5. Summation Notation Σ X i = X 1 +X 2 + … +X n n i=1 Upper limit of summation Lower limit of summation Greek letter Sigma

7. Σ (a i +b i +…+z i ) = Σ a i + Σ b i + … + Σ z i n i=1 n i=1 n i=1 n i=1 The summation of the sum of variables is… the sum of their summations .

8. Rules of Summation Σ cX i = c Σ X i = c(X 1 +X 2 + … +X n ) n i=1 n i=1 If c is a constant, then…

9. Rules of Summation Σ c = nc n i=1 The summation of a constant is the product of upper limit of summation n and constant c .

12. Population mean μ = Σ X i N i=1 N Sample mean x = Σ X i n i=1 n

13. Mean in an Ungrouped Frequency Σ f i X i n i=1 n = (f 1 X 1 +f 2 X 2 + … +f n X n ) where f is the frequency of the occurring score n

16. If n is odd : Md = X (n+1)/2 If n is even : Md = X n/2 + X (n/2)+1 2

24. MD = Σ | X i - X | n i=1 n MAD = Σ | X i - Md | n i=1 n

27. σ 2 = Σ ( X i - μ ) 2 N i=1 N s 2 = Σ ( X i – X ) 2 n i=1 n -1

28. s 2 = n Σ X i 2 - ( Σ X i ) 2 n i=1 n(n -1) n i=1

31. σ = Σ ( X i - μ ) 2 N i=1 N s = Σ ( X i – X ) 2 n i=1 n -1

38. Raw data 74 79 69 72 53 76 62 82 84 87 96 72 79 68 71 50 75 60 81 84 86 91 72 77 66 69 50 75 59 80 82 85 88 72 77 66 69 50 75 60 81 83 85 89 73 78 68 70 50 75 60 81 83 86 89 73 59 65 69 50 75 77 80 82 84 87 73 79 68 71 51 76 62 81 84 87 92 73 79 68 71 52 76 62 82 84 87 94 74 79 68 71 53 76 62 82 84 87 94 50 57 63 69 72 74 77 80 82 84 87

39. Array 50 57 63 69 72 74 77 80 82 84 87 50 59 65 69 72 75 77 80 82 84 87 50 59 66 69 72 75 77 80 82 85 88 50 60 66 69 72 75 77 81 83 85 89 50 60 68 70 73 75 78 81 83 86 89 50 60 68 71 73 75 79 81 84 86 91 51 62 68 71 73 76 79 81 84 87 92 52 62 68 71 73 76 79 82 84 87 94 53 62 68 71 74 76 79 82 84 87 94 53 62 69 72 74 76 79 82 84 87 96

44. Array 50 57 63 69 72 74 77 80 82 84 87 50 59 65 69 72 75 77 80 82 84 87 50 59 66 69 72 75 77 80 82 85 88 50 60 66 69 72 75 77 81 83 85 89 50 60 68 70 73 75 78 81 83 86 89 50 60 68 71 73 75 79 81 84 86 91 51 62 68 71 73 76 79 81 84 87 92 52 62 68 71 73 76 79 82 84 87 94 53 62 68 71 74 76 79 82 84 87 94 53 62 69 72 74 76 79 82 84 87 96

45. Frequency Distribution Table Class Frequency LCB UCB RF <CF >CF 50-54 10 49.5 54.5 0.09 10 110 55-59 3 54.5 59.5 0.03 13 100 60-64 8 59.5 64.5 0.07 21 97 65-69 13 64.5 69.5 0.12 34 89 70-74 17 69.5 74.5 0.15 51 76 75-79 19 74.5 79.5 0.17 70 59 80-84 22 79.5 84.5 0.20 92 40 85-89 13 84.5 89.5 0.12 105 18 90-94 4 89.5 94.5 0.04 109 5 95-99 1 94.5 99.5 0.01 110 1

47. Mean from an FD X = Σ f i X i K i=1 Σ f i K i=1 where X i = class mark of the i th class

48. Median from an FD Md = LCB Md + C n/2 - <CF Md-1 where LCB Md = lower class boundary of median class <CF Md-1 = less than cumulative frequency preceeding the median class f Md

49. Mode from an FD Mo = LCB Mo + C f Mo - f Mo-1 where LCB Mo = lower class boundary of modal class f Mo , f Mo-1 , f Mo+1 = frequency of modal class, class preceding and class succeeding the modal class 2f Mo - f Mo-1 - f Mo+1

50. Mean Deviation from an FD MD = Σ f i |X i - X| n i=1 n where X i = class mark of the i th class n = total number of observations; total frequency, ie. n = Σ f i

51. Variance from an FD s 2 = Σ f i (X i - X) 2 n i=1 (n -1) where X i = class mark of the i th class n = total number of observations; total frequency, ie. n = Σ f i

52. Variance from an FD s 2 = n Σ f i X i 2 - ( Σ f i X i ) 2 n i=1 n(n -1) n i=1 where X i = class mark of the i th class n = total number of observations; total frequency, ie. n = Σ f i