1. 1 STAT 500 – Statistics for Managers STAT 500 Statistics for Managers

2. 2 STAT 500 – Statistics for Managers Objectives for this session Discuss 3 properties of data Identify measures of central tendency Identify measures of dispersion Discuss shape of the data distribution

3. 3 STAT 500 – Statistics for Managers Objectives for this session Discuss 3 properties of data Identify measures of central tendency Identify measures of dispersion Discuss shape of the data distribution

4. 4 STAT 500 – Statistics for Managers Three Properties of Data Central Tendency: It is concerned with the typical or the average case in the distribution of dataset. Dispersion It is concerned with how much variety or heterogeneity or non-uniformity or spread is there in the data distribution. Shape It is concerned with the direction of the spread.

5. 5 STAT 500 – Statistics for Managers Three Properties of Data (Example) Central Tendency: Average salary of UNVA MBA graduates is $65,000 per year. Dispersion The starting salary for UNVA MBA graduates is from $40,000 per year to $80,000 per year. Shape The starting salary is skewed to the left, i.e., the values are spread out more in the left side than in the right side.

6. 6 STAT 500 – Statistics for Managers Objectives for this session Discuss 3 properties of data Identify measures of central tendency Identify measures of dispersion Discuss shape of the data distribution

7. 7 STAT 500 – Statistics for Managers Measures of Central Tendency Mean Median Mode Midhinge Midrange

8. 8 STAT 500 – Statistics for Managers Measures of Central Tendency Mean Median Mode Midhinge Midrange

9. 9 STAT 500 – Statistics for Managers Mean The Most Common Measure of Central Tendency Add all the data values and divide by the number of observations to compute Mean, in other words, it is the arithmetic average. Affected by Extreme Values (Outliers)

10. 10 STAT 500 – Statistics for Managers Mean (Example) Consider the midterm examination scores of 5 students in STAT 500 course at UNVA: 72, 87, 92, 76, 89

11. 11 STAT 500 – Statistics for Managers Measures of Central Tendency Mean Median Mode Midhinge Midrange

12. 12 STAT 500 – Statistics for Managers Median Another Measure of Central Tendency Order the Data from Smallest to Largest Determine Whether the Sample Size (n) is Even or Odd If n is Odd, M = Middle Value If n is Even, M = Average of the Two Middle Values Not Affected by Extreme Values (Outliers)

13. 13 STAT 500 – Statistics for Managers Median (Example) Consider the midterm examination scores of 6 students in STAT 500 course at UNVA: 72, 87, 92, 76, 89, 93 Ordered Data: 72, 76, 87, 89, 92, 93 Positioning Point: (n + 1) / 2 = (6 + 1) / 2 = 3.5 Take the average of the third and the fourth value, i.e., (87 + 89) / 2 = 88 Median = 88

14. 14 STAT 500 – Statistics for Managers Measures of Central Tendency Mean Median Mode Midhinge Midrange

15. 15 STAT 500 – Statistics for Managers Mode Another Measure of Central Tendency It is the value that occurs most frequently Suitable for nominal level variables. Not Affected by Extreme Values There may not be a mode and there may be several modes Used for either Numerical or categorical Data

16. 16 STAT 500 – Statistics for Managers Measures of Central Tendency Mean Median Mode Midhinge Midrange

17. 17 STAT 500 – Statistics for Managers Percentiles 20% Lowest Value Highest Value Range of Data Values Percentile: Data Value that Divides the Data into Two Groups with a Stated Percentage Below the Percentile 80% 20 th Percentile

18. 18 STAT 500 – Statistics for Managers Q 1 25th Percentile Q 1 25th Percentile Percentiles 25% Lowest Value Highest Value Q3Q3 M Lower Quartile First Quartile

19. 19 STAT 500 – Statistics for Managers Percentiles 50% Lowest Value Highest Value Q3Q3 Q1Q1 Second Quartile Q 1 25th Percentile Q 1 25th Percentile 25% Lower Quartile First Quartile Median 50th Percentile Median 50th Percentile

20. 20 STAT 500 – Statistics for Managers Percentiles 75% Lowest Value Highest Value Q1Q1 Upper Quartile Third Quartile Q 1 25th Percentile Q 1 25th Percentile 25% Lower Quartile First Quartile Median 50th Percentile Median 50th Percentile 50% Second Quartile Q 3 75th Percentile Q 3 75th Percentile

21. 21 STAT 500 – Statistics for Managers Quartiles 25th, 50th, 75th Percentiles (Divide the Data into Four Parts) 25th, 50th, 75th Percentiles (Divide the Data into Four Parts) Lowest Value Highest Value Q1Q1 M Q3Q3 1. Order the Data from Smallest to Largest 2. Positioning point of Q1: (n+1)/4 Positioning point of Median: (n+1)/2 Positioning point of Q3: 3 (n+1)/4

22. 22 STAT 500 – Statistics for Managers Quartiles Lowest Value Highest Value Q1Q1 M Q3Q3 Positioning point rule: If positioning point is exactly halfway between two integers, average their corresponding values. Otherwise, round the positioning point to the nearest integer.

23. 23 STAT 500 – Statistics for Managers Quartiles Lowest Value Highest Value Q1Q1 M Q3Q3 Q1 Positioning point: (7+1)/4 = 2 => Q1 = 82 75, 82, 87, 99, 102, 105, 107 N = 7

24. 24 STAT 500 – Statistics for Managers Quartiles Lowest Value Highest Value Q1Q1 M Q3Q3 Q1 Positioning point: (7+1)/4 = 2 => Q1 = 82 Q2 Positioning point: (7+1)/2 = 4 => Q2 = M = Median = 99 75, 82, 87, 99, 102, 105, 107 N = 7

25. 25 STAT 500 – Statistics for Managers Quartiles Lowest Value Highest Value Q1Q1 M Q3Q3 Q1 Positioning point: (7+1)/4 = 2 => Q1 = 82 Q2 Positioning point: (7+1)/2 = 4 => Q2 = M = Median = 99 Q3 Positioning point: 3 (7+1)/4 = 6 => Q3 = 105 75, 82, 87, 99, 102, 105, 107 N = 7

26. 26 STAT 500 – Statistics for Managers Midhinge Measure of central tendency Middle of 1 st & 3 rd quartiles (Q1 + Q3) / 2 Not affected by extreme values Midhinge for the example: Q1 = 82 Q3 = 105 Midhinge = 93.5

27. 27 STAT 500 – Statistics for Managers Measures of Central Tendency Mean Median Mode Midhinge Midrange

28. 28 STAT 500 – Statistics for Managers Midrange Measure of central tendency Mid-Point of the smallest & the largest observation Affected by extreme values

29. 29 STAT 500 – Statistics for Managers Objectives for this session 1 (Part 2) Discuss 3 properties of data Identify measures of central tendency Identify measures of dispersion Discuss shape of the data distribution

30. 30 STAT 500 – Statistics for Managers Measures of Dispersion Range InterQuartile Range Variance Standard Deviation Coefficient of Variation

31. 31 STAT 500 – Statistics for Managers Measures of Dispersion Range InterQuartile Range Variance Standard Deviation Coefficient of Variation

32. 32 STAT 500 – Statistics for ManagersRange Measure of dispersion (Please recall the dispersion is concerned with the uniformity of data, e.g., traffic in the high way, cutting of vegetables/meat for cooking, etc.) Measure of dispersion (Please recall the dispersion is concerned with the uniformity of data, e.g., traffic in the high way, cutting of vegetables/meat for cooking, etc.) Difference between largest & smallest observations Difference between largest & smallest observations Ignores how data are distributed Ignores how data are distributed

33. 33 STAT 500 – Statistics for Managers Measures of Dispersion Range InterQuartile Range Variance Standard Deviation Coefficient of Variation

34. 34 STAT 500 – Statistics for Managers InterQuartile Range It is a measure of dispersion Range in middle 50% Not affected by extreme values Computation:Q3 - Q1

35. 35 STAT 500 – Statistics for Managers Measures of Dispersion Range InterQuartile Range Variance Standard Deviation Coefficient of Variation

36. 36 STAT 500 – Statistics for Managers Variance It is measure of dispersion. It measures spread from the mean Calculated as follows s 2 = SUM (x i - xbar) 2 / (n-1) (Use n-1 for sample, and n for population)

37. 37 STAT 500 – Statistics for Managers Measures of Dispersion Range InterQuartile Range Variance Standard Deviation Coefficient of Variation

38. 38 STAT 500 – Statistics for Managers Standard Deviation It is measure of dispersion. It measures spread from the mean. It is the square root of variance It is a popular measure of risk used in finance literature.

39. 39 STAT 500 – Statistics for Managers Variance (Example) Consider a sample of 5 observations.

40. 40 STAT 500 – Statistics for Managers Measures of Dispersion Range InterQuartile Range Variance Standard Deviation Coefficient of Variation

41. 41 STAT 500 – Statistics for Managers Coefficient of Variation It is a relative measure of dispersion. It is expressed in % (standard deviation as a percentage of mean). Formula: (Standard Deviation / Mean) * 100 It helps to compare variations in two sets of data.

42. 42 STAT 500 – Statistics for Managers Coefficient of Variation (Example) Consider Dow Jones Industrial Average and NASDAQ indexes during the year 2001: DJIA – Mean = 10,200 Std Dev = 2000 NASDAQ – Mean = 1,500 Std Dev = 825 CV for DJIA: 19.6% CV for NASDAQ: 75% Conclusion: NASDAQ has wider variation than that of DJIA.

43. 43 STAT 500 – Statistics for Managers Objectives for this session Discuss 3 properties of data Identify measures of central tendency Identify measures of dispersion Discuss shape of the data distribution

44. 44 STAT 500 – Statistics for Managers Box Plot Calculations Presentation of summary measures in a graphical form. Five elements of interest are: Minimum, Q1, Median, Q3, Maximum

45. 45 STAT 500 – Statistics for Managers Q3 Q1 Median Maximum Minimum Box Plot

46. Symmetric Min Q M Q Max 1 3 Right-Skewed Min Q M Q Max 1 3 Left-Skewed Min Q M Q Max 1 3

47. 47 STAT 500 – Statistics for Managers Measures of Shape Symmetric or skewed Mean to the left of the median – left skewed Mean to the right of the median – right skewed Right-Skewed Left-SkewedSymmetric Mean =Median Mean Median Mean

48. 48 STAT 500 – Statistics for Managers  Objectives for this session  Discuss 3 properties of data  Identify measures of central tendency  Identify measures of dispersion  Discuss shape of the data distribution