Understanding Measures of Central Tendency and Calculating Mean from Grouped Data
Learn about measures of central tendency, including mean calculation from both raw and grouped data. Explore the concepts of average position, comparing groups, and using assumed mean methods.
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CENTRAL TENDENCY OF MEASURES PREPARED FOR BA II YEAR
WHAT IS MEANT BY MEASURE OF CENTRAL TENDENCY TO MAKE DATA CONCISE AND REDUCE TO ONE SCORE TO SHOW CHARACTERISTICS OF WHOLE GROUP. THIS ONE SCORE REPRESENTS SCORES OF WHOLE GROUP ON CERTAIN VARIABLE. THIS SCORE IS CALLED MEASURE OF CENTRAL TENDENCY. SCORE REPRESENTING CENTRAL TENDENCY IS THAT AROUND WHICH MAXIMUM NUMBER OF SCORES CENTRALISE. THIS SCORE IS CALLED CENTRAL VALUE. FEW LARGE AND SMALL SCORES SCATTER FAR FROM CENTRAL VALUE.
PURPOSE OF MEASURES OF CENTRAL TENDENCY THE CENTRAL VALUE REPRESENTS AVERAGE POSITION. THE VALUE SHOWS CHARACTERISTIC OF ALL THE GROUP. ONE MAY COMPARE TWO OR MORE GROUPS WITH THE HELP OF CENTRAL VALUE. MODE, MEDIAN AND MEAN ARE MEASURES OF CENTRAL TENDENCY.
MEAN MEAN IS THAT SCORE WHICH CAN BE OBTAINED BY SUMMING ALL THE SCORES AND DIVIDING IT BY NUMBER OF SCORES. THE GENERAL FORMULA OF MEAN IS MEAN, M= X/N X=SUM OF ALL SCORES N= TOTAL NUMBER OF SCORES
EXAMPLE: 15 STUDENTS SCORED FOLLOWING SCORES ON MATHEMATICS : 31, 34, 28, 36, 33, 34, 26, 29, 26, 39, 37, 37, 32, 35, 39. X= 496 N= 15 M= X/N M=496/15=33.07
MEAN FROM GROUPED DATA LONG METHOD: M= fX/N WHERE, f=FREQUENCY OF CLASS X=MID POINT OF CLASS fX= SUMMATION OF MULTIPLICATION OF FREQUENIES OF DIFFERENT CLASS AND MID POINT OF THE CLASS CLASS FREQUENCY f MID POINT X fX M= fX/N fX=1980 N=80 M=1980/80=24. 75 45-49 40-44 35-39 30-34 25-29 20-24 15-19 10-14 5-9 0-4 2 4 5 16 18 12 9 7 4 3 47 42 37 32 27 22 17 12 7 2 94 168 185 512 486 264 153 84 28 6 i=5 N=80 fX=1980
SHORT METHOD OR ASSUMED MEAN METHOD M=A.M.+( fd/N)i A.M.=ASSUMED MEAN F= CLASS FREQUENCY d= DEVIATION OF CLASS fd=SUMMATION OF MULTIPLICATION OF FREQUENCY AND DEVIATION OF CLASS i=CLASS INTERVAL N=TOTAL FREQUENCY d= X-A.M./i
CLASS f d fd 45-49 40-44 35-39 30-34 25-29 1 4 5 12 18 +5 +4 +3 +2 +1 5 16 15 24 18 20-24 15 0 0 15-19 10-14 5-9 0-4 10 7 5 3 -1 -2 -3 -4 -10 -14 -15 -12 i=15 N=80 fd=+27 M=A.M.+( fd/N)I AM=22 fd=+27 N=80 i=5 M=22+(27/80)5 =22+1.69 =23.69
CLASS f d fd 50-54 45-49 40-44 35-39 2 5 8 15 4 3 2 1 12 15 16 15 30-34 32 0 0 25-29 20-24 15-19 10-14 20 12 4 2 -1 -2 -3 -4 -20 -24 -12 -8 i=5 N=100 fd=-10 M=A.M.+( fd/N)i AM=32 fd=-10 N=100 i=5 M=32+(-6/100)5 =31.70
ADVANTAGE OF MEAN THE MEAN IS THE BEST OF THE MEASURES OF CENTRAL TENDENCY. IT IS MOSTLY USED WHEN: WHEN SCORES NEEDED TO BE WEIGHTED ACCORDING TO THEIR SIZE. WHEN SCORES ARE DISTRIBUTED SYMMETRICALLY. WHEN THE MOST STABLE VALUES OF CENTRAL TENDENCY IS NEEDED TO OBTAINED. WHEN FATHER VALUES ARE NOT INFLUENCING THE VALUE OF MEDIAN. WHEN STANDARD DEVIATION AND CORRELATION IS NEEDED TO BE CALCULATED.
MODE MODE IS THAT VALUE THAT COMES MOST OF THE TIME IN THE DATASET. FOR EXAMPLE, 7,9,6,8,9,7,9,5,4,8,9 HAS 9 AS THE VALUE FOR MODE. IT IS DENOTED BY M0. WHEN THERE ARE TWO MODES IN THE DATA THEN VALUES ARE CALLED BI-MODAL AND WHEN THERE ARE MORE THAN TWO DATA THEN VALUES ARE CALLED MULTI MODAL. 4,5,7,7,7,8,9,10,12,12,12,15 IS BIMODAL AS THERE ARE TWO MODES, 7 AND 12. FOR GROUPED DATA THE CLASS HAVING MAXIMUM FREQUENCY, THE MID POINT OF CLASS IS CALLED AS MODE.
FOR GROUPED DATA FOLLOWING FORMULA IS USED FOR CALCULATING MODE: MODE, M0=L+(fm-fb/fm-fa+fm-fb)I where, L=lower limit of modal class fm=frequency of modal class fa=frequency of the class above modal class fb=frequency of the class below modal class i=class interval class f L=59.5 Fm=11 Fb=8 Fa=7 i=10 Mode=59.5+(11-8/(11-7)+(12-8)10 =63 90-99 2 80-89 7 70-79 7 60-69 11 50-59 8 40-49 30-39 20-29 10-19 7 3 1 2 i=10 N=50
MEDIAN MEDIAN DIVIDES THE SCORES OF WHOLE GROUP IN TO TWO EQUAL HALVES. MEDIAN IS THAT SCORE BELOW AND ABOVE WHICH LOWER AND HIGHER SCORES LIE. IT IS NECESSARY TO ARRANGE SCORES EITHER IN ASCENDING OR DESCENDING ORDER. MEDIAN CONSIDERS ORDER OF SCORES AND DIFFERENCE OF TWO SCORES DO NOT MAKE ANY DIFFERENCE. MEDIAN IS DENOTED WITH Md.
MEDIAN FROM UNGROUPED DATA For odd scores: 10,12, 7, 15, 20------7, 10, 12, 15, 20/20,15,12,10,7 FOR EVEN SCORES: 7,9,10,12,15,20 Md=(N/2)TH SCORE+(N/2+1)TH SCORE/2 =6/2TH SCORE+(6/2)+1TH SCORE/2 =(10+12)/2 =22/2 =11
FOR TIED SCORES 3,4,5,5,6,6,6,7,7,8 Md=L+(N/2-NB/f) WHERE, L=LOWER LIMIT OF THAT SCORE WHICH SITUATES IN THE MID POINT N=TOTAL NUMBER OF SCORES NB=NUMBER OF SCORES LIE BELOW L f=FREQUENCY OF THAT SCORE SITUATED ON THE MID POINT L=5.5, N=10, NB=4, f=3 Md=5.5+(5-4)/3=5.83
MEDIAN FROM GROUPED DATA Md=L+((N/2-CfB)/f)*1 CLASS f cf L=59.5 N=100 cfB=45 F=25 i=10 90-99 80-89 70-79 2 10 18 100 98 88 60-69 25 70 50-59 40-49 30-39 20-29 10-19 23 12 7 2 1 45 22 10 3 1 Md=59.5+50-45/25*10 =61.50 i=10 N=100
USE OF MEDIAN TO GET THE MID POINT OF THE ALL SCORES OF THE GROUP IT IS USEFUL WHEN FEW LARGE SCORES INFLUENCE THE RESULT OF MEAN. WHEN INCOMPLETE SCORES ARE PRESENT
