MEASURES OF CENTRAL TENDENCY

MODULE - 6 Statstcs Measures of Cetral Tedecy 25 MEASURES OF CENTRAL TENDENCY I the prevous lesso, we have leart that the data could be summarsed to some extet by presetg t the form of a frequecy table. We have also see how data were represeted graphcally through bar graphs, hstograms ad frequecy polygos to get some broad dea about the ature of the data. Some aspects of the data ca be descrbed quattatvely to represet certa features of the data. A average s oe of such represetatve measures. As average s a umber of dcatg the represetatve or cetral value of the data, t les somewhere betwee the two extremes. For ths reaso, average s called a measure of cetral tedecy. I ths lesso, we wll study some commo measures of cetral tedecy, vz. () Arthmetcal average, also called mea () Meda () Mode OBJECTIVES After studyg ths lesso, you wll be able to defe mea of raw/ugrouped ad grouped data; calculate mea of raw/ugrouped data ad also of grouped data by ordary ad short-cut-methods; defe meda ad mode of raw/ugrouped data; calculate meda ad mode of raw/ugrouped data. 25. ARITHMETIC AVERAGE OR MEAN You must have heard people talkg about average speed, average rafall, average heght, average score (marks) etc. If we are told that average heght of studets s 50 cm, t does ot mea that heght of each studet s 50 cm. I geeral, t gves a message that heght of 634 Mathematcs Secodary Course

Measures of Cetral Tedecy studets are spread aroud 50 cm. Some of the studets may have a heght less tha t, some may have a heght greater tha t ad some may have a heght of exactly 50 cm. 25.. Mea (Arthmetc average) of Raw Data To calculate the mea of raw data, all the observatos of the data are added ad ther sum s dvded by the umber of observatos. Thus, the mea of observatos x, x 2,...x s MODULE - 6 Statstcs x + x 2 +... + x It s geerally deoted by x. so x = x x + 2 +... + x = = x (I) where the symbol Σ s the captal letter SIGMA of the Greek alphabet ad s used to deote summato. To ecoomse the space requred wrtg such legthy expresso, we use the symbol Σ, read as sgma. x = I, s called the dex of summato. Example 25.: The weght of four bags of wheat ( kg) are 03, 05, 02, 04. Fd the mea weght. Soluto: Mea weght ( x ) = 03 + 05 + 02 + 04 4 kg = 44 kg = 03.5 kg 4 Example 25.2: The erolmet a school last fve years was 605, 70, 745, 835 ad 90. What was the average erolmet per year? Soluto: Average erolmet (or mea erolmet) = 605 + 70 + 745 + 835 + 90 3805 = = 76 5 5 Mathematcs Secodary Course 635

MODULE - 6 Statstcs Measures of Cetral Tedecy Example 25.3:The followg are the marks a Mathematcs Test of 30 studets of Class IX a school: 40 73 49 83 40 49 27 9 37 3 9 40 3 73 7 49 73 62 40 62 49 50 80 35 40 62 73 49 3 28 Fd the mea marks. Soluto: Here, the umber of observato () = 30 x = 40, x 2 = 73,..., x 0 = 3 x = 4, x 2 = 40,..., x 20 = 62 x 2 = 49, x 22 = 50,..., x 30 = 28 From the Formula (I), the mea marks of studets s gve by Mea = ( x ) = 30 = x = 40 + 73 +... + 28 = 30 455 30 = 48.5 Example 25.4: Refer to Example 25.. Show that the sum of x x, x 2 x, x 3 x ad x 4 x s 0, where x s are the weghts of the four bags ad x s ther mea. Soluto: x x = 03 03.5 = 0.5, x 2 x = 05 03.5 =.5 x 3 x = 02 03.5 =.5, x 4 x = 04 03.5 = 0.5 So, (x x ) + (x 2 x ) + (x 3 x ) + (x 4 x ) = 0.5 +.5 + (.5) + 0.5 = 0 Example 25.5: The mea of marks obtaed by 30 studets of Secto A of Class X s 48, that of 35 studets of Secto B s 50. Fd the mea marks obtaed by 65 studets Class X. Soluto: Mea marks of 30 studets of Secto A = 48 So, total marks obtaed by 30 studets of Secto A = 30 48 = 440 Smlarly, total marks obtaed by 35 studets of Secto B = 35 50 = 750 Total marks obtaed by both sectos = 440 + 750 = 390 390 Mea of marks obtaed by 65 studets = = 49. approx. 65 Example 25.6: The mea of 6 observatos was foud to be 40. Later o, t was detected that oe observato 82 was msread as 28. Fd the correct mea. 636 Mathematcs Secodary Course

Measures of Cetral Tedecy MODULE - 6 Statstcs Soluto: Mea of 6 observatos = 40 So, the sum of all the observatos = 6 40 = 240 Sce oe observato 82 was msread as 28, therefore, correct sum of all the observatos = 240 28 + 82 = 294 Hece, correct mea = 294 = 49 6 CHECK YOUR PROGRESS 25.. Wrte formula for calculatg mea of observatos x, x 2..., x. 2. Fd the mea of frst te atural umbers. 3. The daly sale of sugar for 6 days a certa grocery shop s gve below. Calculate the mea daly sale of sugar. Moday Tuesday Wedesday Thursday Frday Saturday 74 kg 2 kg 40 kg 82 kg 70.5 kg 30.5 kg 4. The heghts of 0 grls were measured cm ad the results were as follows: 42, 49, 35, 50, 28, 40, 49, 52, 38, 45 Fd the mea heght. 5. The maxmum daly temperature ( o C) of a cty o 2 cosecutve days are gve below: 32.4 29.5 26.6 25.7 23.5 24.6 24.2 22.4 24.2 23.0 23.2 28.8 Calcualte the mea daly temperature. 6. Refer to Example 25.2. Verfy that the sum of devatos of x from ther mea ( x ) s 0. 7. Mea of 9 observatros was foud to be 35. Later o, t was detected that a observato whch was 8, was take as 8 by mstake. Fd the correct mea of the observatos. 8. The mea marks obtaed by 25 studets a class s 35 ad that of 35 studets s 25. Fd the mea marks obtaed by all the studets. Mathematcs Secodary Course 637

MODULE - 6 Statstcs 25..2 Mea of Ugrouped Data Measures of Cetral Tedecy We wll expla to fd mea of ugrouped data through a example. Fd the mea of the marks (out of 5) obtaed by 20 studets. 2 0 5 8 5 5 2 8 0 5 0 2 2 2 5 2 8 0 5 0 Ths data s the form of raw data. We ca fd mea of the data by usg the formula (I), x.e.,. But ths process wll be tme cosumg. We ca also fd the mea of ths data by frst makg a frequecy table of the data ad the applyg the formula: mea = x = = fx (II) = where f s the frequecy of the th observato x. Frequecy table of the data s : f Marks Number of studets (x ) (f ) 2 4 5 5 8 3 0 5 2 2 5 Σf = 20 To fd mea of ths dstrbuto, we frst fd f x, by multplyg each x wth ts correspodg frequecy f ad apped a colum of f x the frequecy table as gve below. Marks Number of studets f x (x ) (f ) 2 4 2 4 = 8 5 5 5 5 = 25 8 3 3 8 = 24 0 5 5 0 = 50 2 2 2 2 = 24 5 5 = 5 Σf = 20 Σf x = 46 638 Mathematcs Secodary Course

Measures of Cetral Tedecy fx 46 Mea = = = 7. 3 f 20 Example 25.7: The followg data represets the weekly wages ( rupees) of the employees: Weekly wages 900 000 00 200 300 400 500 ( `) Number of 2 3 4 3 4 5 employees Fd the mea weekly wages of the employees. Soluto: I the followg table, etres the frst colum are x s ad etres secod colume are f s,.e., correspodg frequeces. Recall that to fd mea, we requre the product of each x wth correspodg frequecy f. So, let us put them a colum as show the followg table: MODULE - 6 Statstcs Weekly wages ( `) Number of employees f x (x ) (f ) 900 2 0800 000 3 3000 00 4 5400 200 3 5600 300 2 5600 400 5400 500 5 7500 Σf = 80 Σf x = 93300 Usg the Formula II, Mea weekly wages = f x f = ` = ` 66.25 93300 80 Sometmes whe the umercal values of x ad f are large, fdg the product f ad x becomes tedus ad tme cosumg. We wsh to fd a short-cut method. Here, we choose a arbtrary costat a, also called the assumed mea ad subtract t from each of the values x. The reduced value, d = x a s called the devato of x from a. Thus, x = a + d Mathematcs Secodary Course 639

MODULE - 6 Statstcs Measures of Cetral Tedecy ad f x = af + f d = f x = af + = = f d [Summg both sdes over from to r] Hece x f + N = fd, where Σf = N x = a + f d N [sce Σf = N] (III) Ths meghod of calcualto of mea s kow as Assumed Mea Method. I Example 25.7, the values x were very large. So the product f x became tedous ad tme cosumg. Let us fd mea by Assumed Mea Method. Let us take assumed mea a = 200 Weekly wages Number of Devatos f d ( `) (x ) employees (f ) d = x 200 900 2 300 3600 000 3 200 2600 00 4 00 400 200 3 0 0 300 2 00 + 200 400 200 + 2200 500 5 300 + 500 Σf = 80 Σf d = 2700 Usg Formula III, Mea = a f d N + = 200 + 80 ( 2700) = 200 33.75 = 66.25 So, the mea weekly wages = ` 66.25 Observe that the mea s the same whether t s calculated by Drect Method or by Assumed Mea Method. 640 Mathematcs Secodary Course

Measures of Cetral Tedecy Example 25.8: If the mea of the followg data s 20.2, fd the value of k x 0 5 20 25 30 f 6 8 20 k 6 Soluto: Mea = f x 760 + 25k = 40 + k 760 + 25k So, 40 + k = 20.2 (Gve) or 760 +25k = 20.2 (40 + k) or 7600 + 250k = 8080 + 202k or k = 0 f 60 + 20 + 400 + 25k + 80 = 40 + k MODULE - 6 Statstcs CHECK YOUR PROGRESS 25.2. Fd the mea marks of the followg dstrbuto: Marks 2 3 4 5 6 7 8 9 0 Frequecy 3 5 9 4 8 6 9 3 2 2. Calcualte the mea for each of the followg dstrbutos: () x 6 0 5 8 22 27 30 f 2 36 54 72 62 42 22 () x 5 5.4 6.2 7.2 7.6 8.4 9.4 f 3 4 28 23 8 3 3. The weghts ( kg) of 70 workers a factory are gve below. Fd the mea weght of a worker. Weght ( kg) Number of workers 60 0 6 8 62 4 63 6 64 5 65 7 Mathematcs Secodary Course 64

MODULE - 6 Statstcs Measures of Cetral Tedecy 4. If the mea of followg data s 7.45 determe the value of p: x 5 6 7 8 9 20 f 3 8 0 p 5 4 25..3 Mea of Grouped Data Cosder the followg grouped frequecy dstrbuto: Daly wages ( `) Number of workers 50-60 5 60-70 8 70-80 5 80-90 0 90-200 2 What we ca fer from ths table s that there are 5 workers earg daly somewhere from ` 50 to ` 60 (ot cluded 60). We doot kow what exactly the eargs of each of these 5 workers are Therefore, to fd mea of the grasped frequecy dstrbuto, we make the followg assumptos: Frequecy ay class s cetred at ts class mark or md pot Now, we ca say that there are 5 workers earg a daly wage of ` ` 55 each, 8 workers earg a daly wage of ` 60 +70 2 50 +60 2 = = ` 65, 5 workers aerg 70 +60 a daly wage of ` = ` 75 ad so o. Now we ca calculate mea of the gve 2 data as follows, usg the Formula (II) Daly wages ( `) Number of Class marks (x ) f x workers (f ) 50-60 5 55 775 60-70 8 65 320 70-80 5 75 2625 80-90 0 85 850 90-200 2 95 390 Σf = 40 Σf x = 6960 642 Mathematcs Secodary Course

Measures of Cetral Tedecy fx 6960 Mea = = = 74 f 40 So, the mea daly wage = ` 74 Ths method of calculate of the mea of grouped data s Drect Method. We ca also fd the mea of grouped data by usg Formula III,.e., by Assumed Mea Method as follows: We take assumed mea a = 75 MODULE - 6 Statstcs Daly wages Number of Class marks Devatos f d ( `) workers (f ) (x ) d = x 75 50-60 5 55 20 00 60-70 8 65 0 80 70-80 5 75 0 0 80-90 0 85 + 0 00 90-200 2 95 + 20 40 Σf = 40 Σf d = 40 So, usg Formula III, Mea = a N + f d = 75 + ( 40) 40 = 75 = 74 Thus, the mea daly wage = ` 74. Example 25.9: Fd the mea for the followg frequecy dstrbuto by () Drect Method, () Assumed Mea Method. Class Frequecy 20-40 9 40-60 60-80 4 80-00 6 00-20 8 20-40 5 40-60 2 Total 75 Mathematcs Secodary Course 643

MODULE - 6 Statstcs Measures of Cetral Tedecy Soluto: () Drect Method Class Frequecy (f ) Class marks (x ) f x 20-40 9 30 270 40-60 50 550 60-80 4 70 980 80-00 6 90 540 00-20 8 0 880 20-40 5 30 950 40-60 2 50 800 Σf = 75 Σf x = 6970 fx 6970 So, mea = = = 92. 93 f 75 () Assumed mea method Let us take assumed mea = a = 90 Class Frequecy (f ) Class marks (x ) Devato f d d = x 90 20-40 9 30 60 540 40-60 50 40 440 60-80 4 70 20 280 80-00 6 90 0 0 00-20 8 0 + 20 60 20-40 5 30 + 40 600 40-60 2 50 + 60 720 Ν = Σf = 75 Σf d = 220 Mea = a N + f d 220 = 90 + = 92.93 75 Note that mea comes out to be the same both the methods. I the table above, observe that the values colum 4 are all multples of 20. So, f we dvde these value by 20, we would get smaller umbers to multply wth f. Note that, 20 s also the class sze of each class. x a So, let u =, where a s the assumed mea ad h s the class sze. h 644 Mathematcs Secodary Course

Measures of Cetral Tedecy Now we calculate u ths way ad the u f ad ca fd mea of the data by usg the formula MODULE - 6 Statstcs Mea = f U x = a + h f (IV) Let us fd mea of the data gve Example 25.9 Take a = 90. Here h = 20 Class Frequecy Class Devato u = f u (f ) marks (x ) d = x 90 20-40 9 30 60 3 27 40-60 50 40 2 22 60-80 4 70 20 4 80-00 6 90 0 0 0 00-20 8 0 + 20 8 20-40 5 30 + 40 2 30 40-60 2 50 + 60 3 36 Σf = 75 Σf u = Usg the Formula (IV), Mea = f u x = a + h f = 90 + 20 75 220 = 90 + = 92.93 75 Calculatg mea by usg Formula (IV) s kow as Step-devato Method. Note that mea comes out to be the same by usg Drect Method, Assumed Method or Step Devato Method. Example 25.0: Calcualte the mea daly wage from the followg dstrbuto by usg Step devato method. Daly wages ( `) 50-60 60-70 70-80 80-90 90-200 Numbr of workers 5 8 5 0 2 Mathematcs Secodary Course 645

MODULE - 6 Statstcs Measures of Cetral Tedecy Soluto: We have already calculated the mea by usg Drect Method ad Assumed Method. Let us fd mea by Step devato Method. Let us take a = 75. Here h = 0 Daly wages Number of Class Devato u = ( `) workers (f ) marks (x ) d = x 90 x a h 50-60 5 55 20 2 0 60-70 8 65 0 8 70-80 5 75 0 0 0 80-90 0 85 0 0 90-200 2 95 20 2 4 Σf = 40 Σf u = 4 f u Usg Formula (IV), Mea daly wages f u = a + h f 4 = 75 + 0 = ` 74 40 Note: Here aga ote that the mea s the same whether t s calculated usg the Drect Method, Assumed mea Method or Step devato Method. CHECK YOUR PROGRESS 25.3. Followg table shows marks obtaed by 00 studets a mathematcs test Marks 0-0 0-20 20-30 30-40 40-50 50-60 Number of 2 5 25 25 7 6 studets Calculate mea marks of the studets by usg Drect Method. 2. The followg s the dstrbuto of bulbs kept boxes: Number of 50-52 52-54 54-56 56-58 58-60 bulbs Number of 5 00 26 05 30 boxes Fd the mea umber of bulbs kept a box. Whch method of fdg the mea dd you choose? 3. The weekly observatos o cost of lvg dex a certa cty for a partcular year are gve below: 646 Mathematcs Secodary Course

Measures of Cetral Tedecy Cost of lvg 40-50 50-60 60-70 70-80 80-90 90-200 dex Number of 5 8 20 9 6 4 weeks Calculate mea weekly cost of lvg dex by usg Step devato Method. 4. Fd the mea of the followg data by usg () Assumed Mea Method ad () Step devato Method. Class 50-200 200-250 250-300 300-350 350-400 Frequecy 48 32 35 20 0 MODULE - 6 Statstcs 25.2 MEDIAN I a offce there are 5 employees: a supervosor ad 4 workers. The workers draw a salary of ` 5000, ` 6500, ` 7500 ad ` 8000 per moth whle the supervsor gets ` 20000 per moth. 5000 + 6500 + 7500 + 8000 + 20000 I ths case mea (salary) = ` 5 47000 = ` = ` 9400 5 Note that 4 out of 5 employees have ther salares much less tha ` 9400. The mea salary ` 9400 does ot gve eve a approxmate estmate of ay oe of ther salares. Ths s a weakess of the mea. It s affected by the extreme values of the observatos the data. Ths weekess of mea drves us to look for aother average whch s uaffected by a few extreme values. Meda s oe such a measure of cetral tedecy. Meda s a measure of cetral tedecy whch gves the value of the mddlemost observato the data whe the data s arraged ascedg (or descedg) order. 25.2. Meda of Raw Data Meda of raw data s calculated as follows: () Arrage the (umercal) data a ascedg (or descedg) order + () Whe the umber of observatos () s odd, the meda s the value of th 2 observato. Mathematcs Secodary Course 647

MODULE - 6 Statstcs Measures of Cetral Tedecy () Whe the umber of observatos () s eve, the meda s the mea of the ad + th observatos. 2 th 2 Let us llustrate ths wth the help of some examples. Example 25.: The weghts ( kg) of 5 dogs are as follows: 9, 26, 0, 22, 36, 3, 20, 20, 0, 2, 25, 6, 2, 4, 9 Fd the meda weght. Soluto: Let us arrage the data the ascedg (or descedg) order: 9, 0, 0, 2, 3, 4, 6, 9, 20, 20, 2, 22, 25, 36 Here, umber of observatos = 5 Meda + 5 + So, the meda wll be th,.e., th,.e., 8th observato whch s 9 kg. 2 2 Remark: The meda weght 9 kg coveys the formato that 50% dogs have weghts less tha 9 kg ad aother 50% have weghts more the 9 kg. Example 25.2: The pots scored by a basket ball team a seres of matches are as follows: 6,, 6, 26, 4, 4, 3, 8, 9, 23, 47, 9, 7, 8, 7, 28 Fd the meda of the data. Soluto: Here umber of observatos = 6 6 6 So, the meda wll be the mea of th ad + th,.e., mea of 6th ad 9th 2 2 observatos, whe the data s arraged ascedg (or descedg) order as:, 4, 6, 7, 8, 8, 9, 9, 3, 4, 6, 7, 23, 26, 28, 47 9 + 3 So, the meda = = 2 8th term 9th term Remark: Here aga the meda coveys the formato that the values of 50% of the observatos are less tha ad the values of 50% of the observatos are more tha. 648 Mathematcs Secodary Course

Measures of Cetral Tedecy 25.2.2 Meda of Ugrouped Data We llustrate caluculato of the meda of ugrouped data through examples. Example 25.3: Fd the meda of the followg data, whch gves the marks, out of 5, obtae by 35 studets a mathematcs test. Marks obtaed 3 5 6 5 4 3 7 2 0 Number of Studets 4 6 5 7 3 2 3 3 Soluto: Frst arrage marks ascedg order ad prepare a frequecy table as follows: Marks obtaed 3 5 6 7 0 2 3 4 5 Number of Studets 4 6 5 3 7 3 2 3 (frequecy) MODULE - 6 Statstcs + 35 + Here = 35, whch s odd. So, the meda wll be th,.e., th,.e., 8th 2 2 observato. To fd value of 8th observato, we prepare cumulatve frequecy table as follows: Marks obtaed Number of studets Cumulatve frequecy 3 4 4 5 6 0 6 5 5 7 3 8 0 9 7 26 2 3 29 3 2 3 4 3 34 5 35 From the table above, we see that 8th observato s 7 So, Meda = 7 Example 25.4: Fd the meda of the followg data: Weght ( kg) 40 4 42 43 44 45 46 48 Number of 2 5 7 8 3 26 6 3 studets Mathematcs Secodary Course 649

MODULE - 6 Statstcs Measures of Cetral Tedecy Soluto: Here = 2 + 5 + 7 + 8 + 3 + 26 + 6 + 3 = 70, whch s eve, ad weght are already arraged the ascedg order. Let us prepare cumulatve frequecy table of the data: Weght Number of studets Cumulatve ( kg) (frequecy) frequecy 40 2 2 4 5 7 42 7 4 43 8 22 44 3 35 45 26 6 46 6 67 48 3 70 35th observato 36th observato Sce s eve, so the meda wll be the mea of th ad + 2 th observatos, 2.e., 35th ad 36th observatos. From the table, we see that 35 the observato s 44 ad 36th observato s 45 So, Meda = 44 + 45 2 = 44.5 CHECK YOUR PROGRESS 25.4. Followg are the goals scored by a team a seres of matches, 0, 3, 2, 4, 5, 2, 4, 4, 2, 5 Determe the meda score. 2. I a dagostc test mathematcs gve to 2 studets, the followg marks (out of 00) are recorded 46, 52, 48, 39, 4, 62, 55, 53, 96, 39, 45, 99 Calculate the meda for ths data. 650 Mathematcs Secodary Course

Measures of Cetral Tedecy 3. A far de s throw 00 tmes ad ts outcomes are recorded as show below: Outcome 2 3 4 5 6 Frequecy 7 5 6 8 6 8 Fd the meda outcome of the dstrbutos. 4. For each of the followg frequecy dstrbutos, fd the meda: (a) x 2 3 4 5 6 7 f 4 9 6 4 6 MODULE - 6 Statstcs (b) x 5 0 5 20 25 30 35 40 f 3 7 2 20 28 3 28 26 (c) x 2.3 3 5. 5.8 7.4 6.7 4.3 25.3 MODE f 5 8 4 2 3 5 7 Look at the followg example: A compay produces readymade shrts of dfferet szes. The compay kept record of ts sale for oe week whch s gve below: sze ( cm) 90 95 00 05 0 5 Number of shrts 50 25 90 385 270 28 From the table, we see that the sales of shrts of sze 05 cm s maxmum. So, the compay wll go ahead producg ths sze the largest umber. Here, 05 s othg but the mode of the data. Mode s also oe of the measures of cetral tedecy. The observato that occurs most frequetly the data s called mode of the data. I other words, the observato wth maxmum frequecy s called mode of the data. The readymade garmets ad shoe dustres etc, make use of ths measure of cetral tedecy. Based o mode of the demad data, these dustres decde whch sze of the product should be produced large umbers to meet the market demad. 25.3. Mode of Raw Data I case of raw data, t s easy to pck up mode by just lookg at the data. Let us cosder the followg example: Mathematcs Secodary Course 65

MODULE - 6 Statstcs Measures of Cetral Tedecy Example 25.5: The umber of goals scored by a football team 2 matches are:, 2, 2, 3,, 2, 2, 4, 5, 3, 3, 4 What s the modal score? Soluto: Just by lookg at the data, we fd the frequecy of 2 s 4 ad s more tha the frequecy of all other scores. So, mode of the data s 2, or modal score s 2. Example 25.6: Fd the mode of the data: 9, 6, 8, 9, 0, 7, 2, 5, 22, 5 Soluto: Arragg the data creasg order, we have 6, 7, 8, 9, 9, 0, 2, 5, 5, 22 We fd that the both the observatos 9 ad 5 have the same maxmum frequecy 2. So, both are the modes of the data. Remarks:. I ths lesso, we wll take up the data havg a sgle mode oly. 2. I the data, f each observato has the same frequecy, the we say that the data does ot have a mode. 25.3.2 Mode of Ugrouped Data Let us llustrate fdg of the mode of ugrouped data through a example Example 25.7: Fd the mode of the followg data: Weght ( kg) 40 4 42 43 44 45 46 48 Number of Studets 2 6 8 9 0 22 3 5 Soluto: From the table, we see that the weght 45 kg has maxmum frequecy 22 whch meas that maxmum umber of studets have ther weght 45 kg. So, the mode s 45 kg or the modal weght s 45 kg. CHECK YOUR PROGRESS 25.5. Fd the mode of the data: 5, 0, 3, 7, 2, 9, 6, 2,, 2 2. The umber of TV sets each of 5 households are foud as gve below: 2, 2, 4, 2,,,, 2,,, 3, 3,, 3, 0 What s the mode of ths data? 652 Mathematcs Secodary Course

Measures of Cetral Tedecy 3. A de s throw 00 tmes, gvg the followg results Outcome 2 3 4 5 6 Frequecy 5 6 6 5 7 20 Fd the modal outcome from ths dstrbuto. 4. Followg are the marks (out of 0) obtaed by 80 studets a mathematcs test: Marks 0 2 3 4 5 6 7 8 9 0 obtaed Number of 5 2 3 5 9 5 6 9 3 2 studets Determe the modal marks. MODULE - 6 Statstcs LET US SUM UP Mea, meda ad mode are the measures of cetral tedecy. Mea (Arthmetc average) of raw data s gve by x = = x where x, x 2..., x are observatos. Mea of ugrouped data s gve by x = = = = f x f f x N where f s the frequecy of the th observato x. Mea of ugrouped data ca also be foud by usg the formula x = a f d N where d = x a, a beg the assumed mea Mea of grouped data () To fd mea of the grouped frequecy dstrbuto, we take the assumpto: Frequecy ay class s cetred at ts class mark or md pot. + Mathematcs Secodary Course 653

MODULE - 6 Statstcs Measures of Cetral Tedecy () Drect Method x = = = f x f where x s are the class marks ad f are the correspodg freqeuces of x s. () Assumed Mea Method x = a + = f d N where a s the assumed mea, ad d = x a. (v) Step devato method x = a + = = fu h f where a s the assumed mea, u = x a h ad h s the class sze. Meda s a measure of cetral tedecy whch gves the value of the mddle most obserato the data, whe the data s arraged ascedg (or descedg) order. Meda of raw data + () Whe the umber of observatos () s odd, the meda s the value of th 2 observato. () Whe the umber of observatos () s eve, the meda s the mea of the th 2 ad + th observatos. 2 Meda of ugrouped data Meda of ugrouped data ca be foud from the cumulatve frequecy table (arragg data creasg or decreasg order) usg () ad () above. The value of observato wth maxmum frequecy s called the mode of the data. 654 Mathematcs Secodary Course

Measures of Cetral Tedecy MODULE - 6 Statstcs TERMINAL EXERCISE. Fd the mea of frst fve prme umbers. 2. If the mea of 5, 7, 9, x, ad 2 s 9, fd the value of x. 3. Followg are the marks obtaed by 9 studets a class 5, 36, 63, 46, 38, 43, 52, 42 ad 43 () Fd the mea marks of the studets. () What wll be the mea marks f a studet scorg 75 marks s also cluded the class. 4. The mea marks of 0 studets a class s 70. The studets are dvded to two groups of 6 ad 4 respectvely. If the mea marks of the frst group s 60, what wll be the mea marks of the secod group? 5. If the mea of the observatos x, x 2...,x s x, show that f(x x ) = 0 6. There are 50 umbers. Each umber s subtracted from 53 ad the mea of the umbers so obtaed s foud to be 3.5. Determe the mea of the gve umbers. 7. Fd the mea of the followg data: (a) x 5 9 3 7 22 25 f 3 5 2 8 7 5 (b) x 6 8 28 22 24 26 f 3 5 7 5 4 8. Fd the mea of the followg data (a) Classes 0-20 20-30 30-40 40-50 50-60 60-70 Frequeces 2 3 5 7 5 3 (b) Classes 00-200 200-300 300-400 400-500 500-600 600-700 Frequeces 3 5 8 6 5 3 (c) The ages ( moths) of a group of 50 studets are as follows. Fd the mea age. Age 56-58 58-60 60-62 62-64 64-66 66-68 Number of 2 4 8 6 4 6 studets = Mathematcs Secodary Course 655

MODULE - 6 Statstcs Measures of Cetral Tedecy 9. Fd the meda of the followg data: (a) 5, 2, 6, 8, 20, 25, 0 (b) 6, 2, 9, 0, 6, 28, 25, 3, 5, 7 (c) 5, 3, 8, 22, 29, 2, 4, 7, 6 0. The followg data are arraged ascedg order ad the meda of the data s 60. Fd the value of x. 26, 29, 42, 53, x, x + 2, 70, 75, 82, 93. Fd the meda of the followg data: (a) x 25 30 35 45 50 55 65 70 85 f 5 4 2 2 3 4 7 3 (b) x 35 36 37 38 39 40 4 42 f 2 3 5 4 7 6 4 2 2. Fd the mode of the followg data: (a) 8, 5, 2, 5, 3, 5, 3, (b) 9, 8, 7, 6, 7, 5, 4, 5, 7, 9 3. Fd the mode of the followg data whch gves lfe tme ( hours) of 80 bulbs selected at radom from a lot. Lfe tme ( hours) 300 500 700 900 00 Number of bulbs 0 2 20 27 4. I the mea of the followg data s 7, fd the value of p: x 4 p 6 7 9 f 2 4 6 0 6 2 5. For a selected group of people, a surace compay recorded the followg data: Age ( years) 0-0 0-20 20-30 30-40 40-50 50-60 60-70 70-80 Number of deaths 2 2 55 95 7 42 6 7 Determe the mea of the data. 6. If the mea of the observatos: x +, x + 4, x + 5, x + 8, x + s 0, the mea of the last three observatos s (A) 2.5 (B) 2.2 (C) 3.5 (D) 4.2 656 Mathematcs Secodary Course

Measures of Cetral Tedecy 7. If each observato the data s creased by 2, tha ther mea (A) remas the same (B) becomes 2 tmes the orgal mea (C) s decreased by 2 (D) s creased by 2 8. Mode of the data: 5, 4, 9, 20, 4, 5, 4, 8, 4, 5, 7, 4, 8 s (A) 20 (B) 8 (C) 5 (D) 4 MODULE - 6 Statstcs ANSWERS TO CHECK YOUR PROGRESS 25.. x / = 2. 5.5 3. 86.33 kg 4. 42.8 cm 5. 25.68 o C 7. 42 8. 29.7 25.2. 5.84 2. () 8.99 () 6.57 3..68 4. 0 25.3. 28.80 2. 55.9 3. 67.9 4. 244.66 25.4. 3 2. 50 3. 4 4. (a) 4 (b) 30 (c) 5.8 25.5. 2 2. 3. 6 4. 7 ANSWERS TO TERMINAL EXERCISE. 5.6 2. 0 3. () 46 () 48.9 4. 85 6. 56.5 7. (a) 5.775 (b) 2.75 8. (a) 42.6 (b) 396.67 (c) 63 moths (approx) 9. (a) 6 (b) 4 (c) 4 0. 59. (a) 45 (b) 24 2. (a) 5 (b) 7 3. 900 4. 5 5. 39.86 years 6. (A) 7. (D) 8. D Mathematcs Secodary Course 657