Review of Measures of Central Tendency, Dispersion & Association

1. Review of Measures of Central Tendency, Dispersion & Association • Graphical Excellence • Measures of Central Tendency • Mean, Median, Mode • Measures of Dispersion • Variance, Standard Deviation, Range • Measures of Association • Covariance, Correlation Coefficient • Relationship of basic stats to OLS

2. Graphical Excellence • The graph presents large data sets concisely and coherently. • The ideas and concepts to be delivered are clearly understood to the viewer. • The graph encourages the viewer to compare variables. • The display induces the viewer to address the substance of the data and not the form of the graph. • There is no distortion of what the data reveal.

3. Things to be cautious about when observing a graph: • Is there a missing scale on one axis. • Do not be influenced by a graph’s caption. • Are changes presented in absolute values only, or in percent form too.

4. Numerical Descriptive Measures • Measures of Central Tendency • Mean, Median, Mode • Measures of Dispersion • Variance, Standard Deviation • Measures of Association • Covariance, Correlation Coefficient

5. Sum of the measurements Number of measurements Mean = Arithmetic mean • This is the most popular and useful measure of central location Sample mean Population mean Sample size Population size

6. Example 2 Suppose the telephone bills of example 2.1 represent population of measurements. The population mean is • Example 1 The mean of the sample of six measurements 7, 3, 9, -2, 4, 6 is given by 6 7 3 9 4 4.5 42.19 15.30 53.21 43.59

7. When many of the measurements have the same value, the measurement can be summarized in a frequency table. Suppose the number of children in a sample of 16 employees were recorded as follows: NUMBER OF CHILDREN 0 1 2 3 NUMBER OF EMPLOYEES 3 4 7 2 16 employees • Example 3

8. Example 4 Seven employee salaries were recorded (in 1000s) : 28, 60, 26, 32, 30, 26, 29. Find the median salary. Suppose one employee’s salary of $31,000 was added to the group recorded before. Find the median salary. Odd number of observations 26,26,28,29,30,32,60 The median • The median of a set of measurements is the value that falls in the middle when the measurements are arranged in order of magnitude. Even number of observations First, sort the salaries. Then, locate the values in the middle First, sort the salaries. Then, locate the value in the middle There are two middle values! 29.5, 26,26,28,29, 30,32,60,31 26,26,28,29, 30,32,60,31 26,26,28,29, 30,32,60,31 26,26,28,29,30,32,60,31

9. The mode • The mode of a set of measurements is the value that occurs most frequently. • Set of data may have one mode (or modal class), or two or more modes. For large data sets the modal class is much more relevant than the a single- value mode. The modal class

10. Example 5 • The manager of a men’s store observes the waist size (in inches) of trousers sold yesterday: 31, 34, 36, 33, 28, 34, 30, 34, 32, 40. • The mode of this data set is 34 in. This information seems valuable (for example, for the design of a new display in the store), much more than “ the median is 33.2 in.”.

11. Relationship among Mean, Median, and Mode • If a distribution is symmetrical, the mean, median and mode coincide • If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ. A positively skewed distribution (“skewed to the right”) Mode Mean Median

12. ` • If a distribution is symmetrical, the mean, median and mode coincide • If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ. A negatively skewed distribution (“skewed to the left”) A positively skewed distribution (“skewed to the right”) Mode Mean Mean Mode Median Median

13. Measures of variability(Looking beyond the average) • Measures of central location fail to tell the whole story about the distribution. • A question of interest still remains unanswered: How typical is the average value of all the measurements in the data set? or How much spread out are the measurements about the average value?

14. Observe two hypothetical data sets Low variability data set The average value provides a good representation of the values in the data set. High variability data set This is the previous data set. It is now changing to... The same average value does not provide as good presentation of the values in the data set as before.

15. ? ? ? The range • The range of a set of measurements is the difference between the largest and smallest measurements. • Its major advantage is the ease with which it can be computed. • Its major shortcoming is its failure to provide information on the dispersion of the values between the two end points. But, how do all the measurements spread out? The range cannot assist in answering this question Range Largest measurement Smallest measurement

16. The variance • This measure of dispersion reflects the values of all the measurements. • The variance of a population of N measurements x1, x2,…,xN having a mean m is defined as • The variance of a sample of n measurementsx1, x2, …,xn having a mean is defined as

17. Sum = 0 Sum = 0 Consider two small populations: Population A: 8, 9, 10, 11, 12 Population B: 4, 7, 10, 13, 16 9-10= -1 11-10= +1 8-10= -2 12-10= +2 Thus, a measure of dispersion is needed that agrees with this observation. Let us start by calculating the sum of deviations The sum of deviations is zero in both cases, therefore, another measure is needed. A 8 9 10 11 12 …but measurements in B are much more dispersed then those in A. The mean of both populations is 10... 4-10 = - 6 16-10 = +6 B 7-10 = -3 13-10 = +3 4 7 10 13 16

18. Sum = 0 Sum = 0 9-10= -1 The sum of squared deviations is used in calculating the variance. See example next. 11-10= +1 8-10= -2 12-10= +2 The sum of deviations is zero in both cases, therefore, another measure is needed. A 8 9 10 11 12 4-10 = - 6 16-10 = +6 B 7-10 = -3 13-10 = +3 4 7 10 13 16

19. Let us calculate the variance of the two populations Why is the variance defined as the average squared deviation? Why not use the sum of squared deviations as a measure of dispersion instead? After all, the sum of squared deviations increases in magnitude when the dispersion of a data set increases!!

20. sA2 = SumA/N = 10/5 = 2 sB2 = SumB/N = 8/2 = 4 Which data set has a larger dispersion? Let us calculate the sum of squared deviations for both data sets However, when calculated on “per observation” basis (variance), the data set dispersions are properly ranked Data set B is more dispersed around the mean A B 1 2 3 1 3 5 SumA = (1-2)2 +…+(1-2)2 +(3-2)2 +… +(3-2)2= 10 5 times 5 times ! SumB = (1-3)2 + (5-3)2 = 8

21. Example 6 • Find the mean and the variance of the following sample of measurements (in years). 3.4, 2.5, 4.1, 1.2, 2.8, 3.7 • Solution A shortcut formula =[3.42+2.52+…+3.72]-[(17.7)2/6] = 1.075 (years)2

22. The standard deviation of a set of measurements is the square root of the variance of the measurements. • Example 4.9 • Rates of return over the past 10 years for two mutual funds are shown below. Which one have a higher level of risk? Fund A: 8.3, -6.2, 20.9, -2.7, 33.6, 42.9, 24.4, 5.2, 3.1, 30.05 Fund B: 12.1, -2.8, 6.4, 12.2, 27.8, 25.3, 18.2, 10.7, -1.3, 11.4

23. Solution • Let us use the Excel printout that is run from the “Descriptive statistics” sub-menu (use file Xm04-10) Fund A should be considered riskier because its standard deviation is larger

24. The coefficient of variation • The coefficient of variation of a set of measurements is the standard deviation divided by the mean value. • This coefficient provides a proportionate measure of variation. A standard deviation of 10 may be perceived as large when the mean value is 100, but only moderately large when the mean value is 500

25. Interpreting Standard Deviation • The standard deviation can be used to • compare the variability of several distributions • make a statement about the general shape of a distribution.

26. Measures of Association • Two numerical measures are presented, for the description of linear relationship between two variables depicted in the scatter diagram. • Covariance - is there any pattern to the way two variables move together? • Correlation coefficient - how strong is the linear relationship between two variables

27. The covariance mx (my) is the population mean of the variable X (Y) N is the population size. n is the sample size.

28. If the two variables move the same direction, (both increase or both decrease), the covariance is a large positive number. • If the two variables move in two opposite directions, (one increases when the other one decreases), the covariance is a large negative number. • If the two variables are unrelated, the covariance will be close to zero.

29. The coefficient of correlation • This coefficient answers the question: How strong is the association between X and Y.

30. Strong positive linear relationship +1 0 -1 COV(X,Y)>0 or r or r = No linear relationship COV(X,Y)=0 Strong negative linear relationship COV(X,Y)<0

31. If the two variables are very strongly positively related, the coefficient value is close to +1 (strong positive linear relationship). • If the two variables are very strongly negatively related, the coefficient value is close to -1 (strong negative linear relationship). • No straight line relationship is indicated by a coefficient close to zero.

32. Example 7 • Compute the covariance and the coefficient of correlation to measure how advertising expenditure and sales level are related to one another. • Base your calculation on the data provided in example 2.3

33. x y xy x2 y2 • Use the procedure below to obtain the required summations Similarly, sy = 8.839

34. Excel printout • Interpretation • The covariance (10.2679) indicates that advertisement expenditure and sales levelare positively related • The coefficient of correlation (.797) indicates that there is a strong positive linear relationship between advertisement expenditure and sales level. Covariance matrix Correlation matrix

35. The Least Squares Method • We are seeking a line that best fit the data • We define “best fit line” as a line for which the sum of squared differences between it and the data points is minimized. The y value of point i calculated from the equation of the line The actual y value of point i

36. Errors Y X Different lines generate different errors, thus different sum of squares of errors.

37. The coefficients b0 and b1 of the line that minimizes the sum of squares of errors are calculated from the data.