What is the deivation of the formulae for finding median and mode of grouped data? (as listed in class 10 mathematics book)

Asked by reetatyagi8 | 6th Aug, 2015, 10:08: AM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
C o n s i d e r space t a k i n g space c l a s s space i n t e r v a l s space a l o n g space x minus a x i s space a n d space c u m u l a t i v e space f r e q u e n c e space a l o n g space y minus a x i s a n d space d r a w space a n space o g i v e space c u r v e space a s space a b o v e. W e space k n o w space t h a t space M e d i a n equals open parentheses N over 2 close parentheses to the power of t h end exponent space t e r m F equals c u m u l a t i v e space f r e q u e n c y space o f space t h e space c l a s s space p r e c e d i n g space t h e space m e d i a n space c l a s s. f equals f r e q u e n c y space o f space t h e space m e d i a n space c l a s s h equals c l a s s space w i d t h L subscript m equals L o w e r space l i m i t space o f space t h e space m e d i a n space c l a s s O g i v e space i s space a space c u r v e space a n d space w e space a r e space a s s u m i n g space t h a t space d a t a space a r e space u n i f o r m l y space d i s t r i b u t e d space w i t h i n space t h e space m e d i a n space c l a s s. A n d space w e space a r e space t r y i n g space t o space f i t space t h e space d a t a space i n space a space s t r a i g h t space l i n e. T h u s space w e space a r e space a p p r o x i m a t i n g space t h e space c u r v e space A S R D space a s space t h e space s t r a i g h t space l i n e space A C D. T h i s space m e t h o d space o f space f i t t i n g space t h e space d a t a space u sin g space l i n e a r space p o l y n o m i a l s space i s space c a l l e d space c u r v e space f i t t i n g comma space a space m e t h o d space o f space l i n e a r space i n t e r p o l a t i o n. T h r o u g h space N over 2 comma space d r a w space a space s t r a i g h t space l i n e space t o space m e e t space t h e space c u r v e space a t space S. N o w space c o n s i e r space t h e space t r i a n g l e s space A B C space a n d space A E D. W E space n e e d space t o space f i n d space x subscript 0. H e r e comma space angle B A C equals angle E A D angle A B C equals angle A E D equals 90 degree T h u s comma space b y space A A space s i m i l a r i t y comma space w e space h a v e comma triangle A B C tilde triangle A E D H e n c e space t h e space s i d e s space o f space t h e space t r i a n g l e space a r e space p r o p o r t i o n a l. rightwards double arrow fraction numerator A B over denominator U V end fraction equals fraction numerator B C over denominator D E end fraction rightwards double arrow fraction numerator space x subscript 0 over denominator h end fraction equals fraction numerator begin display style N over 2 end style minus F over denominator f end fraction rightwards double arrow space x subscript 0 equals open square brackets fraction numerator begin display style N over 2 minus F end style over denominator f end fraction close square brackets cross times h T h e r e f o r e comma space M e d i a n equals L subscript m plus x subscript 0 equals L subscript m plus open square brackets fraction numerator begin display style N over 2 minus F end style over denominator f end fraction close square brackets cross times h

Answered by Vimala Ramamurthy | 6th Aug, 2015, 12:35: PM

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