STATISTICS ImPORTANT TERmS, DEFINITIONS AND RESULTS l The mean x of n values x 1, x 2, x 3,... x n s gven by x1+ x2 + x3 +... + xn x = n l mean of grouped data (wthout class-ntervals) () Drect method : If the frequences of n observatons x 1, x 2, x 3,... x n be f 1, f 2, f 3,... f respectvely, then the n mean x s gven by x = xf 1 1+ x 2 f 2 + x 3 f 3 +... + x n f f + f + f +... + f 1 2 3 () Devaton method or Assumed mean method n n Σfx = Σf Σf ( x a) Σfd In ths case, the mean x s gven by x = a + = a +, Σf Σf Where, a = assumed mean, Σf = total frequency, d = x a Σf (x a) = sum of the products of devatons and correspondng frequences. l mean of grouped data (wth class-ntervals) In ths case the class marks are treated as x. Class mark = () Drect method Lower class lmt + Upper class lmt 2. If the frequences correspondng to the class marks x 1, x 2, x 3,... x n be f 1, f 2, f 3,... f n respectvely, then fx 1 1+ fx 2 2 + fx 3 3+... + fnxn Σfx mean x s gven by x = = f + f + f +... + f Σf 1 2 3 () Devaton or Assumed mean method In ths case the mean x s gven by x = a + Σ fd, Σf Where, a = assumed mean, Σf = total frequency and d = x a () Step Devaton method In ths case we use the followng formula. x a Σf h Σfu x = a + h = a + h Σf Σf, Where, a = assumed mean, Σf = total frequency, h = class-sze and u = x a h n l Mode s that value among the observatons whch occurs most often.e., the value of the observaton havng the maxmum frequency. l If n a data more than one value have the same maxmum frequency, then the data s sad to be multmodal. l In a grouped frequency dstrbuton, the class whch has the maxmum frequency s called the modal class. 1,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
l We use the followng formula to fnd the mode of a grouped frequency dstrbuton. f1 f0 Mode (M o ) = l + h, where 2 f1 f0 f2 l = lower lmt of modal class, h = sze of the class-nterval, 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. l Medan s the value of the mddle most tem when the data are arranged n ascendng or descendng order of magntude. l medan of ungrouped data () If the number of tems n n the data s odd, then n + 1 Medan = value of th tem. 2 () If the total number of tems n n the data s even, then Medan = 1 n n value of th tem + value of + 1 th tem 2 2 2 l Cumulatve frequency of a partcular value of the varable (or class) s the sum total of all the frequences up to that value (or the class). l There are two types of cumulatve frequency dstrbutons. () cumulatve frequency dstrbuton of less than type. () cumulatve frequency dstrbuton of more than type. l medan of grouped data wth class-ntervals In ths case, we frst fnd the half of the total frequences,.e., n 2. The class n whch n 2 medan class and the medan les n ths class. We use the followng formula for fndng the medan. n cf 2 Medan (M e ) = l + h, f les s called the Where, l = lower lmt of the medan class, n = number of observatons, cf = cumulatve frequency of the class precedng the medan class, f = frequency of the medan class, 2 h = class sze. l The three measures mean, mode and medan are connected by the followng relatons. Mode = 3 medan 2 mean or medan = mode mean + 2 or mean = 3medan mode 3 3 2 2 l The graphcal representaton of a cumulatve frequency dstrbuton s called an ogve or cumulatve frequency curve. l We can draw two types of ogves for a frequency dstrbuton. These are less than ogve and more than ogve. l For less than ogve, we plot the ponts correspondng to the ordered pars gven by (upper lmt, correspondng less than cumulatve frequency). After jonng these ponts by a free hand curve, we get an ogve of less than type. l For more than ogve, we plot the ponts correspondng to the ordered pars gven by (lower lmt, correspondng more than cumulatve frequency). After jonng these ponts by a free hand curve, we get an ogve of more than type.,sector-5,devendra NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
l Ogve can be used to estmate the medan of data. There are two methods to do so. Frst method : Mark a pont correspondng to n, where n s the total frequency, on cumulatve frequency 2 axs (y-axs). From ths pont, draw a lne parallel to x-axs to cut the ogve at a pont. From ths pont, draw a lne perpendcular to the x-axs to get another pont. The abscssa of ths pont gves medan. Second method : Draw both the ogves (less than ogve and more than ogve) on the same graph paper whch cut each other at a pont. From ths pont, draw a lne perpendcular to the x-axs, to get another pont. The abscssa of ths pont gves medan. SUMMATIVE ASSESSMENT multiple CHOICE QUESTIONS [1 mark] A. Important Questons 1. If 35 s the upper lmt of the class-nterval of class-sze 10, then the lower lmt of the class-nterval s : (a) 20 (b) 25 (c) 30 (d) none of these 2. In the assumed mean method, f A s the assumed mean, than devaton d s : (a) x + A (b) x A (c) A x (d) none of these 3. Mode s : (a) mddle most value (b) least frequent value (c) most frequent value (d) none of these 4. The correct formula for fndng the mode of a grouped frequency dstrbuton s : (a) Mode = h + (c) Mode = l f 1 f 0 l 2 f 1 f 0 f (b) Mode = f + f1 f0 1 2h f f 2 f1 f0 2 f f f 1 0 2 h (d) Mode = l + 0 2 f1 f0 2 f f f 1 0 2 l h fu 5. For fndng mean of a data, f we use x = a + f h, then t s called : (a) the drect method (b) the step devaton method (c) the assumed mean method (d) none of these 6. In the formula x = a + fd, for fndng the mean of a grouped data, d f s are devaton from : (a) lower lmts of the classes (c) md-ponts of the classes (b) upper lmts of the classes (d) frequences of the class-marks 7. Whle computng mean of grouped data, we assume that the frequences are : (a) evenly dstrbuted over all the classes (b) centred at the class-marks of the classes (c) centred at the upper lmts of the classes (d) centred at the lower lmts of the classes 8. If x s are the md-ponts of the class-ntervals of a grouped data, f s are the correspondng frquences and x s the mean, then ( f x x ) s equal to : (a) 0 (b) 1 (c) 1 (d) 2 9. In the formula x fu = a + h, for fndng the mean of a grouped frequency dstrbuton, u f s equal to : x a (a) + x (b) h(x a) (c) a a x (d) h h h 10. The formula for medan of a grouped data s : n cf (a) Medan = l + h (b) Medan = l + f 3 n cf 2 f h,sector-5,devendra NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
(c) Medan = 2l + n cf 2 f h (d) none of these 11. In the formula, medan = l + n cf 2 f h, h s : (a) class-mark (b) class-sze (c) heght (d) none of these 12. The curve drawn by takng upper lmts along x-axs and cumulatve frequency along y-axs s : (a) frequency polygon (b) more than ogve (c) less than ogve (d) none of these 13. For more than ogve the x-axs represents : (a) upper lmts of class-ntervals (b) md-values of class-ntervals (c) lower lmts of class-ntervals (d) frequency 14. Ogve s the graph of : (a) lower lmts and frequency (b) upper lmts and frequency (c) lower/upper lmts and cumulatve frequency (d) none of these 15. The curve less than ogve s always : (a) ascendng (b) descendng (c) sometmes ascendng and sometmes descendng (d) none of these B. Questons From CBSE Examnaton Papers 1. In the fgure the value of the medan of the data usng the graph of less than ogve and more than ogve s : (a) 5 (b) 40 (c) 80 (d) 15 2. If mode = 80 and mean = 110, then the medan s : (a) 110 (b) 120 (c) 100 (d) 90 3. The lower lmt of the modal class of the followng data s : C.I. 0 10 10 20 20 30 30 40 40 50 Frequency 5 8 13 7 6 (a) 10 (b) 30 (c) 20 (d) 50 4. The mean of the followng data s : 45, 35, 20, 30, 15, 25, 40 : (a) 15 (b) 25 (c) 35 (d) 30 5. The mean and medan of a data are 14 and 15 respectvely. The value of mode s : (a) 16 (b) 17 (c) 13 (d) 18 6. For a gven data wth 50 observatons the less than ogve and the more then ogve ntersect at (15.5, 20). The medan of the data s : (a) 4.5 (b) 20 (c) 50 (d) 15.5 4,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
7. The emprcal relatonshp among the Medan, Mode and Mean of a data s : (a) mode = 3 medan + 2 mean (b) mode = 3 medan 2 mean (c) mode = 3 mean 2 medan (d) mode = 3 mean + 2 medan 8. For a symmetrcal dstrbuton, whch s correct? (a) Mean > Mode > Medan (b) Mean < Mode < Medan Mean + Medan (c) Mode = 2 (d) Mean = Medan = Mode 9. Whch of the followng s not a measure of central tendency? (a) Mean (b) Medan (c) Range (d) Mode 10. The class mark of a class nterval s : (a) Lower lmt + Upper lmt (b) Upper lmt Lower lmt (c) 1 2 (Lower lmt + Upper lmt) (d) 1 4 (Lower lmt + Upper lmt) 11. If mode of a data s 45, mean s 27, then medan s : (a) 30 (b) 27 (c) 23 (d) None of these 12. For the followng dstrbuton : marks Below 10 Below 20 Below 30 Below 40 Below 50 Below 60 No. of students 3 12 27 57 75 80 The modal class s : (a) 10 20 (b) 20 30 (c) 30 40 (d) 50 60 13. For a gven data wth 60 observatons the less than ogve and more than ogve ntersect at (66.5, 30). The medan of the data s : (a) 66.5 (b) 30 (c) 60 (d) 36.5 14. The abscssa of the pont of ntersecton of the less than type and of the more than type cumulatve frequency curves of a grouped data gves ts : (a) mean (b) medan (c) mode (d) all the three above 15. A data has 25 observatons (arranged n descendng order). Whch observaton represents the medan? (a) 12th (b) 13th (c) 14th (d) 15th 16. If mode of the followng data s 7, then value of k n 2, 4, 6, 7, 5, 6, 10, 6, 7, 2k + 1, 9, 7, 13 s : (a) 3 (b) 7 (c) 4 (d) 2 17. The mean and medan of a data are 14 and 16 respectvely. The value of mode s : (a) 13 (b) 16 (c) 18 (d) 20 18. The upper lmt of the medan class of the followng dstrbuton s : Class 0 5 6 11 12 17 18 23 24 29 Frequency 13 10 15 8 11 (a) 17 (b) 17.5 (c) 18 (d) 18.5 19. The measures of central tendency whch can t be found graphcally s : (a) mean (b) medan (c) mode (d) none of these 20. The measure of central tendency whch takes nto account all data tems s : (a) mode (b) mean (c) medan (d) none of these SHORT ANSWER TYPE QUESTIONS [2 marks] A. Important Questons 1. The followng are the marks of 9 students n a class. Fnd the medan marks : 21, 24, 27, 30, 32, 34, 35, 38, 48. 2. Fnd the medan of the daly wages of ten workers from the followng data : 5,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
8, 9, 11, 14, 15, 17, 18, 20, 22, 25. 3. Fnd the mode of the gven data : 120, 110, 130, 110, 120, 140, 130, 120, 140, 120. 4. Fnd the mode of the followng data : 25, 16, 19, 48, 19, 20, 34, 15, 19, 20, 21, 24, 19, 16, 22, 16, 18, 20, 16, 19. 5. Fnd the value of x, f the mode of the followng data s 25. 15, 20, 25, 18, 14, 15, 25, 15, 18, 16, 20, 25, 20, x, 18. 6. Calculate the mean for the followng dstrbuton : x : 5 6 7 8 9 f : 4 8 14 11 3 7. Fnd the mode of the followng data : 15, 8, 26, 24, 15, 18, 20, 15, 24, 19, 15. 8. Is t correct to say that an ogve s a graphcal representaton of a frequency dstrbuton? Gve reason. 9. Is t true to say that mean, medan and mode of a grouped data wll always be dfferent. Justfy your answer. 10. Wll the medan class and modal class of a grouped data always be dfferent? Justfy your answer. 11. A student draws a cumulatve frequency curve for the marks obtaned by 40 students of a class as shown. Fnd the medan marks obtaned by the students of the class. 12. The mean of ungrouped data and the mean calculated when the same data s grouped are always the same. Do you agree wth the statement? Gve reason. 13. What s the lower lmt of the modal class of the followng frequency dstrbuton? Age (n years) 0 10 10 20 20 30 30 40 40 50 50 60 No. of patents 16 13 6 11 27 18 14. Fnd the sum of the devatons of the varate values 3, 4, 6, 7, 8, 14 from ther mean. 15. If the mean of the followng dstrbuton s 6, fnd the value of p : x 2 4 6 10 p + 5 y 3 2 3 1 2 16. If x s the mean of ten natural numbers x1, x2,..., x10, show that : ( x x) + ( x x) + ( x x) +... + ( x x) =. 1 2 3 10 0 17. For a partcular year, the followng s the dstrbuton of the ages (n yrs.) of prmary school teachers n a state : Age (n yrs) 16 20 21 25 26 30 31 35 36 40 41 45 46 50 No. of teachers 11 32 51 49 27 6 4 Fnd how many teachers are of age less than 31 years. 18. If f = 11, f x = 2p+ 52 and the mean of the dstrbuton s 6, fnd the value of p. 19. 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. No. of days : 0 6 6 10 10 14 14 20 20 28 28 38 38 40 No. of students 11 10 7 4 4 3 1 20. The table gven below shows the frequency dstrbuton of the scores obtaned by 200 canddates n an MBA examnaton. 6,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
Score 200 250 250 300 300 350 350 400 400 450 450 500 500 550 550 600 No. of 30 15 45 20 25 40 10 15 Canddates Draw a cumulatve frequency curve by usng less than seres. 21. The shrt szes worn by a group of 200 persons, who bought the shrt from a store are as follows : Shrt sze 37 38 39 40 41 42 43 44 No. of persons 15 25 39 41 36 17 15 12 Fnd the modal shrt sze worn by the group. 22. Fnd the medan wage of a worker engaged at a constructon ste whose data are gven below : Wages (n Rs.) 3500 3800 4100 4500 5500 6500 7000 No. of workers 12 13 25 17 15 12 6 23. Fnd the medan for the followng data : marks (out of 20) 5 9 10 12 13 16 18 20 No. of students 4 5 6 42 11 6 4 2 24. Fnd f and F. marks 0 10 10 30 30 60 60 80 80 90 No. of students (frequency) c.f. 5 15 f 8 2 N = 60 5 F 50 58 60 N = f = 60 B. Questons From CBSE Examnaton Papers 1. Convert the followng data nto more than type dstrbuton. Class Intervals 50 55 55 60 60 65 65 70 70 75 75 80 Frequency 2 8 12 24 38 16 2. Fnd the mean of the followng data. Class Intervals 1 3 3 5 5 7 7 9 9 11 Frequency 7 8 2 2 1 3. Fnd the modal class and the medan class for the followng dstrbuton. C.I. 0 10 10 20 20 30 30 40 40 50 Frequency 6 10 12 8 7 4. Fnd the modal class and the medan class for the followng dstrbuton : Class Intervals 135 140 140 145 145 150 150 155 155 160 160 165 Frequency 4 7 18 11 6 5 5. The mean of the followng data s 7.5. Fnd the value of P. x 3 5 7 9 11 13 f 6 8 15 P 8 4 6. Fnd the mean of the followng frequency dstrbuton table. C.I. 0 10 10 20 20 30 30 40 40 50 Frequency 5 12 10 14 9 7. The medan class of a frequency dstrbuton s 125 145. The frequency and cumulatve frequency of the 7,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
class precedng to the medan class are 20 and 22 respectvely. Fnd the sum of the frequences, f the medan s 137. 8. 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 : 1 3 3 5 5 7 7 9 9 11 No. of famles : 7 8 2 2 1 Fnd the mode for the data above : 9. Fnd the mode of gven dstrbuton : C.I. 0 10 10 20 20 30 30 40 40 50 Frequency 2 12 22 8 6 10. Wrte the frequency dstrbuton table for the followng data : marks Below 10 Below 20 Below 30 Below 40 Below 50 Below 60 No. of students 0 12 20 28 33 40 11. The marks obtaned by 60 students, out of 50 n a Mathematcs examnaton, are gven below : marks 0 10 10 20 20 30 30 40 40 50 No. of students 5 13 12 20 10 Wrte the above dstrbuton as less than type cumulatve frequency dstrbuton. 12. Fnd the mode of the gven data : Class Intervals 0 20 20 40 40 60 60 80 Frequency 15 6 18 10 13. Fnd the medan of the followng gven data : x 6 7 5 2 10 9 3 f 9 12 8 13 11 14 7 14. Wrte the frequency dstrbuton table for the followng data : marks Above 0 Above 10 Above 20 Above 30 Above 40 Above 50 No. of students 30 28 21 15 10 0 15. Construct the frequency dstrbuton table for the gven data : Marks Obtaned than 10 than 20 than 30 than 40 than 50 than 60 No. of students 14 22 37 58 67 75 16. Fnd the mode of the gven data : Class Intervals 3 6 6 9 9 12 12 15 15 18 18 21 21 24 Frequency 2 5 10 23 21 12 3 17. Fnd the mean of the followng frequency dstrbuton : [2003, 2007] Class 0 10 10 20 20 30 30 40 40 50 Frequency 8 12 10 11 9 18. The wckets taken by a bowler n 10 crcket matches are as follows : 2 6 4 5 0 2 1 3 2 3 Fnd the mode of the data. 19. Fnd the medan for the followng frequency dstrbuton : Class Intervals 10 19 20 29 30 39 40 49 50 59 60 69 70 79 Frequency 2 4 8 9 4 2 1 20. Fnd the medan of the followng data. 8,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
marks 0 10 10 30 30 60 60 80 80 100 Frequency 5 15 30 8 2 21. Convert the gven cumulatve frequency table nto frequency dstrbuton table : Marks Number of students 0 and above 120 20 and above 108 40 and above 90 60 and above 75 80 and above 50 100 and above 24 120 and above 9 140 and above 0 22. For the data gven below draw more than ogve graph and fnd the value of medan. Producton (n tons) 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 Total No. of labourers 8 18 23 37 47 26 16 7 182 23. If the mean of the followng data s 18.75, fnd the value of p. [2005] x 10 15 8 25 30 f 5 10 7 8 2 24. The mean of the followng frequency dstrbuton s 62.8. Fnd the mssng frequency x. [2007] Class 0 20 20 40 40 60 60 80 80 100 100 120 Frequency 5 8 x 12 7 8 25. What s the lower lmt of the modal class of the followng frequency dstrbuton? [2009] Age (n yrs) 0 10 10 20 20 30 30 40 40 50 50 60 No. of patents 6 13 6 11 27 18 26. Fnd the medan class of the followng data : [2008] marks obtaned 0 10 10 20 20 30 30 40 40 50 50 60 Frequency 8 10 12 22 30 18 SHORT ANSWER TYPE QUESTIONS A. Important Questons [3 marks] 1. Usng short-cut method (Devaton method), calculate the mean of the followng frequency dstrbuton. Daly earnngs (n Rs.) 950 1000 1050 1100 1250 1500 1600 No. of shopkeepers 24 18 13 15 20 11 9 2. 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.) 11 13 13 15 15 17 17 19 19 21 21 23 23 25 No. of chldren 7 6 9 13 f 5 4 3. Fnd the value of p f the mean of followng dstrbuton s 20. [V. Imp.] x 15 17 19 20 + p 23 f 2 3 4 5p 6 9,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
4. The followng data gves the nformaton of the observed lfetmes (n hours) of 225 electrcal components : [Imp.] Lfetme (n hours) 0 20 20 40 40 60 60 80 80 100 100 120 Frequency 10 35 52 61 38 29 Determne the modal lfetmes of components. 5. 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 : No. of cars 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 Frequency 7 14 13 12 20 11 15 8 6. If the mode of the followng dstrbuton s 57.5, fnd the value of x. x 30 40 40 50 50 60 60 70 70 80 80 90 90 100 f 6 10 16 x 10 5 2 7. The dstrbuton below gves the weghts of 30 students of a class. Fnd the medan weght of the students. Weght (n kg) 40 45 45 50 50 55 55 60 60 65 65 70 70 75 Number of students 2 3 8 6 6 3 2 8. Usng assumed mean method, fnd the mean of the followng data : x 240 250 260 270 280 290 300 f 15 25 32 47 15 12 4 9. Fnd the value of p, f the value of the followng dstrbuton s 55. x p + 3 30 50 70 89 f p 8 10 15 10 10. Compute the mode of the followng data : Class-nterval 1 5 6 10 11 15 16 20 21 25 26 30 31 35 36 40 41 45 46 50 Frequency 3 8 13 18 28 20 13 8 6 4 11. Calculate the mssng frequency from the followng dstrbuton, t s gven that the medan of the dstrbuton s 24. Class-nterval 0 10 10 20 20 30 30 40 40 50 Frequency 5 25? 18 7 12. Fnd the medan wages for the followng frequency dstrbuton. Wage per day (n Rs.) 61 70 71 80 81 90 91 100 101 110 111 120 No. of workers 5 15 20 30 20 8 B. Questons From CBSE Examnaton Papers 1. Fnd the mean of the followng data : Classes 25 30 30 35 35 40 40 45 45 50 50 55 55 60 Frequency 14 22 16 6 5 3 4 2. Fnd the medan of the followng data : marks Number of Students 0 and above 80 10 and above 77 20 and above 72 30 and above 65 10,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
40 and above 55 50 and above 43 60 and above 28 70 and above 16 80 and above 10 90 and above 8 100 and above 0 3. Fnd the mean of the followng data. Classes-nterval 30 40 40 50 50 60 60 70 70 80 80 90 90 100 Frequency 2 3 8 6 6 3 2 4. Fnd the medan daly expenses from the followng data. Daly Expenses (n Rs.) No. of famles 20 40 6 40 60 9 60 80 11 80 100 14 100 120 20 120 140 15 140 160 10 160 180 8 180 200 7 Total 100 5. The mean of the followng dstrbuton s 62.8 and the sum of all frequences s 50. Compute the mssng frequences f 1 and f 2. Classes 0 20 20 40 40 60 60 80 80 100 100 120 Total Frequency 5 f 1 10 f 2 7 8 50 6. Fnd unknown entres a, b, c, d, e, f n the followng dstrbuton of heghts of students n a class and the total number of students n the class n 50. Heght n c.m. 150 155 155 160 160 165 165 170 170 175 175 180 Frequency 12 b 10 d e 2 Cumulatve frequency a 25 c 43 48 f 7. Fnd the mean marks from the followng data : marks Below 10 Below 20 Below 30 Below 40 Below 50 Below 60 Number of students 4 10 18 28 40 70 8. Fnd the medan of the followng data C.I. 0 10 10 20 20 30 30 40 40 50 Total Frequency 8 16 36 34 6 100 9. Draw a less than ogve for the followng frequency dstrbuton : Classes 150 155 155 160 160 165 165 170 170 175 175 180 Frequency 6 10 22 34 16 12 11,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
10. The gven dstrbuton shows the number of runs scored by some top batsmen of the world n one-day nternatonal crcket matches. Fnd the mode of the data. Runs Scored 3000 4000 4000 5000 5000 6000 6000 7000 No. of batsmen 4 18 9 7 Runs Scored 7000 8000 8000 9000 9000 10000 10000 11000 No. of batsmen 6 3 1 1 11. Durng the medcal check up of 35 students of a class, ther weghts were recorded as follows. Draw a less than type ogve for the gven data. Hence obtan medan weght from the graph. Weght (n kg) 12 Number of Students less than 38 0 less than 40 3 less than 42 5 less than 44 9 less than 46 14 less than 48 28 less than 50 32 less than 52 35 12. Fnd mean of the followng frequency dstrbuton usng step devaton method. Classes 0 10 10 20 20 30 30 40 40 50 Frequency 7 10 15 8 10 13. Fnd the mssng frequency for the gven frequency dstrbuton table, f the mean of the dstrbuton s 18. Classes 11 13 13 15 15 17 17 19 19 21 21 23 23 25 Frequency 3 6 9 13 f 5 4 14. Fnd the mode of the followng frequency dstrbuton : marks 10 20 20 30 30 40 40 50 50 60 60 70 70 80 No. of students 4 8 10 12 10 4 2 15. Fnd the medan of the followng data. Classes 500 600 600 700 700 800 800 900 900 1000 Frequency 40 28 35 22 25 16. Fnd the mssng frequences f 1 and f 2 f mean of 50 observatons s 38.2. Classes Interval 0 10 10 20 20 30 30 40 40 50 50 60 60 70 Frequency 4 4 f 1 10 f 2 8 5 17. The medan of the dstrbuton gven below s 35. Fnd the value of x and y, f the sum all frequences s 170. Varable 0 10 10 20 20 30 30 40 40 50 50 60 60 70 Frequency 10 20 x 40 y 25 15 18. The mean of the followng data s 53, fnd the mssng frequences. Age n years 0 20 20 40 40 60 60 80 80 100 Total No. of people 15 f 1 21 f 2 17 100 19. The daly expendture of 100 famles are gven below. Calculate f 1 and f 2, f the mean daly expendture s Rs. 188.,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
Expendture 140 160 160 180 180 200 200 220 220 240 No. of famles 5 25 f 1 f 2 5 20. Compute the medan for the followng data : Class nterval Cumulatve frequency than 20 than 30 than 40 than 50 13 than 60 than 70 than 80 than 90 than 100 0 4 16 30 46 66 82 92 100 21. Fnd the mssng frequences n the followng frequency dstrbuton table, f N = 100 and medan s 32. marks obtaned 0 10 10 20 20 30 30 40 40 50 50 60 Total No. of students 10? 25 30? 10 100 22. Fnd the medan of the followng data Class Interval 40 45 45 50 50 55 55 60 60 65 65 70 70 75 Frequency 2 3 8 6 6 3 2 23. Fnd the mean of the followng data usng step devaton method. Class Interval 0 10 10 20 20 30 30 40 40 50 50 60 60 70 Frequency 4 4 7 10 12 8 5 24. Fnd the mode of followng frequency dstrbuton : Class Interval 0 10 10 20 20 30 30 40 40 50 50 60 60 70 Frequency 5 8 15 20 14 8 5 25. Fnd the medan of the followng data : Heght (n cm) than 120 than 140 than 160 than 180 than 200 Number of students 12 26 34 40 50 26. Fnd mode of the gven data : C.I. 0 10 10 20 20 30 30 40 40 50 50 60 Frequency 16 13 6 11 27 18 27. The mean of the followng frequency dstrbuton s 57.6 and the sum of observatons s 50. Fnd the mssng frequences f 1 and f 2. [2004] Class 0 20 20 40 40 60 60 80 80 100 100 120 Total Frequency 7 f 1 12 f 2 8 5 50 28. Fnd the mode from the followng frequency dstrbuton : [2004] Class nterval 25 50 50 7 75 100 100 125 125 150 150 175 175 200 Frequency 12 17 20 30 25 19 11 29. Fnd the modal marks from the followng table : [2004] marks 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 No. of students 5 18 30 45 40 15 10 6 30. Fnd the mode from the followng data : [2005] Heght (n cm) 80 90 90 100 100 110 110 120 120 130 No. of students 7 11 5 4 10,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
LONG ANSWER TYPE QUESTIONS A. Important Questons 1. Fnd the mssng frequences f 1 and f 2 n the followng frequency dstrbuton, f t s known that the mean of the dstrbuton s 50 and the total frequency s 150. [HOTS] x 10 30 50 70 90 f 17 f 1 32 f 2 19 [4 marks] 2. 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 years. Age (n years) Number of polcy holders Below 20 2 Below 25 6 Below 30 24 Below 35 45 Below 40 78 Below 45 89 Below 50 92 Below 55 98 Below 60 100 3. Fnd the mean marks of the students from the followng frequency dstrbuton. marks No. of students less than 10 5 less than 20 9 less than 30 17 less than 40 29 less than 50 45 less than 60 60 less than 70 70 less than 80 78 less than 90 83 less than 100 85 4. Calculate the mode from the followng data : monthly salary (n Rs) No. of employees less than 5000 90 less than 10000 240 less than 15000 340 less than 20000 420 less than 25000 490 less than 30000 500 14,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
5. Compute the medan for the followng data : marks (more than or equal to) No. of students 80 150 90 141 100 124 110 105 120 60 130 27 140 12 150 0 6. The medan value for the followng frequency dstrbuton s 35 and the sum of all the frequences s 170. Fnd the values of x and y. C.I. 0 10 10 20 20 30 30 40 40 50 50 60 60 70 Frequency 10 20 x 40 y 25 15 7. Fnd the mode of the marks obtaned by 80 students n a class test n mathematcs as gven below : marks No. of students less than 10 3 less than 20 8 less than 30 24 less than 40 36 less than 50 49 less than 60 69 less than 70 75 less than 80 80 8. The followng table shows the ages of the patents admtted n a hosptal durng a year. Age (n years) (more than or equal to) 5 15 25 35 45 55 No. of patents 80 74 63 42 19 5 Fnd the mode and mean of the data gven above. Compare and nterpret the two measures of central tendency. 9. The mode of the followng dstrbuton s 65.625 hours. Fnd the value of p. Lfetme (n hours) 0 20 20 40 40 60 60 80 80 100 100 120 Frequency 10 35 52 61 p 29 10. Fnd the medan for the followng data : marks Below 10 Below 20 Below 30 15 Below 40 Below 50 Below 60 Below 70 Below 80 Number of students 12 32 57 80 92 116 164 200 11. A survey regardng the heght (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 than 140 4 than 145 11 than 150 29,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
than 155 40 than 160 46 than 165 51 Fnd the medan heght. 12. The followng table gves the dstrbuton of the lfetme of neon lamps. [Imp.] Lfetme (n hours) (more than or equal to) 1500 2000 2500 3000 3500 4000 4500 5000 No. of lamps 400 386 330 270 184 110 48 0 Fnd the medan lfetme of a lamp. 13. The annual profts earned by 38 shops n a market s represented n followng table. [Imp.] Proft (n lakhs of Rs) (more than or equal to) 5 10 15 20 25 30 35 No. of shops 38 28 16 14 10 7 3 Draw both the ogves for the above data and hence obtan the medan. 14. From the followng data, draw the two types of cumulatve frequency curves and determne the medan. Heght (n cm) Frequency 140 144 3 144 148 9 148 152 24 152 156 31 156 160 42 160 164 64 164 168 75 168 172 82 172 176 86 176 180 34 B. Questons From CBSE Examnaton Papers 1. Convert the followng data to a less than type dstrbuton and draw ts ogve. Also fnd the medan from the graph. Class Interval 100 120 120 140 140 160 160 180 180 200 Frequency 12 14 8 6 10 2. Convert the followng data nto a more than type dstrbuton and draw ts ogve. Also fnd the medan of the data from the graph. Class Interval 100 120 120 140 140 160 160 180 180 200 Frequency 12 14 8 6 10 3. Draw more than ogve for the followng frequency dstrbuton and hence obtan the medan. 16 Class Interval 5 10 10 15 15 20 20 25 25 30 30 35 35 40 Frequency 2 12 2 4 3 4 3 4. Draw less than ogve for the followng frequency dstrbuton and hence obtan the medan.,sector-5,devendra NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
marks obtaned 10 20 20 30 30 40 40 50 50 60 60 70 70 80 No. of students 3 4 3 3 4 7 9 5. If the medan of the followng data s 525. Fnd the values of x and y f the sum of the frequences s 100. Class Interval 0 100 100 200 200 300 300 400 400 500 Frequency 2 5 x 12 17 Class Interval 500 600 600 700 700 800 800 900 900 1000 Frequency 20 y 9 7 4 6. Calculate the mode of the followng frequency dstrbuton table. marks Above 25 Above 35 Above 45 Above 55 Above 65 Above 75 Above 85 Number of students 52 47 37 17 8 2 0 7. Durng medcal check up of 35 students of a class, ther weghts were recorded. Weght Number of students than 38 0 than 40 3 than 42 5 than 44 9 than 46 14 than 48 28 than 50 32 than 52 35 Draw less than type ogve for the gven data. Hence obtan the medan weght from graph and verfy the result by usng formula. 8. Change the followng data nto less than type dstrbuton and draw ts ogve. Hence fnd the medan of the data. marks obtaned 30 39 40 49 50 59 60 69 70 79 80 89 90 99 No. of students 5 7 8 10 5 8 7 9. Draw less than and more than ogve for the followng dstrbuton and hence obtan the medan. marks 30 40 40 50 50 60 60 70 70 80 80 90 90 100 No. of students 14 6 10 20 30 8 12 10. The followng dstrbuton gves the annual proft earned by 30 shops of a shoppng complex. Proft (n Lakh Rs.) 0 5 5 10 10 15 15 20 20 25 No. of shops 3 14 5 6 2 Change the above dstrbuton to more than type dstrbuton and draw ts ogve. 11. Followng dstrbuton shows the marks obtaned by the class of 100 students. marks 10 20 20 30 30 40 40 50 50 60 60 70 No. of students 10 15 30 32 8 5 Draw less than ogve for the above data. Fnd medan graphcally and verfy the result by actual method. 17,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
12. Fnd the medan by drawng both ogves. Class Interval 50 60 60 70 70 80 80 90 90 100 Frequency 3 5 9 12 6 13. If the medan of the dstrbuton gven below s 28.5, fnd the values of x and y. Class Intervals 0 10 10 20 20 30 30 40 40 50 50 60 Total Frequency 5 x 20 15 y 5 60 14. The mean of the followng data s 50. Fnd the mssng frequences f 1 and f 2. C.I. 0 20 20 40 40 60 60 80 80 100 Total No. of students 17 f 1 32 f 2 19 120 15. Draw a less than ogve for the followng data : marks Number of students than 20 0 than 30 4 than 40 16 than 50 30 than 60 46 than 70 66 than 80 82 than 90 92 than 100 100 Fnd the medan of the data from the graph and verfy the result usng the formula. 16. The followng table gves the dstrbuton of expendtures of dfferent famles on educaton. Fnd the mean expendture on educaton of a famly. [2004] Expendture (n Rs.) 18 Number of famles 1000 1500 24 1500 2000 40 2000 2500 33 2500 3000 28 3000 3500 30 3500 4000 22 4000 4500 16 4500 5000 7 17. 100 surnames were randomly pcked up from a local telephone drectory and the frequency dstrbuton of the number of letters n the Englsh alphabet n the surnames was obtaned as follows : [2008] Number of letters 1 4 4 7 7 10 10 13 13 16 16 19 Number of surnames 6 30 40 16 4 4,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261
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. 18. Fnd the mean, mode and medan of the followng data : [2008] Classes 0 10 10 20 20 30 30 40 40 50 50 60 60 70 Frequency 5 10 18 30 20 12 5 19. The followng table gves the daly ncome of 50 workers of a factory : [2008] Daly ncome (n Rs.) 100 120 120 140 140 160 160 180 180 200 No. of workers 12 14 8 6 10 Fnd the mean, medan and mode of the above data. 20. The medan of the followng data s 52.5. Fnd the values of x and y f the total frequency s 100. [2009] Class Interval Frequency 0 10 2 10 20 5 20 30 x 30 40 12 40 50 17 50 60 20 60 70 y 70 80 9 80 90 7 90 100 4 21. Fnd the mode, medan and mean for the followng data : [2009] marks obtaned 25 35 35 45 45 55 55 65 65 75 75 85 No. of students 7 31 33 17 11 1 19,SECTOR-5,DEVENDRA NAGAR,HOUSE NO-D/156,RAIPUR,C.G,PH-9752509261