Statistics Assignment 2-2

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STAMFORD UNIVERSITYDepartment Of Business Administration Master of Business Administration Program 744, Satmosjid Road Dhanmondi, Dhaka

Statistics For Business (MAT 421) Topic:

Submitted by: Engr. Mohd. Abdus Sattar ID. MBA 044 12301 Trimester: Spring-2011

Assignment on Statistics

1. Standard DeviationExample-1: Find (i) Variance, and (ii) Standard Deviation, from the following data: 51, 52, 53, 54, 55 Solution:X 51 52 53 54 55 X- x -2 -1 0 1 2265 = 53 5

(X- x ) 4 1 0 1 4 10 (i) Standard Deviation, =( x x) 2 n 10 5

=

x = 265x =

xn

=

(ii) Variance,

= 2 = 1.414 =2

Example-2: Sales on a number of shops are given in thousand taka as follows:Sales 0-5 5-10 10-15 15-20 20-25 Solution: Sales 0-5 5-10 10-15 15-20 20-25 x

f 1 2 4 2 1 f 1 2 4 2 1 10 =

Requird: (i) Variance, (ii) Standard Deviation, (iii) Coefficient of Variation, CV

Midpoint (x) 2.5 7.5 12.5 17.5 22.5 125 10

fx 2.5 15 50 35 22.5 125

X- x -10 -5 0 5 10

f(X- x ) 100 50 0 50 100 300

fxn

=

= 12.5

(i) Standard Deviation, =

f ( x x) 2 n3 0

(ii) Variance, = 30 (iii) CV =x100 x 5.477 x100 = 12.5

=

300 = 10

= 5.477 =43.82%

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Assignment on Statistics

Example-3: Sales of company in January and February/2011 are recorded in thousand taka and presented as follows: Sales 30-35 35-40 40-45 45-50 50-55 55-60 Solution: Sales Midpoint (X) 30-35 35-40 40-45 45-50 50-55 55-60 32.5 37.5 42.5 47.5 52.5 57.5 (i) x1=

Frequency January 1 6 9 12 5 1 February 2 8 14 3 2 1

Requird: (i) Variance, for each month (ii) Standard Deviation, for each month (iii) Coefficient of Variation, CV for each month (iv) Which months average sale is more? v) Which months sale is more variable?

January (f1) 1 6 9 12 5 1 341530 34

Februaryf1(X- x 1)

(f1X) 32.5 225 382.5 570 262.5 57.5 1,530= 45

X- x

1

(f2) 2 8 14 3 2 1 30(i) x2=

(f2X) 65.0 300 595 142.5 105 57.5

X- x

2

f2(X- x 2)

-12.5 -7.5 -2.5 2.5 7.5 12.5

156.25 337.5 56.25 75 281.25 156.25

-9.67 -4.67 0.33 5.33 10.33 15.3311265 30

187.02 174.47 1.52 85.23 213.42 235.11

=

f 1xn

=

=

f 2xn

=

= 42.17

(ii) Standard Deviation, =3 .2 1 5

f ( x x) 2 n

=

(ii) S.D. = 29 .89 =5.45

(iii) Variance, = 29.89 (iii) CV = =12.92%x x100

=5.59

=

(iii) Variance, = 31.25 (iii) CV =x100

5.45 x100 42.17

5.59 x100 =12.42% x 45 iv) Average Sale of January is more. (v) Feb. months sale is more variable because CV of Feb > CV of Jan

=

2. Quartile, Decile & Percentile DeviationExample 4: Sales of a number of shops in thousand taka are given as follows:/opt/scribd/conversion/tmp/scratch2603/57131478.doc****Page 2

Assignment on Statistics

3,11, 13, 12, 14, 9, 8, 18, 17 & 24. Required: (i) Ordered Set, XL , XH, R (ii) Five number summary (XL , Q1, Q2, Q3, XH) (iii) Inter-Quartile Range (IQR), (iv) Quartile Deviation or Semi Inter-Quartile Range (SIQR) (v) D3 , D7 (vi) P29, P61 (vi) Inter-fractile Range(IFR) between P29 and P61

Solution: (i) (ii) Ordered Set = 3, 8, 9, 11, 12, 13, 14, 17, 18, 24 XL = 3, XH =24, R = 24-3 =21 Q1 = P25 = X3 = 9 Q2 = P50 = (X5 + X6)= 12.5 Q3 = P75 = X8 = 17 [ i = .25n= 0.25 x 10 = 2.5, Say 3] [ i = .50n= 0.5 x 10 = 5] [ i = .75n= 0.75 x 10 = 7.5, Say 8]

So Five number summary is 3, 9, 12.5, 17, 24

(iii) (iv) (v) (vi) (vii)

Inter-Quartile Range (IQR) = Q3 - Q1 = 17-9 =8 Quartile Deviation or Semi Inter-Quartile Range (SIQR) = ( Q3- Q1) = 4 D3 = P30 = (X3 + X4)= (9 +11)= 10 D7 = P70 = (X7 + X8)= (14 +17)= 15.5 P29 = X3 = 9 P61 = X7 = 14 [ i = .30n= 0.3 x 10 = 3] [ i = .70n= 0.7 x 10 = 7] [ i = .29n= 0.29 x 10 = 2.9 Say 3] [ i = .61n= 0.61 x 10 = 6.1 Say 7]

Inter-fractile Range (IFR) between P29 and P61 = P61 P29 = 14-9 = 5

Example 5: Sales of a number of shops in thousand taka are given in the following Frequency Distribution: Sales f Required: (i) XL , XH, R (ii) Five number summary (XL , Q1, Q2, Q3, XH) (iii) Inter-Quartile Range (IQR), (iv) Quartile Deviation or Semi Inter-Quartile 40-45 1 Range (SIQR) (v) D3 , D7 (vi) P29, P61 (vi) Inter-fractile Range(IFR) between 45-50 4 P29 and P6150-55 55-60 60-65 65-70 8 9 5 3

Solution:

/opt/scribd/conversion/tmp/scratch2603/57131478.doc****Page 3

Assignment on Statistics Sales 40-45 45-50 50-55 55-60 60-65 65-70 f 1 4 8 9 5 3 fc 1 5 13 22 27 30

(i) (ii)

XL=40, XH=70, R = 70-40 = 30 Q2 = Median, Me= L1 +0.5n fc xc fm

0.5 n = 0.5 x 30 = 15; So, Q2 lies on series 55-60 i.e. Median group is 55-60.0.5n fc xc fm

So, Q2

= L1 + = 55 +

15 13 x5 9

= 56.11 In the same way, 0.25 n = 0.25 x 30 = 7.5; So, Q 1 lies on series 50-55 i.e. Quartile groupis 50-55. So, Q1

= L1 + = 50 +

0.25 n fc xc fm

7.5 5 x5 8

= 51.56 0.75 n = 0.75 x 30 = 22.5; So, Q3 lies on series 60-65 i.e. 3rd Quartile group is 60-65.So, Q3

= L1 + = 60 +

0.75 n fc xc fm

22 .5 22 x5 5

= 60.5 Five Number Summary (iii) (iv) (v)

= XL , Q1, Q2, Q3, XH

= 40, 51.56, 56.11, 60.5, 70

Inter-Quartile Range (IQR) = Q3- Q1 = 60.5-51.56 = 8.94 Quartile Deviation or Semi Inter-Quartile Range (SIQR) = ( Q3- Q1) = 4.47 D3 = P30

/opt/scribd/conversion/tmp/scratch2603/57131478.doc****Page 4

Assignment on Statistics

0.30 n = 0.30 x 30 = 9; So, D3 lies on series 50-55So, D3

= L1 + = 50 + = 52.5

0.30 n fc xc fm

9 5 x5 8

D7 = P70 0.70 n = 0.70 x 30 = 21; So, D7 lies on series 55-60So, D7

= L1 + = 55 +

0.70 n fc xc fm

21 13 x5 9

(vi)

= 59.44 0.29n = 0.29 x 30 = 8.7; So, P29 lies on series 50-55So, P29

= L1 + = 50 +

0.29 n fc xc fm

8.7 5 x5 8

= 52.31

0.61n = 0.61 x 30 =18.3; So, P29 lies on series 55-60So, P61

= L1 + = 55 +

0.61 n fc xc fm

18 .3 13 x5 9

= 57.94 (vii) Inter-fractile Range (IFR) between P29 and P61 = P61 P29 = 57.94-52.31 = 5.63

Levin & Rubin Pg 111: SC-3-11 First we arrange the data in ascending order: 59 85 65 87 67 88 68 91 71 92 72 93 75 94 79 95 81 100 83 100

P80 = (X16 + X17)= (93 +94)= 93.5/opt/scribd/conversion/tmp/scratch2603/57131478.doc****Page 5

[ i = .80n= 0.80 x 20 = 16 ]

Assignment on Statistics

Levin & Rubin Pg 111: SC-3-12 XL = 3,600, XH =20,300, R = 20,300-3,600 =16,700miles Q1 = P25 = (X10 + X11)= (8,100 +8,300)= 8,200 miles Q3 = P75 = (X30 + X31)= (12,700 +12,900)= 12,800miles Inter-Quartile Range (IQR) = Q3- Q1 = 12,800-8,200 = 4,600miles Levin & Rubin Pg 111: 3-52 First we arrange the data in ascending order:33 74 45 75 52 76 54 77 55 84 61 91 66 91 68 93 69 97 72 99

[ i = .25n= 0.25 x 40 = 10] [ i = .75n= 0.75 x 40 = 30]

Q1 = P25 = (X5 + X6)= (55 +61)= 58 Q3 = P75 = (X15 + X16)= (84 +91)= 87.5 Inter-Quartile Range (IQR) = Q3- Q1 = 87.5-58 = 29.5 Levin & Rubin Pg 111: 3-53 First we arrange the data in ascending order:2145 3249 2200 3268 2228 3362 2268 3469 2549 3661 2598 3692

[ i = .25n= 0.25 x 20 = 5] [ i = .75n= 0.75 x 20 = 15]

2653 3812

2668 3842

2697 3891

2841 3897

a) XL = 2,145, XH =3,897, R = 3,897-2,145 =1752 b) P20 = (X4 + X5)= (2,268 +2,549)= 2,408.5 P80 = (X16 + X17)= (3,692 +3,812)= 3752 c) Q1 = P25 = (X5 + X6)= (2,549 +2,598)=2,573.5 Q3 = P75 = (X15 + X16)= (3,661 +3,692)= 3676.5 Inter-Quartile Range (IQR) = Q3- Q1 = 3676.5-2573.5 = 1103 Levin & Rubin Pg 111: 3-54 First we arrange the data in ascending order:/opt/scribd/conversion/tmp/scratch2603/57131478.doc****Page 6

[ i = .20n= 0.20 x 20 = 4] [ i = .80n= 0.80 x 20 = 16] [ i = .25n= 0.25 x 20 = 5] [ i = .75n= 0.75 x 20 = 15]

Inter-fractile Range (IFR) between P20 and P80 = P80 P20 = 3752-2408.5 = 1,343.5

Assignment on Statistics69 88 94 78 88 95 82 89 96 84 89 97 84 89 98 86 92 99 87 92 99 87 94 102 88 94 102 88 94 105

P70 = (X21 + X22)= (94 +95)= 94.5 Degrees Levin & Rubin Pg 111: 3-55 First we arrange the data in ascending order:51 126 83 127 92 129 93 132 93 133 95 135

[ i = .70n= 0.70 x 30 = 21]

101 143

115 147

123 157

125 185

XL = 51, XH =185, R = 185-51 =134 Comment: It is one of the measures of dispersion. But there are more useful measures are available. Levin & Rubin Pg 111: 3-56 First we arrange the data in ascending order:0.10 0.59 0.12 0.66 0.23 0.67 0.32 0.69 0.45 0.77 0.48 0.83 0.50 0.89 0.51 0.95 0.53 1.10 0.58 1.20

XL = 0.10, XH =1.20, R = 1.20-0.10 =1.10 minutes Q1 = P25 = (X5 + X6)= (0.45 +0.48)=0.465 minutes Q3 = P75 = (X15 + X16)= (0.77 +0.83)= 0.800 minutes Inter-Quartile Range (IQR) = Q3- Q1 = 0.800-0.465 = 0.335 minutes [ i = .25n= 0.25 x 20 = 5] [ i = .75n= 0.75 x 20 = 15]

1. What is Dispersion? What are the different measures of dispersion? Dispersion: The extent to which the observations in a sample or in a population vary about their mean is known as Dispersion. A quantity that measures the dispersion in a sample or in a population is known as the measure of dispersion./opt/scribd/conversion/tmp/scratch2603/57131478.doc****Page 7

Assignment on Statistics The main measures of dispersions are the range, the semi-inter quartile range or the quartile deviation, the mean deviation or the average deviation, the variance and the standard deviation. 2. Distinguish between Absolute and Relat