STATISTICS. , the mean deviation about their mean x is given by. x x M.D (M) =

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1 Chapter 5 STATISTICS 5. Overvew I earler classes, you have studed measures of cetral tedecy such as mea, mode, meda of ugrouped ad grouped data. I addto to these measures, we ofte eed to calculate a secod type of measure called a measure of dsperso whch measures the varato the observatos about the mddle value mea or meda etc. Ths chapter s cocered wth some mportat measures of dsperso such as mea devato, varace, stadard devato etc., ad fally aalyss of frequecy dstrbutos. 5.. Measures of dsperso (a) RageThe measure of dsperso whch s easest to uderstad ad easest to calculate s the rage. Rage s defed as: Rage Largest observato Smallest observato (b) Mea Devato () Mea devato for ugrouped data: For observato x, x,..., x, the mea devato about ther mea x s gve by M.D ( x ) x x Mea devato about ther meda M s gve by () M.D (M) x M () Mea devato for dscrete frequecy dstrbuto Let the gve data cosst of dscrete observatos x, x,..., x occurrg wth frequeces f, f,..., f, respectvely. I ths case ()

2 STATISTICS 7 M.D ( x ) f x x f x x f (3) M.D (M) f x M where f. () Mea devato for cotuous frequecy dstrbuto (Grouped data). M.D ( x ) f x x (4) (5) (c) (d) (e) (f) M.D (M) f x M where x are the mdpots of the classes, x ad M are, respectvely, the mea ad meda of the dstrbuto. Varace : Let x, x,..., x be observatos wth x as the mea. The varace, deoted by σ, s gve by σ ( x x) (7) Stadard Devato: If σ s the varace, the σ, s called the stadard devato, s gve by σ ( x x) Stadard devato for a dscrete frequecy dstrbuto s gve by σ where f s are the frequeces of x s ad (6) (8) f ( x x) (9) f. Stadard devato of a cotuous frequecy dstrbuto (grouped data) s gve by

3 7 EXEMPLAR PROBLEMS MATHEMATICS σ f ( x x) (0) where x are the mdpots of the classes ad f ther respectve frequeces. Formula (0) s same as (g) Aother formula for stadard devato : () σ ( ) σ x where h s the wdth of class tervals ad y mea. h f x f x ( f y ) () f y x A ad A s the assumed h 5.. Coeffcet of varato It s sometmes useful to descrbe varablty by expressg the stadard devato as a proporto of mea, usually a percetage. The formula for t as a percetage s 5. Solved Examples Coeffcet of varato Stadard devato 00 Mea Short Aswer Type Example Fd the mea devato about the mea of the followg data: Sze (x): Frequecy (f): Soluto Mea x f x f M.D. ( x ) f x x 3(7) + 3(5) + 4(3) + 4() + 7() + 4(3) + 3(5) + 4(7) f 4

4 STATISTICS Example Fd the varace ad stadard devato for the followg data: 57, 64, 43, 67, 49, 59, 44, 47, 6, 59 Soluto Mea ( x ) Varace (σ ) ( x x) Stadard devato (σ) σ Example 3 Show that the two formulae for the stadard devato of ugrouped data. ( x x) σ ad are equvalet. Soluto We have ( x x) x σ x + ( x xx x ) + + x xx x x x x + ( x ) ( ) + x x x x x x

5 74 EXEMPLAR PROBLEMS MATHEMATICS Dvdg both sdes by ad takg ther square root, we get σ σ. Example 4 Calculate varace of the followg data : Class terval Frequecy Mea ( x ) fx f Soluto Varace (σ ) f ( x x) 3( 7) + 6( 3) + 4() + 7(5) f Log Aswer Type Example 5 Calculate mea, varato ad stadard devato of the followg frequecy dstrbuto: Classes Frequecy

6 STATISTICS 75 Soluto Let A, the assumed mea, be 5.5. Here h 0 Classes x y x f f y fy x fy f Mea x ( 0) (0.4).5 h Varace (σ ) fy ( fy ) [70(4) ( 8) ] 70 (4) S.D. (σ) 6.7

7 76 EXEMPLAR PROBLEMS MATHEMATICS Example 6 Lfe of bulbs produced by two factores A ad B are gve below: Legth of lfe Factory A Factory B ( hours) (umber of bulbs) (umber of bulbs) The bulbs of whch factory are more cosstet from the pot of vew of legth of lfe? Soluto Here h 00, let A (assumed mea) 800. Legth of lfe Md values(x ) y x A 0 Factory A Factory B ( hour) f f y fy f f y fy For factory A Mea ( x ) hours 0 S.D (46)

8 STATISTICS 77 Therefore, Coeffcet of varato (C.V.) For factory B Mea S.D. S.D x (56) ( 36) 0 0 S.D. 0 Therefore, Coeffcet of varato Mea 770 Sce C.V. of factory B > C.V. of factory A Factory B has more varablty whch meas bulbs of factory A are more cosstet. Objectve Type Questos Choose the correct aswer out of the four optos gve agast each of the Examples 7 to 9 (M.C.Q.). Example 7 The mea devato of the data, 9, 9, 3, 6, 9, 4 from the mea s (A).3 (B).57 (C) 3.3 (D) 3.57 Soluto (B) s the correct aswer M.D. ( x ) x x Example 8 Varace of the data, 4, 5, 6, 8, 7 s The varace of 4, 8, 0,, 6, 34 wll be (A) 3.3 (B) 5.33 (C) (D) Soluto (C) s the correct aswer. Whe each observato s multpled by, the varace s also multpled by. Example 9 A set of values x, x,..., x has stadard devato 6. The stadard devato of values x + k, x + k,..., x + k wll be (A) σ (B) σ + k (C) σ k (D) kσ Soluto (A) s correct aswer. If each observato s creased by a costat k, the stadard devato s uchaged.

9 78 EXEMPLAR PROBLEMS MATHEMATICS 5.3 EXERCISE Short Aswer Type. Fd the mea devato about the mea of the dstrbuto: Sze Frequecy Fd the mea devato about the meda of the followg dstrbuto: Marks obtaed o. of studets Calculate the mea devato about the mea of the set of frst atural umbers whe s a odd umber. 4. Calculate the mea devato about the mea of the set of frst atural umbers whe s a eve umber. 5. Fd the stadard devato of the frst atural umbers. 6. The mea ad stadard devato of some data for the tme take to complete a test are calculated wth the followg results: umber of observatos 5, mea 8. secods, stadard devato 3.5 secods. Further, aother set of 5 observatos x, x,..., x 5, also secods, s ow avalable ad we have 5 x 79 ad dervato based o all 40 observatos. 5 x 554. Calculate the stadard 7. The mea ad stadard devato of a set of observatos are x ad s, respectvely whle the mea ad stadard devato of aother set of observatos are x ad s, respectvely. Show that the stadard devato of the combed set of ( + ) observatos s gve by S.D. ( ) + ( ) ( ) + + ( + ) s s x x

10 STATISTICS Two sets each of 0 observatos, have the same stadard dervato 5. The frst set has a mea 7 ad the secod a mea. Determe the stadard devato of the set obtaed by combg the gve two sets. 9. The frequecy dstrbuto: x A A 3A 4A 5A 6A f where A s a postve teger, has a varace of 60. Determe the value of A. 0. For the frequecy dstrbuto: x f Fd the stadard dstrbuto.. There are 60 studets a class. The followg s the frequecy dstrbuto of the marks obtaed by the studets a test: Marks Frequecy x x x (x + ) x x + where x s a postve teger. Determe the mea ad stadard devato of the marks.. The mea lfe of a sample of 60 bulbs was 650 hours ad the stadard devato was 8 hours. A secod sample of 80 bulbs has a mea lfe of 660 hours ad stadard devato 7 hours. Fd the overall stadard devato. 3. Mea ad stadard devato of 00 tems are 50 ad 4, respectvely. Fd the sum of all the tem ad the sum of the squares of the tems. ad the total umber of tem s 4. If for a dstrbuto ( x 5) 3, ( x 5) 43 8, fd the mea ad stadard devato. 5. Fd the mea ad varace of the frequecy dstrbuto gve below: x x < 3 3 x < 5 5 x < 7 7 x < 0 f 6 4 5

11 80 EXEMPLAR PROBLEMS MATHEMATICS Log Aswer Type 6. Calculate the mea devato about the mea for the followg frequecy dstrbuto: Class terval Frequecy Calculate the mea devato from the meda of the followg data: Class terval Frequecy Determe the mea ad stadard devato for the followg dstrbuto: Marks Frequecy The weghts of coffee 70 jars s show the followg table: Weght ( grams) Frequecy Determe varace ad stadard devato of the above dstrbuto. 0. Determe mea ad stadard devato of frst terms of a A.P. whose frst term s a ad commo dfferece s d.

12 STATISTICS 8. Followg are the marks obtaed, out of 00, by two studets Rav ad Hasha 0 tests. Rav Hasha Who s more tellget ad who s more cosstet?. Mea ad stadard devato of 00 observatos were foud to be 40 ad 0, respectvely. If at the tme of calculato two observatos were wrogly take as 30 ad 70 place of 3 ad 7 respectvely, fd the correct stadard devato. 3. Whle calculatg the mea ad varace of 0 readgs, a studet wrogly used the readg 5 for the correct readg 5. He obtaed the mea ad varace as 45 ad 6 respectvely. Fd the correct mea ad the varace. Objectve Type Questos Choose the correct aswer out of the gve four optos each of the Exercses 4 to 39 (M.C.Q.). 4. The mea devato of the data 3, 0, 0, 4, 7, 0, 5 from the mea s (A) (B).57 (C) 3 (D) Mea devato for observatos x, x,..., x from ther mea x s gve by (A) ( x x) (B) x x (C) ( x ) x (D) ( x x ) 6. Whe tested, the lves ( hours) of 5 bulbs were oted as follows: 357, 090, 666, 494, 63 The mea devatos ( hours) from ther mea s (A) 78 (B) 79 (C) 0 (D) Followg are the marks obtaed by 9 studets a mathematcs test: 50, 69, 0, 33, 53, 39, 40, 65, 59 The mea devato from the meda s: (A) 9 (B) 0.5 (C).67 (D) 4.76

13 8 EXEMPLAR PROBLEMS MATHEMATICS 8. The stadard devato of the data 6, 5, 9, 3,, 8, 0 s (A) 5 7 (B) 5 7 (C) 6 (D) 6 9. Let x, x,..., x be observatos ad x be ther arthmetc mea. The formula for the stadard devato s gve by (A) ( x x) (B) ( x x) (C) ( x x) (D) x + x 30. The mea of 00 observatos s 50 ad ther stadard devato s 5. The sum of all squares of all the observatos s (A) (B) (C) 5500 (D) Let a, b, c, d, e be the observatos wth mea m ad stadard devato s. The stadard devato of the observatos a + k, b + k, c + k, d + k, e + k s (A) s (B) ks (C) s + k (D) 3. Let x, x, x 3, x 4, x 5 be the observatos wth mea m ad stadard devato s. The stadard devato of the observatos kx, kx, kx 3, kx 4, kx 5 s (A) k + s (B) s k (C) ks (D) s 33. Let x, x,... x be observatos. Let w lx + k for,,..., where l ad k are costats. If the mea of x s s 48 ad ther stadard devato s, the mea of w s s 55 ad stadard devato of w s s 5, the values of l ad k should be (A) l.5, k 5 (B) l.5, k 5 (C) l.5, k 5 (D) l.5, k Stadard devatos for frst 0 atural umbers s (A) 5.5 (B) 3.87 (C).97 (D) Cosder the umbers,, 3, 4, 5, 6, 7, 8, 9, 0. If s added to each umber, the varace of the umbers so obtaed s s k

14 STATISTICS 83 (A) 6.5 (B).87 (C) 3.87 (D) Cosder the frst 0 postve tegers. If we multply each umber by ad the add to each umber, the varace of the umbers so obtaed s (A) 8.5 (B) 6.5 (C) 3.87 (D) The followg formato relates to a sample of sze 60: x 8000, x 960 The varace s (A) 6.63 (B) 6 (C) (D) Coeffcet of varato of two dstrbutos are 50 ad 60, ad ther arthmetc meas are 30 ad 5 respectvely. Dfferece of ther stadard devato s (A) 0 (B) (C).5 (D) The stadard devato of some temperature data C s 5. If the data were coverted to ºF, the varace would be (A) 8 (B) 57 (C) 36 (D) 5 Fll the blaks Exercses from 40 to Coeffcet of varato... Mea If x s the mea of values of x, the ( x x) s always equal to. If a has ay value other tha x, the ( x x) s tha ( x a) 4. If the varace of a data s, the the stadard devato of the data s. 43. The stadard devato of a data s of ay chage org, but s o the chage of scale. 44. The sum of the squares of the devatos of the values of the varable s whe take about ther arthmetc mea. 45. The mea devato of the data s whe measured from the meda. 46. The stadard devato s to the mea devato take from the arthmetc mea.