Median is 24 i.e. (23+25 / 2) Page 1 of 9
Applied Statistics in Business & Economics Mode Mode is the observation with highest frequency/occurrence. E.g. In Observations: 10, 20, 30, 30, 40 Mode is 30 because it occurs twice which is highest in the set. If there are more than one observation values which have the highest occurrence then there is NO Mode to the data. Eg. 10, 20, 20, 30, 30, 40, Here 20 and 30 both have Frequency of 2 which is the highest, in this case there is No Mode to the data. Midrange Midrange is the average of just the minimum value and maximum value. Xmin + Xmax Midrange = 2 E.g. In Observations: 10, 20, 30, 30, 40 Midrange is (10+40) / 2 i.e. 25 Geometric Mean - Trimmed Mean Similar to Arithmetic Mean, but a few extreme values are excluded. Page 2 of 9
Applied Statistics in Business & Economics Measures of Dispersion Dispersion, also known as scatter spread or variation, measures the extent to which the items vary from some central value and they measure only the degree but not the direction of variation. Significance of Measuring Dispersion To determine the reliability of an average. To facilitate comparison. To facilitate control. To facilitate the use of other statistical measures. Range Difference between minimum and maximum value Range = XMax – Xmin Mean Deviation Mean Deviation is the arithmetic mean of the absolute deviations of all items of the distribution from a measure of central tendency. If nothing is specified, ‘Mean Deviation’ means ‘Mean Deviation’ about the Arithmetic Mean. Steps to Compute Mean Deviation 1. Calculate the Arithmetic Mean 2. Take the absolute deviations of each observation from the Mean ( Say |D| ) 3. Calculate the sum of all these deviations i.e. | D | 4. Calculate the Mean Deviation by dividing this sum by total number of observation. |D| Mean Deviation = N Coefficient of Mean Deviation Mean Deviation Coefficient of Mean Deviation = Mean Standard Deviation ( ) Standard Deviation is the Square Root of the Arithmetic Mean of the Squares of deviations of all items of the distribution from the Arithmetic Mean. Page 3 of 9
Applied Statistics in Business & Economics Properties of Standard Deviation The Sum of the Squares of the Deviations of Items from Arithmetic Mean is minimal. If all the observations are added by the same constant C, then the standard deviation remains unchanged. If all the observations are multiplied by the same constant C, then the standard deviation will be | C | times the standard deviation. Standard Deviation is the square root of Variance. = x2 N Where x = X - X Variance Variance is the arithmetic mean of the squares of deviations of all items of distributions from the arithmetic mean in other words variance is the square of standard deviation. 2 V= Smaller the value of variance, lesser is the variability in population or greater the uniformity in population and vice-versa. Coefficient of Variation CV = X 100% X Page 4 of 9
Applied Statistics in Business & Economics Regression Analysis Regression is the measure of average relationship between 2 or more variables in terms of the original unit of the data. Regression analysis is a statistical tool to study the nature & extent of functional relationship between 2 or more variables and to estimate the unknown values of dependent variables from known values of independent variable. The terms dependence & independence does not necessarily indicate a cause- effect relationship between the variables. Regression Analysis is a valuable tool in business economics & business research. Linear Regression Incase if a Linear Regression model involving 2 variables there are 2 regression lines possible. Regression of X on Y and Regression of Y on X. Regression of X on Y X = a + bY Where, X = Dependent variable Y = Independent variable a = X Intercept. (Value of dependent variable when value of independent variable is zero). b = Slope of the Line. (The amount of change in the amount of dependent variable per unit change in independent variable). The value of constants a & b for the given data of X & Y can be calculated by solving the following two algebraic equations (simultaneous equations) called NORMAL EQUATIONS. X = aN + bY XY = aY + bY2 Where N is the number of pairs of X & Y variables and denotes the respective summation. Page 5 of 9
Applied Statistics in Business & Economics Regression of Y on X Y = a + bX Where, X = Independent variable Y = Dependent variable a = Y Intercept. (Value of dependent variable when value of independent variable is zero). b = Slope of the Line. (The amount of change in the amount of dependent variable per unit change in independent variable). The value of constants a & b for the given data of X & Y can be calculated by solving the following two algebraic equations (simultaneous equations) called NORMAL EQUATIONS. Y = aN + bX XY = aX + bX2 Page 6 of 9
Applied Statistics in Business & Economics Correlation Correlation is the relationship that exists between 2 or more variable. Correlation Analysis is a statistical technique to measure the degree and direction of relation between the variables. If both the variables incase in the same direction then the correlation is said to be positive. Whereas if both the variables vary in opposite direction, the correlation is said to be negative. When only 2 variables are considered, it is called simple correlation where as if 3 or more variables are considered then it is multiple correlations. Covariance Given a set of N pairs , our observation relating to 2 variables X & Y, the covariance pf X & Y are denoted by COV(X,Y) and is given by the formula: (X – X ) ( Y- Y ) COV(X,Y) = N Covariance may be +ve, –ve or zero and take any value from - to + Coefficient Of Correlation (Karl Pearson’s) Given a set of ‘N’ pairs of observations relating to 2 variables X & Y, the coefficient of Correlation between X & Y is denoted by the symbol ‘ r ‘ And is given by the formula COV(X,Y) r= x . y Where x and y are the standard deviations of variables X & Y respectively. Spearman’s Rank Correlation Spearman’s Rank Correlation uses ranks rather than actual observations and makes no assumptions about the population from which actual observations are drawn. The correlation coefficient between 2 series of ranks is called rank correlation coefficient. Page 7 of 9
Applied Statistics in Business & Economics It is denoted by ‘ R ‘ and is given by the formula 6 D2 r= 1- N3 - N Where D is the difference of ranks between paired items in 2 series and N is the number of pairs of ranks. ‘R’ lies between -1 and +1 ( -1 <= R >= +1 ), it can be interpreted in the same fashion as Karl Pearson’s Coefficient of Correlation. The Sum of difference of ranks will always be Zero. i.e. D = 0 Spearman’s Rank Correlation Coefficient is very useful when dealing with Qualitative data. Incase of tied ranks, average rank is allotted to each of these items and the factor (m3 – m) / 12 is added to D2 for each instance of such tie. Coefficient of Determination The coefficient of determination is defined as the ratio of explained variance to the total variance. The coefficient of determination is calculated by squaring the coefficient of correlation. Coefficient of Determination = r2 For illustration , if r = 0.8 then r2 = 0.64 which means that 64% of the variation in the dependent variable has been explained by the independent variable. r2 takes the values between 0 and 1. 0 <= r2 <= 1 Coefficient of Non Determination It is defined as the ratio between unexplained variance & total variance. It is denoted by K2 and its value is calculated by subtracting r2 from 1. K2 = 1 = r2 Page 8 of 9
Applied Statistics in Business & Economics Exercise For the following set of data X 10 20 30 40 50 60 70 80 90 Y 20 22 30 45 50 65 67 78 85 Central Tendencies 1. Find Mean of X & Y 2. Find Median of Y 3. Find Mode of Y 4. Find Midrange of Y Dispersion 5. Find Mean Deviation of Y 6. Find Coefficient of Mean Deviation of Y 7. Find Standard Deviation of X & Y 8. Find Variance of Y 9. Find Coefficient of Variation of Y Regression Analysis 10. Find Both Regression Equations ( X on Y and Y on X) 11. Find the value of Y when X is 100, 150 and 200 12. Find the value of X when Y is 100, 150 and 200 Correlation 13. Find Covariance 14. Find Coefficient of Correlation (Karl Pearson’s) Page 9 of 9
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