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Coverage ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Introduction to Notations ,[object Object],[object Object]
Sigma ,[object Object],Σ
Summation Notation ,[object Object],[object Object],Σ  X i  = X 1 +X 2 + … +X n n i=1
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
Rules of Summation ,[object Object],Σ  (X i +Y i ) =  Σ  X i +  Σ  Y i   n i=1 n i=1 n i=1 the  sum of  their  summations
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 .
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…
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 .
MEASURES OF CENTRAL TENDENCY ,[object Object]
Mean ,[object Object],[object Object]
Population mean  μ  =  Σ  X i N i=1 N Sample mean  x  =  Σ  X i n i=1 n
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
Properties - Mean ,[object Object],[object Object],[object Object],[object Object]
Median ,[object Object],[object Object]
If n is  odd : Md   = X (n+1)/2 If n is  even : Md   = X n/2  + X (n/2)+1 2
Properties - Median ,[object Object],[object Object],[object Object],[object Object],[object Object]
Mode ,[object Object],[object Object],[object Object]
Properties - Mode ,[object Object],[object Object],[object Object]
MEASURES OF VARIABILITY AND DISPERSION ,[object Object]
Range ,[object Object],[object Object],[object Object],[object Object]
Properties - Range ,[object Object],[object Object],[object Object],[object Object]
Absolute Deviation ,[object Object],[object Object]
MD   =  Σ  | X i  - X | n i=1 n MAD   =  Σ  | X i  - Md | n i=1 n
Properties – Absolute Deviation ,[object Object],[object Object]
Variance ,[object Object],[object Object]
σ 2   =  Σ  ( X i  -  μ  ) 2 N i=1 N s 2   =  Σ  ( X i  – X ) 2 n i=1 n -1
s 2   = n Σ  X i 2  - ( Σ  X i  ) 2 n i=1 n(n -1) n i=1
Properties – Variance ,[object Object],[object Object]
Standard Deviation ,[object Object],[object Object]
σ   =  Σ  ( X i  -  μ  ) 2 N i=1 N s   =  Σ  ( X i  – X ) 2 n i=1 n -1
Why n-1? ,[object Object],[object Object],[object Object],[object Object],[object Object]
Properties – Standard Deviation ,[object Object],[object Object],[object Object]
Properties – Standard Deviation ,[object Object],[object Object]
Choosing a measure ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Choosing a measure ,[object Object],[object Object],[object Object],[object Object]
FREQUENCY DISTRIBUTION ,[object Object]
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
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
Frequency Distribution Table ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Frequency Distribution Table ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Constructing an FDT ,[object Object],[object Object],[object Object],[object Object],[object Object]
Constructing an FDT ,[object Object],[object Object],[object Object],[object Object]
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
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
Other Terms ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
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
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
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
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
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
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

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Statistics in Research

  • 1.
  • 2.
  • 3.
  • 4.
  • 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
  • 6.
  • 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 .
  • 10.
  • 11.
  • 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
  • 14.
  • 15.
  • 16. If n is odd : Md = X (n+1)/2 If n is even : Md = X n/2 + X (n/2)+1 2
  • 17.
  • 18.
  • 19.
  • 20.
  • 21.
  • 22.
  • 23.
  • 24. MD = Σ | X i - X | n i=1 n MAD = Σ | X i - Md | n i=1 n
  • 25.
  • 26.
  • 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
  • 29.
  • 30.
  • 31. σ = Σ ( X i - μ ) 2 N i=1 N s = Σ ( X i – X ) 2 n i=1 n -1
  • 32.
  • 33.
  • 34.
  • 35.
  • 36.
  • 37.
  • 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
  • 40.
  • 41.
  • 42.
  • 43.
  • 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
  • 46.
  • 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