Calculate five yearly moving averages of the number of students who have studied in a school given below: Year 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 No. of students 442 427 467 502 512 515 520 527 515 541

Given, Year 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 No. of students 442 427 467 502 512 515 520 527 515 541 The first 5 -yearly moving average is 442 + 427 + 467 + 502 + 512 5 = 2350 5 = 470 and is placed against 3 rd year i.e, 1995 . The second 5 -yearly moving average is 427 + 467 + 502 + 512 + 515 5 = 2423 5 = 484 . 6 and is placed against 4 th year i.e, 1996 . The third 5 -yearly moving average is 467 + 502 + 512 + 515 + 520 5 = 2516 5 = 503 . 2 and is placed against 5 th year i.e, 1997 . The fourth 5 -yearly moving average is 502 + 512 + 515 + 520 + 527 5 = 2576 5 = 515 . 2 and is placed against 6 th year i.e, 1998 . The fifth 5 -yearly moving average is 512 + 515 + 520 + 527 + 515 5 = 2589 5 = 517 . 8 and is placed against 7 th year i.e, 1999 . The sixth 5 -yearly moving average is 515 + 520 + 527 + 515 + 541 5 = 2618 5 = 523 . 6 and is placed against 8 th year i.e, 2000 . Thus, the 5 -yearly moving average are 470 , 484 . 6 , 503 . 2 , 515 . 2 , 517 . 8 and 523 . 6 .

Calculate five yearly moving averages of the number of students who have studied in a school given below: Year Production (in tonnes) 1986 351 1987 366 1988 361 1989 362 1990 400 1991 419 1992 410 1993 420 1994 450 1995 500

The first 5 -yearly moving average is 351 + 366 + 361 + 362 + 400 5 = 1840 5 = 368 and is placed against 3 rd year i.e, 1988 . The second 5 -yearly moving average is 366 + 361 + 362 + 400 + 419 5 = 1908 5 = 381 . 6 and is placed against 4 th year i.e, 1989 . The third 5 -yearly moving average is 361 + 362 + 400 + 419 + 410 5 = 1952 5 = 390 . 4 and is placed against 5 th year i.e, 1990 . The fourth 5 -yearly moving average is 362 + 400 + 419 + 410 + 420 5 = 2011 5 = 402 . 2 and is placed against 6 th year i.e, 1991 . The fifth 5 -yearly moving average is 400 + 419 + 410 + 420 + 450 5 = 2099 5 = 419 . 8 and is placed against 7 th year i.e, 1992 . The sixth 5 -yearly moving average is 419 + 410 + 420 + 450 + 500 5 = 2199 5 = 439 . 8 and is placed against 8 th year i.e, 1993 . Thus, the 5 -yearly moving average are 368 , 381 . 6 , 390 . 4 , 402 . 2 , 419 . 8 and 439 . 8 .

Daily absence from a school during three weeks is recorded as follows: Monday Tuesday Wednesday Thursday Friday Week 1 23 28 21 33 40 Week 2 38 52 43 58 63 Week 3 52 54 61 51 51 Draw a graph, illustrating these figures. Calculate 5 day moving averages and plot them on the same graph.

The given data can be rearranged in the following way: Days Number of Absence Week 1 Monday 23 Tuesday 28 Wednesday 21 Thursday 33 Friday 40 Week 3 Monday 38 Tuesday 52 Wednesday 43 Thursday 58 Friday 63 Week 3 Monday 52 Tuesday 54 Wednesday 61 Thursday 51 Friday 51 The first 5 -day moving average is 23 + 28 + 21 + 33 + 40 5 = 145 5 = 29 and is placed against 3 rd day of 1 st week i.e, Wednesday. The second 5 -day moving average is 28 + 21 + 33 + 40 + 38 5 = 160 5 = 32 and is placed against 4 th day of 1 st week i.e, Thursday. The third 5 -day moving average is 21 + 33 + 40 + 38 + 52 5 = 184 5 = 36 . 8 and is placed against 5 th day of 1 st week i.e, Friday. The fourth 5 -day moving average is 33 + 40 + 38 + 52 + 43 5 = 206 5 = 41 . 2 and is placed against 1 st day of week 2 i.e, Monday. The fifth 5 -day moving average is 40 + 38 + 52 + 43 + 58 5 = 231 5 = 46 . 2 and is placed against 2 nd day of week 2 i.e, Tuesday. The sixth 5 -day moving average is 38 + 52 + 43 + 58 + 63 5 = 274 5 = 54 . 8 and is placed against 3 rd day of week 2 i.e, Wednesday. The seventh 5 -day moving average is 52 + 43 + 58 + 63 + 52 5 = 268 5 = 53 . 6 and is placed against 4 th day of week 2 i.e, Thursday. The eighth 5 -day moving average is 43 + 58 + 63 + 52 + 54 5 = 270 5 = 54 and is placed against 5 th day of week 2 i.e, Friday. The ninth 5 -day moving average is 58 + 63 + 52 + 54 + 61 5 = 288 5 = 57 . 6 and is placed against 1 st day of week 3 i.e, Monday. The tenth 5 -day moving average is 63 + 52 + 54 + 61 + 51 5 = 281 5 = 56 . 2 and is placed against 2 nd day of week 3 i.e, Tuesday. The eleventh 5 -day moving average is 52 + 54 + 61 + 51 + 51 5 = 269 5 = 53 . 8 and is placed against 3 rd day of week 3 i.e, Wednesday. Thus, the 5 -day moving average are 29 , 32 , 36 . 8 , 41 . 2 , 46 . 2 , 54 . 8 , 53 . 6 , 54 , 57 . 6 , 56 . 2 and 53 . 8 which are graphically represented as below: The moving averages are shown by dotted line and the actual figures are shown by regular line.

The table given below shows that the daily attendance in thousands at a certain exhibition over a period of two weeks: Week 1 52 48 64 68 52 70 72 Week 2 55 47 61 65 58 75 81 Calculate seven day moving averages and illustrate these and original information on the same graph using the same scales.

The given data can be rearranged in the following way: Days Number of Absence Week 1 Monday 52 Tuesday 48 Wednesday 64 Thursday 68 Friday 52 Saturday 70 Sunday 72 Week 2 Monday 55 Tuesday 47 Wednesday 61 Thursday 65 Friday 58 Saturday 75 Sunday 81 The first 7 -day moving average is 52 + 48 + 64 + 68 + 52 + 70 + 72 7 = 426 7 = 60 . 8571 ≅ 60 . 86 and is placed against 4 th day of 1 st week i.e, Thursday. The second 7 -day moving average is 48 + 64 + 68 + 52 + 70 + 72 + 55 7 = 429 7 = 61 . 2857 ≅ 61 . 28 and is placed against 5 th day of 1 st week i.e, Friday. The third 7 -day moving average is 64 + 68 + 52 + 70 + 72 + 55 + 47 7 = 428 7 = 61 . 1428 ≅ 61 . 14 and is placed against 6 th day of 1 st week i.e, Saturday. The fourth 7 -day moving average is 68 + 52 + 70 + 72 + 55 + 47 + 61 7 = 425 7 = 60 . 7142 ≅ 60 . 71 and is placed against 7 th day of 1 st week i.e, Sunday. The fifth 7 -day moving average is 52 + 70 + 72 + 55 + 47 + 61 + 65 7 = 422 7 = 60 . 2857 ≅ 60 . 28 and is placed against 1 st day of 2 nd week i.e, Monday. The sixth 7 -day moving average is 70 + 72 + 55 + 47 + 61 + 65 + 58 7 = 428 7 = 61 . 1428 ≅ 61 . 14 and is placed against 2 nd day of 2 nd week i.e, Tuesday. The seventh 7 -day moving average is 72 + 55 + 47 + 61 + 65 + 58 + 75 7 = 433 7 = 61 . 8571 ≅ 61 . 86 and is placed against 3 rd day of 2 nd week i.e, Wednesday. The eighth 7 -day moving average is 55 + 47 + 61 + 65 + 58 + 75 + 81 7 = 442 7 = 63 . 1428 ≅ 63 . 14 and is placed against 4 th day of 2 nd week i.e, Thursday. Thus, the 7 -day moving average are 60 . 86 , 61 . 28 , 61 . 14 , 60 . 71 , 60 . 28 , 61 . 14 , 61 . 86 and 63 . 14 which are graphically represented as below: The moving averages are shown by dotted line and the actual figures are shown by regular line.

Assuming a four yearly cycle, calculate the trend by the method of moving averages from the following data: Year 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 Value 12 25 39 54 70 87 105 100 82 65

The given time period is even which indicates that we have to determine centred moving averages. The first quarterly moving average is 12 + 25 + 39 + 54 4 = 130 4 = 32 . 5 . The second quarterly moving average is 25 + 39 + 54 + 70 4 = 188 4 = 47 . Similarly, the third, fourth, fifth, sixth and seventh quarterly moving averages are 39 + 54 + 70 + 87 4 = 250 4 = 62 . 5 , 54 + 70 + 87 + 105 4 = 316 4 = 79 , 70 + 87 + 105 + 100 4 = 362 4 = 90 . 5 , 87 + 105 + 100 + 82 4 = 374 4 = 93 . 5 , and 105 + 100 + 82 + 65 4 = 352 4 = 88 respectively. Now, The first quarterly centre moving average is 32 . 5 + 47 2 = 39 . 75 . The second quarterly centre moving average is 47 + 62 . 5 2 = 54 . 75 Similarly, the third, fourth, fifth and sixth quarterly centre moving averages are 62 . 5 + 79 2 = 70 . 75 , 79 + 90 . 5 2 = 84 . 75 , 90 . 5 + 93 . 5 2 = 92 and 93 . 5 + 88 2 = 90 . 75 respectively. Thus, the quarterly moving average are 39 . 75 , 54 . 75 , 70 . 75 , 84 . 75 , 92 and 90 . 75 .

The profit of a soft drink firm (in thousands of rupees) during each month of the year is as given below: Months Profit (in thousands of Rupees) January 3 . 6 February 4 . 3 March 4 . 3 April 3 . 4 May 4 . 4 June 5 . 4 July 3 . 4 August 2 . 4 September 3 . 4 October 1 . 8 November 0 . 8 December 1 . 2 Calculate the four monthly moving averages and plot these and the original data on a graph sheet.

The given time period is even which indicates that we have to determine centred moving averages. The first quarterly moving average is 3 . 6 + 4 . 3 + 4 . 3 + 3 . 4 4 = 15 . 6 4 = 3 . 9 . The second quarterly moving average is 4 . 3 + 4 . 3 + 3 . 4 + 4 . 4 4 = 16 . 4 4 = 4 . 1 . Similarly, the third, fourth, fifth, sixth, seventh, eighth and ninth quarterly moving averages are 4 . 3 + 3 . 4 + 4 . 4 + 5 . 4 4 = 17 . 5 4 = 4 . 375 , 3 . 4 + 4 . 4 + 5 . 4 + 3 . 4 4 = 16 . 6 4 = 4 . 15 , 4 . 4 + 5 . 4 + 3 . 4 + 2 . 4 4 = 15 . 6 4 = 3 . 9 , 5 . 4 + 3 . 4 + 2 . 4 + 3 . 4 4 = 14 . 6 4 = 3 . 65 , 3 . 4 + 2 . 4 + 3 . 4 + 1 . 8 4 = 11 4 = 2 . 75 , 2 . 4 + 3 . 4 + 1 . 8 + 0 . 8 4 = 8 . 4 4 = 2 . 1 and 3 . 4 + 1 . 8 + 0 . 8 + 1 . 2 4 = 7 . 2 4 = 1 . 8 respectively. Now, The first quarterly centre moving average is 3 . 9 + 4 . 1 2 = 4 . The second quarterly centre moving average is 4 . 1 + 4 . 375 2 = 4 . 2375 Similarly, the third, fourth, fifth, sixth, seventh and eighth quarterly centre moving averages are 4 . 375 + 4 . 15 2 = 4 . 2625 , 4 . 15 + 3 . 9 2 = 4 . 025 , 3 . 9 + 3 . 65 2 = 3 . 775 , 3 . 65 + 2 . 75 2 = 3 . 2 , 2 . 75 + 2 . 1 2 = 2 . 425 and 2 . 1 + 1 . 8 2 = 1 . 95 respectively. Thus, the quarterly moving average are 4 , 4 . 2375 , 4 . 2625 , 4 . 025 , 3 . 775 , 3 . 2 , 2 . 425 and 1 . 95 which are represented as the dotted line and the actual figures by regular line in the graph below:

The number of road accidents in the city due to rash driving, over a period of 3 years, is given in the following table: Year Jan-Mar April-June July-Sept. Oct-Dec. 2010 70 60 45 72 2011 79 56 46 84 2012 90 64 45 82 Calculate four quarterly moving averages and illustrate them and original figures on one graph using the same axes for both.

The given time period is even which indicates that we have to determine centred moving averages. The first quarterly moving average is 70 + 60 + 45 + 72 4 = 247 4 = 61 . 75 . The second quarterly moving average is 60 + 45 + 72 + 79 4 = 256 4 = 64 . Similarly, the third, fourth, fifth, sixth, seventh, eighth and ninth quarterly moving averages are 45 + 72 + 79 + 56 4 = 252 4 = 63 , 72 + 79 + 56 + 46 4 = 253 4 = 63 . 25 , 79 + 56 + 46 + 84 4 = 265 4 = 66 . 25 , 56 + 46 + 84 + 90 4 = 276 4 = 69 , 46 + 84 + 90 + 64 4 = 284 4 = 71 , 84 + 90 + 64 + 45 4 = 283 4 = 70 . 75 and 90 + 64 + 45 + 82 4 = 281 4 = 70 . 5 respectively. Now, The first quarterly centre moving average is 61 . 75 + 64 2 = 62 . 875 . The second quarterly centre moving average is 64 + 63 2 = 63 . 5 Similarly, the third, fourth, fifth, sixth, seventh and eighth quarterly centre moving averages are 63 + 63 . 25 2 = 63 . 125 , 63 . 25 + 66 . 25 2 = 64 . 75 , 66 . 25 + 69 2 = 67 . 625 , 69 + 71 2 = 70 , 71 + 70 . 75 2 = 70 . 625 and 70 . 75 + 70 . 25 2 = 70 . 5 respectively. Thus, the quarterly moving average are 62 . 87 , 63 . 5 , 63 . 125 , 64 . 75 , 67 . 625 , 70 , 70 . 625 and 70 . 5 which are represented as the dotted line and the actual figures by regular line in the graph below:

Coded monthly sales figures of a particular brand of T.V. for 18 months commencing January 1 , 2005 are as follows: Year Jan. Feb. March April May June 2005 18 16 23 27 28 19 July Aug. Sep. Oct. Nov. Dec. 31 29 35 27 28 24 2006 Jan. Feb. March April May June 24 28 29 30 29 22 Calculate six monthly moving averages and display these and the original figures on the same graph, using the same axes for both.

The given time period is even which indicates that we have to determine centred moving averages. The first six monthly moving average is 18 + 16 + 23 + 27 + 28 + 19 6 = 131 6 = 21 . 83 . The second six monthly moving average is 16 + 23 + 27 + 28 + 19 + 31 6 = 144 6 = 24 . Similarly, the third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, eleventh, twelfth and thirteenth six monthly moving averages are 23 + 27 + 28 + 19 + 31 + 29 6 = 157 6 = 26 . 17 , 27 + 28 + 19 + 31 + 29 + 35 6 = 169 6 = 28 . 17 , 28 + 19 + 31 + 29 + 35 + 27 6 = 169 6 = 28 . 17 , 19 + 31 + 29 + 35 + 27 + 28 6 = 169 6 = 28 . 17 , 31 + 29 + 35 + 27 + 28 + 24 6 = 174 6 = 29 , 29 + 35 + 27 + 28 + 24 + 24 6 = 167 6 = 27 . 83 , 35 + 27 + 28 + 24 + 24 + 28 6 = 166 6 = 27 . 67 , 27 + 28 + 24 + 24 + 28 + 29 6 = 160 6 = 26 . 67 , 28 + 24 + 24 + 28 + 29 + 30 6 = 163 6 = 27 . 17 , 24 + 24 + 28 + 29 + 30 + 29 6 = 164 6 = 27 . 33 and 24 + 28 + 29 + 30 + 29 + 22 6 = 162 6 = 27 respectively. Now, The first six monthly centre moving average is 21 . 83 + 24 2 = 22 . 915 . The second six monthly centre moving average is 24 + 26 . 17 2 = 25 . 085 Similarly, the third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, eleventh and twelfth six monthly centre moving averages are 26 . 17 + 28 . 17 2 = 27 . 17 , 28 . 17 + 28 . 17 2 = 28 . 17 , 28 . 17 + 28 . 17 2 = 28 . 17 , 28 . 17 + 29 2 = 28 . 585 , 29 + 27 . 83 2 = 28 . 415 , 27 . 83 + 27 . 67 2 = 27 . 75 , 27 . 67 + 26 . 67 2 = 27 . 17 , 26 . 67 + 27 . 17 2 = 26 . 92 , 27 . 17 + 27 . 33 2 = 27 . 25 and 27 . 33 + 27 2 = 27 . 165 , respectively. Thus, the quarterly moving average are 22 . 915 , 25 . 085 , 27 . 17 , 28 . 17 , 28 . 17 , 28 . 585 , 28 . 415 , 27 . 75 , 27 . 17 , 26 . 92 , 27 . 25 and 27 . 165 which are represented as the dotted line and the actual figures by regular line in the graph below:

M L Aggarwal Solutions for Chapter: Index Numbers and Moving Averages, Exercise 2: EXERCISE

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Index Numbers and Moving Averages, Exercise 2: EXERCISE

Attempt the practice questions on Chapter 3: Index Numbers and Moving Averages, Exercise 2: EXERCISE with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 2 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Chapter: Index Numbers and Moving Averages, Exercise 2: EXERCISE with Hints & Solutions

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 3 - Index Numbers and Moving Averages>EXERCISE>Q 6

Calculate five yearly moving averages of the number of students who have studied in a school given below:

Year	$1993$	$1994$	$1995$	$1996$	$1997$	$1998$	$1999$	$2000$	$2001$	$2002$
No. of students	$442$	$427$	$467$	$\frac{502 + 512 + 515 + 520 + 527}{5} = \frac{2576}{5} = 515.2$	$512$	$515$	$520$	$527$	$515$	$541$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 3 - Index Numbers and Moving Averages>EXERCISE>Q 7

Calculate five yearly moving averages of the number of students who have studied in a school given below:

Year	Production (in tonnes)
$1986$	$7 th$
$1987$	$5 -yearly$
$1988$	$361$
$1989$	$362$
$1990$	$400$
$1991$	$419$
$1992$	$410$
$1993$	$420$
$1994$	$450$
$1995$	$500$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 3 - Index Numbers and Moving Averages>EXERCISE>Q 8

Daily absence from a school during three weeks is recorded as follows:

	Monday	Tuesday	Wednesday	Thursday	Friday
Week $1$	$23$	$28$	$21$	$33$	$40$
Week $2$	$38$	$52$	$43$	$58$	$63$
Week $3$	$52$	$54$	$61$	$51$	$51$

Draw a graph, illustrating these figures. Calculate 5 day moving averages and plot them on the same graph.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 3 - Index Numbers and Moving Averages>EXERCISE>Q 9

The table given below shows that the daily attendance in thousands at a certain exhibition over a period of two weeks:

Week $1$	$52$	$48$	$64$	$68$	$52$	$70$	$72$
Week $2$	$55$	$47$	$61$	$65$	$58$	$75$	$81$

Calculate seven day moving averages and illustrate these and original information on the same graph using the same scales.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 3 - Index Numbers and Moving Averages>EXERCISE>Q 10

Assuming a four yearly cycle, calculate the trend by the method of moving averages from the following data:

Year	$1984$	$1985$	$1986$	$1987$	$1988$	$1989$	$1990$	$1991$	$1992$	$1993$
Value	$12$	$25$	$39$	$54$	$70$	$87$	$105$	$100$	$82$	$65$

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 3 - Index Numbers and Moving Averages>EXERCISE>Q 11

The profit of a soft drink firm (in thousands of rupees) during each month of the year is as given below:

Months	Profit (in thousands of Rupees)
January	$3.6$
February	$4.3$
March	$4.3$
April	$3.4$
May	$4.4$
June	$5.4$
July	$3.4$
August	$2.4$
September	$3.4$
October	$1.8$
November	$0.8$
December	$1.2$

Calculate the four monthly moving averages and plot these and the original data on a graph sheet.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 3 - Index Numbers and Moving Averages>EXERCISE>Q 12

The number of road accidents in the city due to rash driving, over a period of $3$ years, is given in the following table:

Year	Jan-Mar	April-June	July-Sept.	Oct-Dec.
$2010$	$70$	$60$	$45$	$72$
$2011$	$79$	$56$	$46$	$84$
$2012$	$90$	$64$	$45$	$82$

Calculate four quarterly moving averages and illustrate them and original figures on one graph using the same axes for both.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 3 - Index Numbers and Moving Averages>EXERCISE>Q 13

Coded monthly sales figures of a particular brand of T.V. for $18$ months commencing January $1, 2005$ are as follows:

Year	Jan.	Feb.	March	April	May	June
$2005$	$18$	$16$	$23$	$27$	$28$	$19$
	July	Aug.	Sep.	Oct.	Nov.	Dec.
	$31$	$29$	$35$	$27$	$28$	$24$
$2006$	Jan.	Feb.	March	April	May	June
$2006$	$24$	$28$	$29$	$30$	$29$	$22$

Calculate six monthly moving averages and display these and the original figures on the same graph, using the same axes for both.

M L Aggarwal Solutions for Chapter: Index Numbers and Moving Averages, Exercise 2: EXERCISE

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Index Numbers and Moving Averages, Exercise 2: EXERCISE

Questions from M L Aggarwal Solutions for Chapter: Index Numbers and Moving Averages, Exercise 2: EXERCISE with Hints & Solutions

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