260 MATHEMATICS STATISTICS. There are lies, damned lies and statistics. by Disraeli. are observations with respective frequencies f 1

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1 260 MATHEMATICS STATISTICS Introducton There are les, damned les and statstcs. by Dsrael In Class IX, you have studed the classfcaton of gven data nto ungrouped as well as grouped frequency dstrbutons. You have also learnt to represent the data pctorally n the form of varous graphs such as bar graphs, hstograms (ncludng those of varyng wdths) and frequency polygons. In fact, you went a step further by studyng certan numercal representatves of the ungrouped data, also called measures of central tendency, namely, mean, medan and mode. In ths chapter, we shall extend the study of these three measures,.e., mean, medan and mode from ungrouped data to that of grouped data. We shall also dscuss the concept of cumulatve frequency, the cumulatve frequency dstrbuton and how to draw cumulatve frequency curves, called ogves Mean of Grouped Data The mean (or average) of observatons, as we know, s the sum of the values of all the observatons dvded by the total number of observatons. From Class IX, recall that f x 1, x 2,..., x n are observatons wth respectve frequences f 1, f 2,..., f n, then ths means observaton x 1 occurs f 1 tmes, x 2 occurs f 2 tmes, and so on. Now, the sum of the values of all the observatons = f 1 x 1 + f 2 x f n x n, and the number of observatons = f 1 + f f n. So, the mean x of the data s gven by f1x1+ f2x2+ L + fnxn x = f1 + f2 + L + fn Recall that we can wrte ths n short form by usng the Greek letter Σ (captal sgma) whch means summaton. That s,

2 STATISTICS 261 x = n = 1 n = 1 fx f whch, more brefly, s wrtten as x = Σ fx, f t s understood that vares from Σ f 1 to n. Let us apply ths formula to fnd the mean n the followng example. Example 1 : The marks obtaned by 30 students of Class X of a certan school n a Mathematcs paper consstng of 100 marks are presented n table below. Fnd the mean of the marks obtaned by the students. Marks obtaned (x ) Number of student ( f ) Soluton: Recall that to fnd the mean marks, we requre the product of each x wth the correspondng frequency f. So, let us put them n a column as shown n Table Table 14.1 Marks obtaned (x ) Number of students ( f ) f x Total Σf = 30 Σf x = 1779

3 262 MATHEMATICS Now, Σ fx x = Σ f = = 59.3 Therefore, the mean marks obtaned s In most of our real lfe stuatons, data s usually so large that to make a meanngful study t needs to be condensed as grouped data. So, we need to convert gven ungrouped data nto grouped data and devse some method to fnd ts mean. Let us convert the ungrouped data of Example 1 nto grouped data by formng class-ntervals of wdth, say 15. Remember that, whle allocatng frequences to each class-nterval, students fallng n any upper class-lmt would be consdered n the next class, e.g., 4 students who have obtaned 40 marks would be consdered n the classnterval and not n Wth ths conventon n our mnd, let us form a grouped frequency dstrbuton table (see Table 14.2). Table 14.2 Class nterval Number of students Now, for each class-nterval, we requre a pont whch would serve as the representatve of the whole class. It s assumed that the frequency of each classnterval s centred around ts md-pont. So the md-pont (or class mark) of each class can be chosen to represent the observatons fallng n the class. Recall that we fnd the md-pont of a class (or ts class mark) by fndng the average of ts upper and lower lmts. That s, Upper class lmt + Lower class lmt Class mark = Wth reference to Table 14.2, for the class 10-25, the class mark s,.e., Smlarly, we can fnd the class marks of the remanng class ntervals. We put them n Table These class marks serve as our x s. Now, n general, for the th class nterval, we have the frequency f correspondng to the class mark x. We can now proceed to compute the mean n the same manner as n Example 1.

4 STATISTICS 263 Table 14.3 Class nterval Number of students ( f ) Class mark (x ) f x Total Σ f = 30 Σ f x = The sum of the values n the last column gves us Σ f x. So, the mean x of the gven data s gven by Σfx x = = = 62 Σ f 30 Ths new method of fndng the mean s known as the Drect Method. We observe that Tables 14.1 and 14.3 are usng the same data and employng the same formula for the calculaton of the mean but the results obtaned are dfferent. Can you thnk why ths s so, and whch one s more accurate? The dfference n the two values s because of the md-pont assumpton n Table 14.3, 59.3 beng the exact mean, whle 62 an approxmate mean. Sometmes when the numercal values of x and f are large, fndng the product of x and f becomes tedous and tme consumng. So, for such stuatons, let us thnk of a method of reducng these calculatons. We can do nothng wth the f s, but we can change each x to a smaller number so that our calculatons become easy. How do we do ths? What about subtractng a fxed number from each of these x s? Let us try ths method. The frst step s to choose one among the x s as the assumed mean, and denote t by a. Also, to further reduce our calculaton work, we may take a to be that x whch les n the centre of x 1, x 2,..., x n. So, we can choose a = 47.5 or a = Let us choose a = The next step s to fnd the dfference d between a and each of the x s, that s, the devaton of a from each of the x s..e., d = x a = x 47.5

5 264 MATHEMATICS The thrd step s to fnd the product of d wth the correspondng f, and take the sum of all the f d s. The calculatons are shown n Table Table 14.4 Class nterval Number of Class mark d = x 47.5 f d students ( f ) (x ) Total Σf = 30 Σf d = 435 So, from Table 14.4, the mean of the devatons, d = Σfd. Σf Now, let us fnd the relaton between d and x. Snce n obtanng d, we subtracted a from each x, so, n order to get the mean x, we need to add a to d. Ths can be explaned mathematcally as: Mean of devatons, d = So, d = = = Σfd Σf Σf( x a) Σf Σfx Σfa Σf Σf Σf x a Σf = x a So, x = a + d Σfd.e., x = a + Σ f

6 STATISTICS 265 Substtutng the values of a, Σf d and Σf from Table 14.4, we get 435 x = = = Therefore, the mean of the marks obtaned by the students s 62. The method dscussed above s called the Assumed Mean Method. Actvty 1 : From the Table 14.3 fnd the mean by takng each of x (.e., 17.5, 32.5, and so on) as a. What do you observe? You wll fnd that the mean determned n each case s the same,.e., 62. (Why?) So, we can say that the value of the mean obtaned does not depend on the choce of a. Observe that n Table 14.4, the values n Column 4 are all multples of 15. So, f we dvde the values n the entre Column 4 by 15, we would get smaller numbers to multply wth f. (Here, 15 s the class sze of each class nterval.) So, let u = x h a, where a s the assumed mean and h s the class sze. Now, we calculate u n ths way and contnue as before (.e., fnd f u and then Σ f u ). Takng h = 15, let us form Table Table 14.5 Class nterval f x d = x a u = x a h f u Total Σf = 30 Σf u = 29 Let u = Σfu Σf Here, agan let us fnd the relaton between u and x.

7 266 MATHEMATICS We have, u = x h a Therefore, u = Σf ( x a) h 1 Σfx aσf = Σf h Σf = 1 Σfx Σf a h Σf Σf 1 x a h = [ ] So, hu = x a.e., x = a + hu Σfu So, x = a + h Σf Now, substtutng the values of a, h, Σf u and Σf from Table 14.5, we get x = = = 62 So, the mean marks obtaned by a student s 62. The method dscussed above s called the Step-devaton method. We note that : the step-devaton method wll be convenent to apply f all the d s have a common factor. The mean obtaned by all the three methods s the same. The assumed mean method and step-devaton method are just smplfed forms of the drect method. The formula x = a + hu stll holds f a and h are not as gven above, but are x a any non-zero numbers such that u =. h Let us apply these methods n another example.

8 STATISTICS 267 Example 2 : The table below gves the percentage dstrbuton of female teachers n the prmary schools of rural areas of varous states and unon terrtores (U.T.) of Inda. Fnd the mean percentage of female teachers by all the three methods dscussed n ths secton. Percentage of female teachers Number of States/U.T. Source : Seventh All Inda School Educaton Survey conducted by NCERT Soluton : Let us fnd the class marks, x, of each class, and put them n a column (see Table 14.6): Table 14.6 Percentage of female Number of x teachers States /U.T. ( f ) Here we take a = 50, h = 10, then d = x 50 and x 50 u. 10 We now fnd d and u and put them n Table 14.7.

9 268 MATHEMATICS Table 14.7 Percentage of Number of x d = x 50 x 50 u = f x f d f u female states/u.t. 10 teachers ( f ) Total From the table above, we obtan Σf = 35, Σf x = 1390, Σf d = 360, Σf u = 36. Σfx 1390 Usng the drect method, x = = = Σf 35 Usng the assumed mean method, Σfd x = a + = Σ f Usng the step-devaton method, ( 360) 50 + = x = Σfu 36 a + h = = Σf 35 Therefore, the mean percentage of female teachers n the prmary schools of rural areas s Remark : The result obtaned by all the three methods s the same. So the choce of method to be used depends on the numercal values of x and f. If x and f are suffcently small, then the drect method s an approprate choce. If x and f are numercally large numbers, then we can go for the assumed mean method or step-devaton method. If the class szes are unequal, and x are large numercally, we can stll apply the step-devaton method by takng h to be a sutable dvsor of all the d s.

10 STATISTICS 269 Example 3 : The dstrbuton below shows the number of wckets taken by bowlers n one-day crcket matches. Fnd the mean number of wckets by choosng a sutable method. What does the mean sgnfy? Number of wckets Number of bowlers Soluton : Here, the class sze vares, and the x, s are large. Let us stll apply the stepdevaton method wth a = 200 and h = 20. Then, we obtan the data as n Table Table 14.8 d Number of Number of x d = x 200 u = 20 wckets bowlers taken ( f ) u f Total So, 106 u = Therefore, x = = = Ths tells us that, on an average, the number of wckets taken by these 45 bowlers n one-day crcket s Now, let us see how well you can apply the concepts dscussed n ths secton!

11 270 MATHEMATICS Actvty 2 : Dvde the students of your class nto three groups and ask each group to do one of the followng actvtes. 1. Collect the marks obtaned by all the students of your class n Mathematcs n the latest examnaton conducted by your school. Form a grouped frequency dstrbuton of the data obtaned. 2. Collect the daly maxmum temperatures recorded for a perod of 30 days n your cty. Present ths data as a grouped frequency table. 3. Measure the heghts of all the students of your class (n cm) and form a grouped frequency dstrbuton table of ths data. After all the groups have collected the data and formed grouped frequency dstrbuton tables, the groups should fnd the mean n each case by the method whch they fnd approprate. EXERCISE A survey was conducted by a group of students as a part of ther envronment awareness programme, n whch they collected the followng data regardng the number of plants n 20 houses n a localty. Fnd the mean number of plants per house. Number of plants Number of houses Whch method dd you use for fndng the mean, and why? 2. Consder the followng dstrbuton of daly wages of 50 workers of a factory. Daly wages (n Rs) Number of workers Fnd the mean daly wages of the workers of the factory by usng an approprate method. 3. The followng dstrbuton shows the daly pocket allowance of chldren of a localty. The mean pocket allowance s Rs 18. Fnd the mssng frequency f. Daly pocket allowance (n Rs) Number of chldren f 5 4

12 STATISTICS Thrty women were examned n a hosptal by a doctor and the number of heart beats per mnute were recorded and summarsed as follows. Fnd the mean heart beats per mnute for these women, choosng a sutable method. Number of heart beats per mnute Number of women In a retal market, frut vendors were sellng mangoes kept n packng boxes. These boxes contaned varyng number of mangoes. The followng was the dstrbuton of mangoes accordng to the number of boxes. Number of mangoes Number of boxes Fnd the mean number of mangoes kept n a packng box. Whch method of fndng the mean dd you choose? 6. The table below shows the daly expendture on food of 25 households n a localty. Daly expendture (n Rs) Number of households Fnd the mean daly expendture on food by a sutable method. 7. To fnd out the concentraton of SO 2 n the ar (n parts per mllon,.e., ppm), the data was collected for 30 localtes n a certan cty and s presented below: Concentraton of SO 2 (n ppm) Frequency Fnd the mean concentraton of SO 2 n the ar.

13 272 MATHEMATICS 8. A class teacher has the followng absentee record of 40 students of a class for the whole term. Fnd the mean number of days a student was absent. Number of days Number of students 9. The followng table gves the lteracy rate (n percentage) of 35 ctes. Fnd the mean lteracy rate. Lteracy rate (n %) Number of ctes Mode of Grouped Data Recall from Class IX, a mode s that value among the observatons whch occurs most often, that s, the value of the observaton havng the maxmum frequency. Further, we dscussed fndng the mode of ungrouped data. Here, we shall dscuss ways of obtanng a mode of grouped data. It s possble that more than one value may have the same maxmum frequency. In such stuatons, the data s sad to be multmodal. Though grouped data can also be multmodal, we shall restrct ourselves to problems havng a sngle mode only. Let us frst recall how we found the mode for ungrouped data through the followng example. Example 4 : The wckets taken by a bowler n 10 crcket matches are as follows: Fnd the mode of the data. Soluton : Let us form the frequency dstrbuton table of the gven data as follows: Number of wckets Number of matches

14 STATISTICS 273 Clearly, 2 s the number of wckets taken by the bowler n the maxmum number (.e., 3) of matches. So, the mode of ths data s 2. In a grouped frequency dstrbuton, t s not possble to determne the mode by lookng at the frequences. Here, we can only locate a class wth the maxmum frequency, called the modal class. The mode s a value nsde the modal class, and s gven by the formula: Mode = l f f1 f0 f2 f h where l = lower lmt of the modal class, h = sze of the class nterval (assumng all class szes to be equal), f 1 = frequency of the modal class, f 0 = frequency of the class precedng the modal class, f 2 = frequency of the class succeedng the modal class. Let us consder the followng examples to llustrate the use of ths formula. Example 5 : A survey conducted on 20 households n a localty by a group of students resulted n the followng frequency table for the number of famly members n a household: Famly sze Number of famles Fnd the mode of ths data. Soluton : Here the maxmum class frequency s 8, and the class correspondng to ths frequency s 3 5. So, the modal class s 3 5. Now modal class = 3 5, lower lmt (l ) of modal class = 3, class sze (h) = 2 frequency ( f 1 ) of the modal class = 8, frequency ( f 0 ) of class precedng the modal class = 7, frequency ( f 2 ) of class succeedng the modal class = 2. Now, let us substtute these values n the formula :

15 274 MATHEMATICS Mode = l f f 2 f f f h Therefore, the mode of the data above s = = 3 + = Example 6 : The marks dstrbuton of 30 students n a mathematcs examnaton are gven n Table 14.3 of Example 1. Fnd the mode of ths data. Also compare and nterpret the mode and the mean. Soluton : Refer to Table 14.3 of Example 1. Snce the maxmum number of students (.e., 7) have got marks n the nterval 40-55, the modal class s Therefore, the lower lmt (l ) of the modal class = 40, the class sze ( h) = 15, the frequency ( f 1 ) of modal class = 7, the frequency ( f 0 ) of the class precedng the modal class = 3, the frequency ( f 2 ) of the class succeedng the modal class = 6. Now, usng the formula: f1 f 0 Mode = l + h, 2 f1 f0 f2 7 3 we get Mode = = So, the mode marks s 52. Now, from Example 1, you know that the mean marks s 62. So, the maxmum number of students obtaned 52 marks, whle on an average a student obtaned 62 marks. Remarks : 1. In Example 6, the mode s less than the mean. But for some other problems t may be equal or more than the mean also. 2. It depends upon the demand of the stuaton whether we are nterested n fndng the average marks obtaned by the students or the average of the marks obtaned by most

16 STATISTICS 275 of the students. In the frst stuaton, the mean s requred and n the second stuaton, the mode s requred. Actvty 3 : Contnung wth the same groups as formed n Actvty 2 and the stuatons assgned to the groups. Ask each group to fnd the mode of the data. They should also compare ths wth the mean, and nterpret the meanng of both. Remark : The mode can also be calculated for grouped data wth unequal class szes. However, we shall not be dscussng t. EXERCISE The followng table shows the ages of the patents admtted n a hosptal durng a year: Age (n years) Number of patents Fnd the mode and the mean of the data gven above. Compare and nterpret the two measures of central tendency. 2. The followng data gves the nformaton on the observed lfetmes (n hours) of 225 electrcal components : Lfetmes (n hours) Frequency Determne the modal lfetmes of the components. 3. The followng data gves the dstrbuton of total monthly household expendture of 200 famles of a vllage. Fnd the modal monthly expendture of the famles. Also, fnd the mean monthly expendture : Expendture (n Rs) Number of famles

17 276 MATHEMATICS 4. The followng dstrbuton gves the state-wse teacher-student rato n hgher secondary schools of Inda. Fnd the mode and mean of ths data. Interpret the two measures. Number of students per teacher Number of states / U.T The gven dstrbuton shows the number of runs scored by some top batsmen of the world n one-day nternatonal crcket matches. Runs scored Number of batsmen Fnd the mode of the data A student noted the number of cars passng through a spot on a road for 100 perods each of 3 mnutes and summarsed t n the table gven below. Fnd the mode of the data : Number of cars Frequency

18 STATISTICS Medan of Grouped Data As you have studed n Class IX, the medan s a measure of central tendency whch gves the value of the mddle-most observaton n the data. Recall that for fndng the medan of ungrouped data, we frst arrange the data values of the observatons n n + 1 ascendng order. Then, f n s odd, the medan s the th observaton. And, f n 2 s even, then the medan wll be the average of the n th 2 n and the th observatons. Suppose, we have to fnd the medan of the followng data, whch gves the marks, out of 50, obtaned by 100 students n a test : Marks obtaned Number of students Frst, we arrange the marks n ascendng order and prepare a frequency table as follows : Table 14.9 Marks obtaned Number of students (Frequency) Total 100

19 278 MATHEMATICS Here n = 100, whch s even. The medan wll be the average of the 2 n th and the n + 1 th observatons,.e., the 50th and 51st observatons. To fnd these 2 observatons, we proceed as follows: Table Marks obtaned Number of students 20 6 upto = 26 upto = 50 upto = 78 upto = 93 upto = 97 upto = 99 upto = 100 Now we add another column depctng ths nformaton to the frequency table above and name t as cumulatve frequency column. Table Marks obtaned Number of students Cumulatve frequency

20 STATISTICS 279 From the table above, we see that: 50th observaton s 28 51st observaton s 29 (Why?) So, Medan = = Remark : The part of Table consstng Column 1 and Column 3 s known as Cumulatve Frequency Table. The medan marks 28.5 conveys the nformaton that about 50% students obtaned marks less than 28.5 and another 50% students obtaned marks more than Now, let us see how to obtan the medan of grouped data, through the followng stuaton. Consder a grouped frequency dstrbuton of marks obtaned, out of 100, by 53 students, n a certan examnaton, as follows: Table Marks Number of students From the table above, try to answer the followng questons: How many students have scored marks less than 10? The answer s clearly 5.

21 280 MATHEMATICS How many students have scored less than 20 marks? Observe that the number of students who have scored less than 20 nclude the number of students who have scored marks from 0-10 as well as the number of students who have scored marks from So, the total number of students wth marks less than 20 s 5 + 3,.e., 8. We say that the cumulatve frequency of the class s 8. Smlarly, we can compute the cumulatve frequences of the other classes,.e., the number of students wth marks less than 30, less than 40,..., less than 100. We gve them n Table gven below: Table Marks obtaned Number of students (Cumulatve frequency) Less than 10 5 Less than = 8 Less than = 12 Less than = 15 Less than = 18 Less than = 22 Less than = 29 Less than = 38 Less than = 45 Less than = 53 The dstrbuton gven above s called the cumulatve frequency dstrbuton of the less than type. Here 10, 20, 30, , are the upper lmts of the respectve class ntervals. We can smlarly make the table for the number of students wth scores, more than or equal to 0, more than or equal to 10, more than or equal to 20, and so on. From Table 14.12, we observe that all 53 students have scored marks more than or equal to 0. Snce there are 5 students scorng marks n the nterval 0-10, ths means that there are 53 5 = 48 students gettng more than or equal to 10 marks. Contnung n the same manner, we get the number of students scorng 20 or above as 48 3 = 45, 30 or above as 45 4 = 41, and so on, as shown n Table

22 STATISTICS 281 Table Marks obtaned Number of students (Cumulatve frequency) More than or equal to 0 53 More than or equal to = 48 More than or equal to = 45 More than or equal to = 41 More than or equal to = 38 More than or equal to = 35 More than or equal to = 31 More than or equal to = 24 More than or equal to = 15 More than or equal to = 8 The table above s called a cumulatve frequency dstrbuton of the more than type. Here 0, 10, 20,..., 90 gve the lower lmts of the respectve class ntervals. Now, to fnd the medan of grouped data, we can make use of any of these cumulatve frequency dstrbutons. Let us combne Tables and to get Table gven below: Table Marks Number of students ( f ) Cumulatve frequency (cf) Now n a grouped data, we may not be able to fnd the mddle observaton by lookng at the cumulatve frequences as the mddle observaton wll be some value n

23 282 MATHEMATICS a class nterval. It s, therefore, necessary to fnd the value nsde a class that dvdes the whole dstrbuton nto two halves. But whch class should ths be? To fnd ths class, we fnd the cumulatve frequences of all the classes and 2 n. We now locate the class whose cumulatve frequency s greater than (and nearest to) n n Ths s called the medan class. In the dstrbuton above, n = 53. So, = Now s the class whose cumulatve frequency 29 s greater than (and nearest to) 2 n,.e., Therefore, s the medan class. After fndng the medan class, we use the followng formula for calculatng the medan. where n cf Medan = l + 2 h, f l = lower lmt of medan class, n = number of observatons, cf = cumulatve frequency of class precedng the medan class, f = frequency of medan class, h = class sze (assumng class sze to be equal). n Substtutng the values = 26.5, l = 60, cf = 22, f = 7, h = 10 2 n the formula above, we get Medan = = = 66.4 So, about half the students have scored marks less than 66.4, and the other half have scored marks more 66.4.

24 STATISTICS 283 Example 7 : A survey regardng the heghts (n cm) of 51 grls of Class X of a school was conducted and the followng data was obtaned: Heght (n cm) Number of grls Less than Less than Less than Less than Less than Less than Fnd the medan heght. Soluton : To calculate the medan heght, we need to fnd the class ntervals and ther correspondng frequences. The gven dstrbuton beng of the less than type, 140, 145, 150,..., 165 gve the upper lmts of the correspondng class ntervals. So, the classes should be below 140, , ,..., Observe that from the gven dstrbuton, we fnd that there are 4 grls wth heght less than 140,.e., the frequency of class nterval below 140 s 4. Now, there are 11 grls wth heghts less than 145 and 4 grls wth heght less than 140. Therefore, the number of grls wth heght n the nterval s 11 4 = 7. Smlarly, the frequency of s = 18, for , t s = 11, and so on. So, our frequency dstrbuton table wth the gven cumulatve frequences becomes: Table Class ntervals Frequency Cumulatve frequency Below

25 284 MATHEMATICS Now 51 n = 51. So, n = = Ths observaton les n the class Then, l (the lower lmt) = 145, cf (the cumulatve frequency of the class precedng ) = 11, f (the frequency of the medan class ) = 18, h (the class sze) = 5. Usng the formula, Medan = l + n cf 2 h, we have f Medan = = = So, the medan heght of the grls s cm. Ths means that the heght of about 50% of the grls s less than ths heght, and 50% are taller than ths heght. Example 8 : The medan of the followng data s 525. Fnd the values of x and y, f the total frequency s 100. Class nterval Frequency x y

26 STATISTICS 285 Soluton : Class ntervals Frequency Cumulatve frequency x 7 + x x x x y 56 + x + y x + y x + y x + y It s gven that n = 100 So, 76 + x + y = 100,.e., x + y = 24 (1) The medan s 525, whch les n the class So, l = 500, f = 20, cf = 36 + x, h = 100 Usng the formula : Medan = n 2 cf l + h f, we get 525 = x e., = (14 x) 5.e., 25 = 70 5x.e., 5x = = 45 So, x =9 Therefore, from (1), we get 9 + y =24.e., y =15

27 286 MATHEMATICS Now, that you have studed about all the three measures of central tendency, let us dscuss whch measure would be best suted for a partcular requrement. The mean s the most frequently used measure of central tendency because t takes nto account all the observatons, and les between the extremes,.e., the largest and the smallest observatons of the entre data. It also enables us to compare two or more dstrbutons. For example, by comparng the average (mean) results of students of dfferent schools of a partcular examnaton, we can conclude whch school has a better performance. However, extreme values n the data affect the mean. For example, the mean of classes havng frequences more or less the same s a good representatve of the data. But, f one class has frequency, say 2, and the fve others have frequency 20, 25, 20, 21, 18, then the mean wll certanly not reflect the way the data behaves. So, n such cases, the mean s not a good representatve of the data. In problems where ndvdual observatons are not mportant, and we wsh to fnd out a typcal observaton, the medan s more approprate, e.g., fndng the typcal productvty rate of workers, average wage n a country, etc. These are stuatons where extreme values may be there. So, rather than the mean, we take the medan as a better measure of central tendency. In stuatons whch requre establshng the most frequent value or most popular tem, the mode s the best choce, e.g., to fnd the most popular T.V. programme beng watched, the consumer tem n greatest demand, the colour of the vehcle used by most of the people, etc. Remarks : 1. There s a emprcal relatonshp between the three measures of central tendency : 3 Medan = Mode + 2 Mean 2. The medan of grouped data wth unequal class szes can also be calculated. However, we shall not dscuss t here.

28 STATISTICS 287 EXERCISE The followng frequency dstrbuton gves the monthly consumpton of electrcty of 68 consumers of a localty. Fnd the medan, mean and mode of the data and compare them. Monthly consumpton (n unts) Number of consumers If the medan of the dstrbuton gven below s 28.5, fnd the values of x and y. Class nterval Frequency x y Total A lfe nsurance agent found the followng data for dstrbuton of ages of 100 polcy holders. Calculate the medan age, f polces are gven only to persons havng age 18 years onwards but less than 60 year.

29 288 MATHEMATICS Age (n years) Number of polcy holders Below 20 2 Below 25 6 Below Below Below Below Below Below Below The lengths of 40 leaves of a plant are measured correct to the nearest mllmetre, and the data obtaned s represented n the followng table : Length (n mm) Number of leaves Fnd the medan length of the leaves. (Hnt : The data needs to be converted to contnuous classes for fndng the medan, snce the formula assumes contnuous classes. The classes then change to , ,..., )

30 STATISTICS The followng table gves the dstrbuton of the lfe tme of 400 neon lamps : Lfe tme (n hours) Number of lamps Fnd the medan lfe tme of a lamp surnames were randomly pcked up from a local telephone drectory and the frequency dstrbuton of the number of letters n the Englsh alphabets n the surnames was obtaned as follows: Number of letters Number of surnames Determne the medan number of letters n the surnames. Fnd the mean number of letters n the surnames? Also, fnd the modal sze of the surnames. 7. The dstrbuton below gves the weghts of 30 students of a class. Fnd the medan weght of the students. Weght (n kg) Number of students Graphcal Representaton of Cumulatve Frequency Dstrbuton As we all know, pctures speak better than words. A graphcal representaton helps us n understandng gven data at a glance. In Class IX, we have represented the data through bar graphs, hstograms and frequency polygons. Let us now represent a cumulatve frequency dstrbuton graphcally. For example, let us consder the cumulatve frequency dstrbuton gven n Table

31 290 MATHEMATICS Recall that the values 10, 20, 30,..., 100 are the upper lmts of the respectve class ntervals. To represent the data n the table graphcally, we mark the upper lmts of the class ntervals on the horzontal axs (x-axs) and ther correspondng cumulatve frequences on the vertcal axs (y-axs), choosng a convenent scale. The scale may not be the same on both the axs. Let us now plot the ponts correspondng to the ordered pars gven by (upper lmt, correspondng cumulatve frequency), Fg e., (10, 5), (20, 8), (30, 12), (40, 15), (50, 18), (60, 22), (70, 29), (80, 38), (90, 45), (100, 53) on a graph paper and jon them by a free hand smooth curve. The curve we get s called a cumulatve frequency curve, or an ogve (of the less than type). (See Fg. 14.1) The term ogve s pronounced as ojeev and s derved from the word ogee. An ogee s a shape consstng of a concave arc flowng nto a convex arc, so formng an S-shaped curve wth vertcal ends. In archtecture, the ogee shape s one of the characterstcs of the 14th and 15th century Gothc styles. Next, agan we consder the cumulatve frequency dstrbuton gven n Table and draw ts ogve (of the more than type). Recall that, here 0, 10, 20,..., 90 are the lower lmts of the respectve class ntervals 0-10, 10-20,..., To represent the more than type graphcally, we plot the lower lmts on the x-axs and the correspondng cumulatve frequences on the y-axs. Then we plot the ponts (lower lmt, correspondng cumulatve frequency),.e., (0, 53), (10, 48), (20, 45), (30, 41), (40, 38), (50, 35), (60, 31), (70, 24), (80, 15), (90, 8), on a graph paper, Fg and jon them by a free hand smooth curve. The curve we get s a cumulatve frequency curve, or an ogve (of the more than type). (See Fg. 14.2)

32 STATISTICS 291 Remark : Note that both the ogves (n Fg and Fg. 14.2) correspond to the same data, whch s gven n Table Now, are the ogves related to the medan n any way? Is t possble to obtan the medan from these two cumulatve frequency curves correspondng to the data n Table 14.12? Let us see. One obvous way s to locate n 53 = = on the y-axs (see Fg. 14.3). From ths pont, draw a lne parallel to the x-axs cuttng the curve at a pont. From ths pont, draw a perpendcular to the x-axs. The pont of ntersecton of ths perpendcular wth the x-axs determnes the medan of the data Fg (see Fg. 14.3). Another way of obtanng the medan s the followng : Draw both ogves (.e., of the less than type and of the more than type) on the same axs. The two ogves wll ntersect each other at a pont. From ths pont, f we draw a perpendcular on the x-axs, the pont at whch t cuts the x-axs gves us the medan (see Fg. 14.4). Fg Example 9 : The annual profts earned by 30 shops of a shoppng complex n a localty gve rse to the followng dstrbuton : Proft (n lakhs Rs) Number of shops (frequency) More than or equal to 5 30 More than or equal to More than or equal to More than or equal to More than or equal to More than or equal to 30 7 More than or equal to 35 3

33 292 MATHEMATICS Draw both ogves for the data above. Hence obtan the medan proft. Soluton : We frst draw the coordnate axes, wth lower lmts of the proft along the horzontal axs, and the cumulatve frequency along the vertcal axes. Then, we plot the ponts (5, 30), (10, 28), (15, 16), (20, 14), (25, 10), (30, 7) and (35, 3). We jon these ponts wth a smooth curve to get the more than ogve, as shown n Fg Now, let us obtan the classes, ther frequences and the cumulatve frequency from the table above. Cumulatve frequency Lower lmts of proft (n lakhs Rs) Fg Table Classes No. of shops Cumulatve frequency Usng these values, we plot the ponts (10, 2), (15, 14), (20, 16), (25, 20), (30, 23), (35, 27), (40, 30) on the same axes as n Fg to get the less than ogve, as shown n Fg The abcssa of ther pont of ntersecton s nearly 17.5, whch s the medan. Ths can also be verfed by usng the formula. Hence, the medan proft (n lakhs) s Rs Remark : In the above examples, t may be noted that the class ntervals were contnuous. For drawng ogves, t should be ensured that the class ntervals are contnuous. (Also see constructons of hstograms n Class IX) Fg. 14.6

34 STATISTICS 293 EXERCISE The followng dstrbuton gves the daly ncome of 50 workers of a factory. Daly ncome (n Rs) Number of workers Convert the dstrbuton above to a less than type cumulatve frequency dstrbuton, and draw ts ogve. 2. Durng the medcal check-up of 35 students of a class, ther weghts were recorded as follows: Weght (n kg) Number of students Less than 38 0 Less than 40 3 Less than 42 5 Less than 44 9 Less than Less than Less than Less than Draw a less than type ogve for the gven data. Hence obtan the medan weght from the graph and verfy the result by usng the formula. 3. The followng table gves producton yeld per hectare of wheat of 100 farms of a vllage. Producton yeld (n kg/ha) Number of farms Change the dstrbuton to a more than type dstrbuton, and draw ts ogve Summary In ths chapter, you have studed the followng ponts: 1. The mean for grouped data can be found by : () the drect method : Σfx x = Σ f

35 294 MATHEMATICS Σfd () the assumed mean method : x = a + Σ f Σfu () the step devaton method : x = a + h, Σf wth the assumpton that the frequency of a class s centred at ts md-pont, called ts class mark. 2. The mode for grouped data can be found by usng the formula: f1 f0 Mode = l + h 2 f1 f0 f2 where symbols have ther usual meanngs. 3. The cumulatve frequency of a class s the frequency obtaned by addng the frequences of all the classes precedng the gven class. 4. The medan for grouped data s formed by usng the formula: Medan = l n cf 2 + h, f where symbols have ther usual meanngs. 5. Representng a cumulatve frequency dstrbuton graphcally as a cumulatve frequency curve, or an ogve of the less than type and of the more than type. 6. The medan of grouped data can be obtaned graphcally as the x-coordnate of the pont of ntersecton of the two ogves for ths data. A NOTE TO THE READER For calculatng mode and medan for grouped data, t should be ensured that the class ntervals are contnuous before applyng the formulae. Same condton also apply for constructon of an ogve. Further, n case of ogves, the scale may not be the same on both the axes.