bus-stat-book1

If the mean is a fractional value then it becomes a

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Unformatted text preview: s about the actual mean. The moments about an origin are known as raw moments. 170 The first four raw moments – individual series. ∑( X − A ) 2 ∑ d 2 ∑( X − A ) ∑ d = µ1= = µ2= N N N N 3 3 4 ∑( X − A ) ∑d ∑( X − A ) ∑d4 µ3= = µ4= = N N N N Where A – any origin, d=X-A The first four raw moments – Discrete series (step – deviation method) X−A , A – origin , C – Common point C The first four raw Moments – Continuous series Where d = Where d = m−A A – origin , C – Class internal C 7.8 Relationship between Raw Moments and Central moments: Relation between moments about arithmetic mean and moments about an origin are given below. µ1 = µ 1 –µ 1 = 0 µ2 = µ 2 –µ 12 µ3 = µ 3 – 3µ 1 µ 2 + 2(µ 1)3 µ4 = µ 4 – 4µ 3 µ 1 + 6 µ 2 µ 12 - 3 µ 14 171 Example 17: Calculate first four moments from the following data. X: 0 12 3 4 5 6 7 F: 5 10 15 20 25 20 15 10 Solution: X f 5 10 15 20 25 20 15 10 5 N =125 0 1 2 3 4 5 6 7 8 fx 0 10 30 60 100 100 90 70 40 ∑fx =500 8 5 fd 500 ∑ fx = =4 N 125 0 ∑ fd = = =0 N 125 ∑ fd 3 0 = = =0 N 125 fd2 fd3 fd4 -20 -30 -30 -20 0 20 30 30 20 ∑fd =0 d=x- x (x-4) -4 -3 -2 -1 0 1 2 3 4 ∑d =0 80 90 60 20 0 20 60 90 80 ∑fd2 =500 -320 -270 -120 -20 0 20 120 270 320 ∑fd3 =0 1280 810 240 20 0 20 240 810 1280 ∑fd4 =4700 X= µ1 µ3 ∑ fd 2 500 = =4 µ2 = N 125 ∑ fd 4 4700 = = 37.6 µ4 = N 125 Example 18: From the data given below, first calculate the first four moments about an arbitrary origin and then calculate the first four moments about the mean. X: f: 30-33 2 33-36 36-39 4 26 39-42 42-45 47 15 172 45-48 6 Solution: X Midvalues (m) f 31.5 34.5 37.5 40.5 43.5 46.5 2 4 26 47 15 6 N= 100 30-33 33-36 36-39 39-42 42-45 45-48 µ µ d= (m − 37.5) 3 -2 -1 0 1 2 3 fd -4 -4 0 47 30 18 ∑fd’ =87 fd 2 fd 3 32 -16 8 4 -4 4 0 0 0 47 47 47 240 120 60 486 162 54 3 2 ∑fd’ = ∑fd’ = ∑fd’ 4= 809 309 173 1 = ∑ fd ' 87 261 = 2.61 ×c= ×c = N 100 100 2 = ∑ fd '2 173 × c2 = ×9 N 100 = 1557 = 15.57 100 ∑ fd '3 309 8343 =...
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