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   For a single sum if I use all the 3 methods to find mean will I get the same answer

Asked by seeni2005 27th April 2016, 7:54 PM
Answered by Expert
Answer:
T h e r e space a r e space t h r e e space m e t h o d s space t o space f i n d space t h e space m e a n space o f space a space g r o u p e d space d a t a colon
1. space D i r e c t space m e t h o d
2. space S h o r t c u t space m e t h o d
3. space S t e p space D e v i a t i o n space m e t h o d

C o n s i d e r space a n space e x a m p l e colon
F i n d space t h e space m e a n space o f space t h e space f o l l o w i n g space d i s t r i b u t i o n colon
x colon space 4 space space space space 6 space space space space 9 space space space 10 space space 15
f colon space 5 space space space 10 space space 10 space space space space 7 space space space space 8

S o l u t i o n colon
1.
L e t space u s space c a l c u l a t e space t h e space m e a n space b y space d i r e c t space m e t h o d colon
table row cell x subscript i end cell cell f subscript i end cell cell f subscript i x subscript i end cell row 4 5 20 row 6 10 60 row 9 10 90 row 10 7 70 row 15 8 120 row cell t o t a l end cell cell N equals sum for blank of f subscript i equals 40 end cell cell sum for blank of f subscript i x subscript i equals 360 end cell end table
therefore M e a n equals X with bar on top equals fraction numerator sum for blank of f subscript i x subscript i over denominator sum for blank of f subscript i end fraction equals 360 over 40 equals 9

2.
L e t space u s space c a l c u l a t e space t h e space m e a n space b y space s h o r t c u t space m e t h o d.
L e t space A s s u m e d space m e a n space b e space 9 comma space A equals 9
T h u s comma space w e space h a v e comma
table row cell x subscript i end cell cell f subscript i end cell cell d subscript i equals x subscript i minus A end cell cell f subscript i d subscript i end cell row 4 5 cell negative 5 end cell cell negative 25 end cell row 6 10 cell negative 3 end cell cell negative 30 end cell row 9 10 0 0 row 10 7 1 7 row 15 8 6 48 row blank cell N equals sum for blank of f subscript i equals 40 end cell blank cell sum for blank of f subscript i d subscript i equals 0 end cell end table
therefore M e a n equals A plus fraction numerator sum for blank of f subscript i d subscript i over denominator sum for blank of f subscript i end fraction equals 9 plus 0 over 40 equals 9
3.
L e t space u s space c a l c u l a t e space t h e space m e a n space b y space s t e p space d e v i a t i o n space m e t h o d.
L e t space A s s u m e d space m e a n space b e space 10 comma space A equals 10 space a n d space h equals 2
T h u s comma space w e space h a v e comma
table row cell x subscript i end cell cell f subscript i end cell cell d subscript i equals x subscript i minus A end cell cell u subscript i equals fraction numerator x subscript i minus A over denominator h end fraction end cell cell f subscript i u subscript i end cell row 4 5 cell negative 6 end cell cell negative 3 end cell cell negative 15 end cell row 6 10 cell negative 4 end cell cell negative 2 end cell cell negative 20 end cell row 9 10 cell negative 1 end cell cell negative 1 half end cell cell negative 5 end cell row 10 7 0 0 0 row 15 8 5 cell 5 over 2 end cell 20 row blank cell N equals sum for blank of f subscript i equals 40 end cell blank blank cell sum for blank of f subscript i u subscript i equals negative 20 end cell end table
therefore M e a n equals A plus h open curly brackets fraction numerator sum for blank of f subscript i u subscript i over denominator sum for blank of f subscript i end fraction close curly brackets equals 10 plus 2 open curly brackets fraction numerator negative 20 over denominator 40 end fraction close curly brackets equals 10 minus 1 equals 9

S o space t h e space v a l u e space o f space m e a n space c a l c u l a t e d space i n space a l l space t h e space t h r e e space m e t h o d s space i s space s a m e.
B u t space t h e space v a l u e space o f space m e a n space d e p e n d s space o n space t h e space n u m b e r space o f space o b s e r v a t i o n s comma space A s s u m e d space m e a n space a n d space t h e space d i v i d i n g space f a c t o r space h.
S o space t h e r e space m a y space b e space s o m e space v a r i a t i o n space i n space t h e space v a l u e s space o f space m e a n left parenthesis i n space d e c i m a l s right parenthesis
W e space a r e space u sin g space d i f f e r e n t space m e t h o d s space a c c o r d i n g space t o space o u r space n e e d s space a n d space o u r space m a i n space a i m space i s space t o space r e d u c e space t h e space c a l c u l a t i o n space p a r t.
I n space c o n c i s e space t h e space v a l u e space o f space m e a n space s h o u l d space b e space a p p r o x i m a t e l y space left parenthesis i n space d e c i m a l s right parenthesis space t h r o u g h space a l l space m e t h o d s.
Answered by Expert 28th April 2016, 9:44 AM
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