Download - Statistics assignment 2
Answer to the Question No. 2
8. Population Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(N+1)th = .25(12+1)th =3.25th
So, the first quartile is one quarter of the way from the 3rd observation (22) to the 4th (22).
Q1 = 22 + .25 (22 – 22) = 22+0 = 22
Again, Q3 = .75(N+1)th = .75(12+1)th = 9.75th
So, the third quartile is three quarters of the way from the 9th observation (29) to the 10th (33).
Q3 = 29 + .75(33 – 29) = 29 + 3 = 32
IQR = Q3 – Q1 = 32 – 22 = 10
9. Population Variance:
σ2x =
= − (26.75)2
=
= 29.52
10. Population Standard Deviation:
σx= √σ2 = √29.52 = 5.43
11. Population Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
µ = Population Mean = 26.75 [from (1)], N = 12
σ = standard deviation, σ² = 29.52 [from (9)]
σ²x = Variance, µx = Population Mean = 26.75 [from (1)], N = 12
(21)²+ (22)²+ (27)²+ (36)²+ (22)²+ (29)²+ (22)²+ (23)²+ (22)²+ (28)²+ (36)²+ (33)²12
21 22 22 22 22 23 27 28 29 33 36 36
Here,Q1 = First QuartileQ3 = Third Quartile
∑i=1
N
Xi2
N−μ
x2
894112
−715 .56
∑i=1
N
( x i−μx )
N
∴ MAD = 5712
= 4.75
12. Population Coefficient of Variation:
C. V. = σ xμx
× 100 = 5.4326.75
× 100 = 20.29%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge = Q1+Q 3
2=22+32
2 = 27
Measures of Central Tendency for Sample
Sample:
14. Sample Mean: The sample contains n=6, observations, so the Mean is
X=¿
= 21+27+36+22+29+33
6
= 1686 = 28
15. Sample Median: Arranging n=6, observations in ascending order, we have
Xi Xi - µx = Xi – 26.75 (Xi - µx)21 -5.75 5.7522 -4.75 4.7527 0.25 0.2536 9.25 9.2522 -4.75 4.7529 2.25 2.2522 -4.75 4.7523 -3.75 3.7522 -4.75 4.7528 1.25 1.2536 9.25 9.2533 6.25 6.25
* Sums = 0 Sums = 57
21 22 27 29 33 36
Q₁ = 22, Q₃ = 32 [from (8)]
Here, = Sample Meann = Number of observationXi = Observations
21 27 36 22 29 33
C.V = coefficient of variation
σx = 5.43 [from (10)]
µx = 26.75 [from (1)]
∑i=1
n
X i
n
Ascending order:
Median =
= 28
16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange =XS+XL2
= 21+362
=572 = 28.5
18. Sample Harmonic Mean:
H. M. =n
1a1
+1a2
±−−−−−−∓1
an
¿ 6121
+127
+136
+122
+129
+133
=6
.04+.03+.02+.04+.03+.03 = 31.57
19. Sample Geometric Mean:
G. M. = n√a1×a2×a3×−−−×an
= 6√21×27×36×22×29×33
= 27.47
Measures of Dispersion for Sample
20. Sample Range:
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
[n2 ]th
+[n+22 ]th
2
¿[62 ]
th
+[6+22 ]th
2=3
rd+4th
2=27+292
¿562
Range = XL – XS = 36 – 21 = 15
21. Sample Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(n+1)th = .25(6+1)th =1.75th
So, the first quartile is three quarters of the way from the 1st observation (21) to the 2nd (22).
Q1 = 21 + .75 (22 – 21) = 21+.75 = 21.75
Again, Q3 = .75(n+1)th = .75(6+1)th = 5.25th
So, the third quartile is one quarter of the way from the 5th observation (33) to the 6th (36).
Q3 = 33 + .25(36 – 33) = 33 + .75 = 33.75
IQR = Q3 – Q1 = 33.75 – 21.75 = 12
22. Sample Variance:
Sx2 =∑i=1
n
x i2−n x ²
n−1
=
= 4880−4704
5 = 176
5 = 35.2
23. Sample Standard Deviation: Sx = √S x2 = √35.2 = 5.93
24. Sample Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
Here,Q1 = First QuartileQ3 = Third Quartile
21 22 27 29 33 36
Sx² = Variance, Sample Mean = 28 [from (14)], n= 6
6-1(21)² + (27)² + (36)² + (22)² + (29)² + (33)² - 6(28)²
S = Standard Deviation, S² = 35.2 [from (22)]
Xi Xi - ̅� = Xi - 28 (Xi - )21 -7 727 -1 136 8 822 -6 629 1 133 5 5
* Sum = 0 Sum=28
∑i=1
n
( x i−x )
n
∴ MAD = 286
= 4.66
25. Sample Coefficient of Variation:
C. V. = SxX
× 100 = 5.9328
× 100 = 21.17%
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge = Q1+Q 3
2=21.75+33.75
2
= 27.75
Answer to the Question No. 3
Population:
Measures of Central Tendency for Population
1. Population Mean (Average): The population contains N=10 observations, so the Mean is
µx =
=
=
= 3.49
2. Population Median: Arranging N=10 observations in ascending order, we have
Ascending order:
Median =
=
Here,C.V. = coefficient of variationSx = 5.93 [from (23)]
= 28 [from (14)]
Q₁ = 21.5, Q₃ = 33.75 [from (21)]
Here,µx = Population MeanN = Number of observationXi = Observations
3.6 3.1 3.9 3.7 3.5 3.7 3.4 3.0 3.6 3.4
3.6 + 3.1 + 3.9 + 3.7 + 3.5 + 3.7 + 3.4 + 3.0 + 3.6 + 3.410
3.0 3.1 3.4 3.4 3.5 3.6 3.6 3.7 3.7 3.9
∑i=1
N
x i
N
34 .910
[ N2 ]th
+[ N+22 ]
th
2
[102 ]th
+[10+22 ]th
25th+6 th
2=3 .5+3 .6
2
=
= 3.55
3. Population Mode:
The Mode of a set of observations is the value that occurs most frequently. So, Mode = 3.6
4. Population Midrange:
Midrange = XS+XL2
= 3.0+3.92
=6.92 = 3.45
5. Population Harmonic Mean:
H. M. =N
1a1
+1a2
±−−−−−−−−−−−−−−−−−∓1
aN
¿ 1013.6
+13.1
+13.9
+13.7
+13.5
+13.7
+13.4
+13.0
+13.6
+13.4
=10
.27+.32+.25+.27+.28+ .27+.29+.33+.27+ .29 = 3.52
6. Population Geometric Mean:
G. M. = N√a1×a2×a3×−−−−−−−×aN
= 10√3.6×3.1×3.9×3.7×3.5×3.7×3.4×3.0×3.6×3.4
= 3.47
7. Population Range:
Range = XL – XS = 3.9 – 3.0 = 0.9
8. Population Interquartile Range (IQR):
Ascending Order:
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
Measures of Dispersion for Population
3.0 3.1 3.4 3.4 3.5 3.6 3.6 3.7 3.7 3.9
IQR = Q3 – Q1
Q1 = .25(N+1)th = .25(10+1)th =2.75th
So, the first quartile is three quarters of the way from the 2nd
observation (3.1) to the 3rd (3.4).
Q1 = 3.1 + .75 (3.4 – 3.1) = 3.1+.225 = 3.325
Again, Q3 = .75(N+1)th = .75(10+1)th = 8.25th
So, the third quartile is one quarter of the way from the 8th observation (3.7) to the 9th (3.7).
Q3 = 3.7 + .25(3.7 – 3.7) = 3.7 + 0 = 3.7
IQR = Q3 – Q1 = 3.7 – 3.325 = .375
9. Population Variance:
σ2x =
= − (3.49)2
=
= .069
10. Population Standard Deviation:
σx= √σ2 = √ .069 = .262
11. Population Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD = 2.1210
= .212
Here,Q1 = First QuartileQ3 = Third Quartile
(3.6)² + (3.1)² + (3.9)² + (3.7)² + (3.5)² + (3.7)² + (3.4)²+ (3.0)² + (3.6)² + (3.4)²10
σx² = variance, µx = Population Mean = 3.49 [from (1)], N=10
σ = standard deviation, σ² = .069 [from (9)]
µ = Population Mean = 3.49 [from (1), N=10Xi Xi - µx = Xi - 3.49 (Xi - µx)
3.6 0.11 0.113.1 -0.39 0.393.9 0.41 0.413.7 0.21 0.213.5 0.01 0.013.7 0.21 0.213.4 -0.09 0.093.0 -0.49 0.493.6 0.11 0.113.4 -0.09 0.09
* Sums = 0 Sums=2.12
∑i=1
N
Xi2
N−μ
x2
122 .4910
−12 .18
∑i=1
N
( x i−μx )
N
12. Population Coefficient of Variation:
C. V. = σ xμx
× 100 = .2623.49
× 100 = 7.50%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge = Q1+Q 3
2=3.325+3.7
2 = 3.51
Measures of Central Tendency for Sample
Sample:
14. Sample Mean: The sample contains n=6, observations, so the Mean is
X=¿
= 3.6+3.0+3.7+3.4+3.9
5
= 17.65 = 3.52
15. Sample Median: Arranging n=6, observations in ascending order, we have
Ascending order:
Median =
Here, = Sample Meann = Number of observationXi = Observations
Q₁ = 3.325, Q₃ = 3.7 [from (8)]
3.6 3.0 3.7 3.4 3.9
3.0 3.4 3.6 3.7 3.9
C.V = coefficient of variation
σx = .262 [from (10)]
µx = 3.49 [from (1)]
∑i=1
n
X i
n
[n+1 ]th
2
¿[5+1 ]th
2=3rd
= 3.6
16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange = XS+XL2
= 3.0+3.92
=6.92 = 3.45
18. Sample Harmonic Mean:
H. M. =n
1a1
+1a2
±−−−−−−∓1
an
¿ 513.6
+13.0
+13.7
+13.4
+13.9
=5
.27+.33+.27+.29+ .25 = 3.54
19. Sample Geometric Mean:
G. M. = n√a1×a2×a3×−−−×an
= 5√3.6×3.0×3.7×3.4×3.9
= 3.50
Measures of Dispersion for Sample
20. Sample Range:
Range = XL – XS = 3.9 – 3.0 = 0.9
21. Sample Interquartile Range (IQR):
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(n+1)th = .25(5+1)th =1.5th
So, the first quartile is half quarter of the way from the 1st observation (3.0) to the 2nd (3.4).
Q1 = 3.0 + .5 (3.4 – 3.0) = 3.0+.215 = 3.215
Again, Q3 = .75(n+1)th = .75(5+1)th = 4.5th
So, the third quartile is half quarter of the way from the 4th observation (3.7) to the 5th (3.5).
Q3 = 3.7 + .5(3.9 – 3.7) = 3.7 + .1 = 3.8
IQR = Q3 – Q1 = 3.8 – 3.215 = 0.58522. Sample Variance:
Sx2 = ∑i=1
n
x i2−n x ²
n−1
=
= 62.42−61.95
4 = .468
4 = .117
23. Sample Standard Deviation: Sx = √S x2 = √ .117 = 0.34
24. Sample Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD = 1.285
= .256
Here,Q1 = First QuartileQ3 = Third Quartile
3.0 3.4 3.6 3.7 3.9
Sx² = variance, = sample mean = 3.52 [from (14)], n=5
(3.6)² + (3.0)² + (3.7)² + (3.4)²+ (3.9)² - 5(3.52)²5 - 1
Sx = standard deviation, Sx² = .117 [from (22)]
Xi Xi - ̅� = Xi - 3.52 (Xi - )3.6 0.08 0.083.0 -0.52 0.523.7 0.18 0.183.4 -0.12 0.123.9 0.38 0.38* Sum = 0 Sum=1.28
∑i=1
n
( x i−x )
n
25. Sample Coefficient of Variation:
C. V. = SxX
× 100 = 0.343.52
× 100 = 9.65%
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge = Q1+Q 3
2=3.215+3.8
2
= 3.50
Answer to the Question No – 4
Population:
Measures of Central Tendency for Population
1. Population Mean (Average): The population contains N=10 observations, so the Mean is
µx =
=
=
= 3.125
2. Population Median: Arranging N=8 observations in ascending order, we have
Ascending order:
Median =
=
=
Here,C.V. = coefficient of variationSx = 0.34 [from (23)] = 3.52 [from (14)]
σ₁ = 3.215, σ₃ = 3.8 [from (21)]
2 4 2 3 5 4 3 2
Here,µx = Population MeanN = Number of observationXi = Observations2 + 4 + 2 + 3 + 5 + 4 + 3 + 2
8
2 2 2 3 3 4 4 5
∑i=1
N
x i
N
258
[ N2 ]th
+[ N+22 ]
th
2
[ 82 ]th
+[ 8+22 ]th
2
4 th+5th
2=3+32
= 3
3. Population Mode:
The Mode of a set of observations is the value that occurs most frequently. So, Mode = 2
4. Population Midrange:
Midrange = XS+XL2
= 2+52 =7
2 = 3.5
5. Population Harmonic Mean:
H. M. =N
1a1
+1a2
±−−−−−−−−−−−−−−−−−∓1
aN
¿ 812+14+12+13+15+14+13+12
=8
.5+.25+.5+.33+.2+.25+.33+ .5 = 2.7
6. Population Geometric Mean:
G. M. = N√a1×a2×a3×−−−−−−−×aN
= 8√2×4×2×3×5×4×3×2
= 2.95
7. Population Range:
Range = XL – XS = 5 – 2 = 3
8. Population Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(N+1)th = .25(8+1)th =2.25th
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
Measures of Dispersion for Population
2 2 2 3 3 4 4 5
So, the first quartile is one quarter of the way from the 2nd observation (2) to the 3rd (2).
Q1 = 2 + .25 (2 – 2) = 2 + 0 = 2
Again, Q3 = .75(N+1)th = .75(8+1)th = 6.75th
So, the third quartile is three quarters of the way from the 6th observation (4) to the 7th (4).
Q3 = 4 + .75(4 – 4) = 4 + 0 = 4
IQR = Q3 – Q1 = 4 – 2 = 2
9. Population Variance:
σ2x =
= − (3.125)2
=
= 1.11
10. Population Standard Deviation:
σx= √σ2 = √1.11 = 1.05
11. Population Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD = 2.258
= .90625
σx² = variance, µx = Population Mean = 3.125 [from (1)], N=8
(2)²+ (4)²+ (2)²+ (3)²+ (5)²+ (4)²+ (3)²+ (2)²8
σ = standard deviation, σ² = 3.125 [from (9)]
µ = Population Mean = 3.125 [from (1), N=8
Xi Xi - µx = Xi - 3.125 (Xi - µx)2 -1.125 1.1254 0.875 0.8752 -1.125 1.1253 -0.125 0.1255 1.875 1.8754 0.875 0.8753 -0.125 0.1252 -1.125 1.125* Sums = 0 Sums=7.25
∑i=1
N
Xi2
N−μ
x2
878
−9 .765
∑i=1
N
( x i−μx )
N
12. Population Coefficient of Variation:
C. V. = σ xμx
× 100 = 1.053.125
× 100 = 33.6%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge = Q1+Q 3
2=2+4
2 = 3
Measures of Central Tendency for Sample
Sample:
14. Sample Mean: The sample contains n=4, observations, so the Mean is
X=¿
= 2+4+3+5
4
= 144 = 3.5
15. Sample Median: Arranging n=4, observations in ascending order, we have
Ascending order:
Median =
= 3.5
Here, = Sample Meann = Number of observationXi = Observations
C.V = coefficient of variation
σx = 1.05 [from (10)]
µx = 3.125 [from (1)]
Q₁ = 2, Q₃ = 4 [from (8)]
2 4 3 5
2 3 4 5
∑i=1
n
X i
n
[n2 ]th
+[n+22 ]th
2
¿[42 ]
th
+[4+22 ]th
2=2
nd+3rd
2=3+42
16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange = XS+XL2
= 2+52
=72 = 3.5
18. Sample Harmonic Mean:
H. M. =n
1a1
+1a2
±−−−−−−∓1
an
¿ 412+13+14+15
=4
.5+.33+.25+.2 = 3.125
19. Sample Geometric Mean:
G. M. = n√a1×a2×−−−×an
= 4√2×4×3×5
= 3.309
Measures of Dispersion for Sample
20. Sample Range:
Range = XL – XS = 5 – 2 = 3
21. Sample Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
2 3 4 5
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
Q1 = .25(n+1)th = .25(4+1)th =1.25th
So, the first quartile is three quarters of the way from the 1st observation (2) to the 2nd (3).
Q1 = 2 + .25(3 – 2) = 2 + .25 = 2.25
Again, Q3 = .75(n+1)th = .75(4+1)th = 3.75th
So, the third quartile is threequarters of the way from the 3rd observation (4) to the 4th (5).
Q3 = 4+ .75(5 – 4) = 4 + .75 = 4.75
IQR = Q3 – Q1 = 4.75 – 3.75 = 1
22. Sample Variance:
Sx2 = ∑i=1
n
x i2−n x ²
n−1
=
= 54−493
= 53
= 1.6666
23. Sample Standard Deviation: Sx = √S x2 = √1.6666 = 1.29
24. Sample Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD = 44
= 1
25. Sample Coefficient of Variation:
C. V. = SxX
× 100 = 1.293.5
× 100 = 36.85%
Sx² = variance, = sample mean = 3.5 [from (14)], n=4
(2)²+ (4)²+ (3)²+ (5)² - 4(3.5)²4 - 1
Sx = standard deviation, Sx² = 1.6666 [from (22)]
Here,C.V. = coefficient of variationSx = 1.29 [from (23)]
= 3.5 [from (14)]
Xi Xi - ̅� = Xi - 3.5 (Xi - )2 -1.5 1.53 0.5 0.54 -0.5 0.55 1.5 1.5
* Sum = 0 Sum=4
∑i=1
n
( x i−x )
n
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge = Q1+Q 3
2=2.25+4.75
2
= 3.5
Answer to the Question No. 5
Population:
Measures of Central Tendency for Population
1. Population Mean (Average):
The population contains N=10 observations, so the Mean is
µx =
=
=
= 35.2
2. Population Median: Arranging N=10 observations in ascending order, we have
Ascending order:
Median =
=
=
Q₁ = 2.25, Q₃ = 4.75 [from (21)]
Here,µx = Population MeanN = Number of observationXi = Observations
42 29 21 37 40 33 38 26 39 47
42 + 29 + 21 + 37 + 40 + 33 + 38 + 26 + 39 + 4710
21 26 29 33 37 38 39 40 42 47
∑i=1
N
x i
N
35210
[ N2 ]th
+[ N+22 ]
th
2
[102 ]th
+[10+22 ]th
25th+6 th
2=37+38
2
= 37.5
3. Population Mode:
The Mode of a set of observations is the value that occurs most frequently. So, there is no Mode.
4. Population Midrange:
Midrange = XS+XL2
= 21+472 =682 = 37.5
5. Population Harmonic Mean:
H. M. =N
1a1
+1a2
±−−−−−−−−−−−−−∓1
aN
¿ 10142
+129
+121
+137
+140
+133
+138
+126
+139
+147
=10
.02+ .03+.04+.02+.02+.03+.02+.03+ .02+.02 = 40
6. Population Geometric Mean:
G. M. = N√a1×a2×a3×−−−−−−−×aN
= 10√42×29×21×37×40×33×38×26×39×47
= 34.31
7. Population Range:
Range = XL – XS = 47 – 21 = 26
8. Population Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(N+1)th = .25(10+1)th =2.75th
So, the first quartile is three quarters of the way from the 2nd
observation (26) to the 3rd (29).
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
Measures of Dispersion for Population
21 26 29 33 37 38 39 40 42 47
Q1 = 26 + .75 (29 – 26) = 26+2.25 = 28.25
Again, Q3 = .75(N+1)th = .75(10+1)th = 8.25th
So, the third quartile is one quarter of the way from the 8th observation (40) to the 9th (42).
Q3 = 40 + .25(42 – 40) = 40 + 0.5 = 40.5
IQR = Q3 – Q1 = 40.5 – 28.5 = 12
9. Population Variance:
σ2x =
= − (35.2)2
=
= 56.36
10. Population Standard Deviation:
σx= √σ2 = √56.36 = 7.50
11. Population Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD = 63.610
= 6.36
σx² = variance, µx = Population Mean = 35.2 [from (1)], N=10
(42)² + (29)² + (21)² + (37)² + (40)² + (33)² + (38)² + (26)² + (39)² + (47)²10
σ = standard deviation, σ² = 56.36 [from (9)]
µ = Population Mean = 35.2 [from (1), N=10
Xi Xi - µx = Xi - 35.2 (Xi - µx)42 6.8 6.829 -6.2 6.221 -14.2 14.237 1.8 1.840 4.8 4.833 -2.2 2.238 2.8 2.826 -9.2 9.239 3.8 3.847 11.8 11.8
* Sums = 0 Sums=63.6
∑i=1
N
Xi2
N−μ
x2
1295410
−1239 .04
∑i=1
N
( x i−μx )
N
12. Population Coefficient of Variation:
C. V. = σ xμx
× 100 = 7.5035.2
× 100 = 21.30%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge = Q1+Q 3
2=28.5+40.5
2 = 34.5
Measures of Central Tendency for Sample
Sample:
14. Sample Mean: The sample contains n=5, observations, so the Mean is
X=¿
= 29+21+33+39+47
5
= 1695 = 33.8
15. Sample Median: Arranging n=5, observations in ascending order, we have
Ascending Order:
Median =
16. Sample Mode: There is no Mode.
Here, = Sample Meann = Number of observationXi = Observations
C.V = coefficient of variation
σx = 7.50 [from (10)]
µx = 35.2 [from (1)]
Q₁ = 28.5, Q₃ = 40.5 [from (21)]
29 21 33 39 47
21 29 33 39 47
∑i=1
n
X i
n
[n+1 ]th
2
¿[5+1 ]th
2=3rd=33
17. Sample Midrange:
Midrange =XS+XL2
= 21+472
=682 = 34
18. Sample Harmonic Mean:
H. M. =n
1a1
+1a2
±−−−−−−∓1
an
¿ 5129
+121
+133
+139
+147
=5
..03+.04+.03+ .02+.02 = 35.71
19. Sample Geometric Mean:
G. M. = n√a1×a2×a3×−−−×an
= 5√29×21×33×39×47
= 32.60
Measures of Dispersion for Sample
20. Sample Range:
Range = XL – XS = 47 – 21 = 26
21. Sample Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
21 29 33 39 47
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
Q1 = .25(n+1)th = .25(5+1)th =1.5th
So, the first quartile is half quarter of the way from the 1st observation (21) to the 2nd (29).
Q1 = 21 + .5(29 – 21) = 21 + 4 = 25
Again, Q3 = .75(n+1)th = .75(5+1)th = 4.5th
So, the third quartile is half quarter of the way from the 4th observation (39) to the 5th (47).
Q3 = 39 + .5(47 – 39) = 39 + 4 = 43
IQR = Q3 – Q1 = 43 – 25 = 18
22. Sample Variance:
Sx2 =∑i=1
n
x i2−n x ²
n−1
=
= 6101−5712.2
4 = 388.8
4 = 97.2
23. Sample Standard Deviation: Sx = √S x2 = √97.2 = 9.85
24. Sample Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD = 36.85
= 7.36
25. Sample Coefficient of Variation:
C. V. = SxX
× 100 = 9.8533.8
× 100 = 29.14%
Sx² = variance, = sample mean = 33.8 [from (14)], n=5
(29)²+ (21)²+ (33)²+ (39)²+ (47)² - 5(33.8)²5 - 1
Sx = standard deviation, Sx² = 97.2 [from (22)]
Xi Xi - ̅� = Xi - 33.8 (Xi - )29 -4.8 4.821 -12.8 12.833 -0.8 0.839 5.2 5.247 13.2 13.2
* Sums = 0 Sums=36.8
Here,C.V. = coefficient of variationSx = 9.85 [from (23)]
= 33.8 [from (14)]
∑i=1
n
( x i−x )
n
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge = Q1+Q 3
2=25+43
2
= 34
Answer to the Question No. 6
Population:
Measures of Central Tendency for Population
1. Population Mean (Average) : The population contains N=10 observations, so the Mean is
µx =
=
=
= 5.94
2. Population Median: Arranging N=10 observations in ascending order, we have
Ascending Order:
Median =
=
=
Q₁ = 25, Q₃ = 43 [from (21)]
Here,µx = Population MeanN = Number of observationXi = Observations
10.2 3.1 5.9 7.0 3.7 2.9 6.8 7.3 8.2 4.3
10.2 + 3.1 + 5.9 + 7.0 + 3.7 + 2.9 + 6.8 + 7.3 + 8.2 + 4.310
2.9 3.1 3.7 4.3 5.9 6.8 7.0 7.3 8.2 10.2
∑i=1
N
x i
N
59 .410
[ N2 ]th
+[ N+22 ]
th
2
[102 ]th
+[10+22 ]th
2
5th+6 th
2=5 .9+6 .8
2
= 6.35
3. Population Mode: The Mode of a set of observations is the value that occurs most frequently. So, there is no Mode.
4. Population Midrange:
Midrange = XS+XL2
= 2.9+10.22=13.12 = 6.55
5. Population Harmonic Mean:
H. M. =
N1a1
+1a2
±−−−−−−−−−−−−−∓1
aN
¿ 10110.2
+13.1
+15.9
+17.0
+13.7
+12.9
+16.8
+17.3
+18.2
+14.3
=10
.09+.32+.16+.14+ .27+.34+.14+.13+.12+.23 = 4.85
6. Population Geometric Mean: G. M. = N√a1×a2×a3×−−−−−−−×aN
= 10√10.2×3.1×5.9×7.0×3.7×2.9×6.8×7.3×8.2×4.3 = 5.48
7. Population Range:
Range = XL – XS = 10.2 – 2.9 = 7.3
8. Population Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
Measures of Dispersion for Population
2.9 3.1 3.7 4.3 5.9 6.8 7.0 7.3 8.2 10.2
Q1 = .25(N+1)th = .25(10+1)th =2.75th
So, the first quartile is three quarters of the way from the 2nd observation (3.1) to the 3rd (3.7).
Q1 = 3.1 + .75(3.7 – 3.1) = 3.1+.45 = 3.55
Again, Q3 = .75(N+1)th = .75(10+1)th = 8.25th
So, the third quartile is one quarter of the way from the 8th observation (7.3) to the 9th (8.2).
Q3 = 7.3 + .25(8.2 – 7.3) = 7.3 + .225 = 7.525
IQR = Q3 – Q1 = 7.525 – 3.55 = 3.975
9. Population Variance:
σ2x =
= − (5.94)2
= = 5.20
10. Population Standard Deviation: σx= √σ2 = √5.20 = 2.28
11. Population Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD = 19.610
= 1.96
σx² = variance, µx = Population Mean = 5.94 [from (1)], N=10
(10.2)² + (3.1)² + (5.9)² + (7.0)² + (3.7)² + (2.9)² + (6.8)²+ (7.3)² + (8.2)² + (4.3)²10
σ = standard deviation, σ² = 5.20 [from (9)]
µ = Population Mean = 5.94 [from (1), N=10
Xi Xi - µx = Xi - 5.94 (Xi - µx)10.2 4.26 4.263.1 -2.84 2.845.9 -0.04 0.047.0 1.06 1.063.7 -2.24 2.242.9 -3.04 3.046.8 0.86 0.867.3 1.36 1.368.2 2.26 2.264.3 -1.64 1.64
* Sums = 0 Sums 19.6
∑i=1
N
Xi2
N−μ
x2
404 .8210
−35 .28
∑i=1
N
( x i−μx )
N
12. Population Coefficient of Variation:
C. V. = σ xμx
× 100 = 2.285.94
× 100 = 38.38%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge = Q1+Q 3
2=3.55+7.525
2 = 5.5375
Measures of Central Tendency for Sample
Sample:
14. Sample Mean: The sample contains n=5, observations, so the Mean is
X=¿
= 3.1+5.9+7.0+4.3+8.2
5
= 28.55
= 5.7
15. Sample Median: Arranging n=5, observations in ascending order, we have
Ascending order:
Median = = 5.9
16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange =XS+XL2
Here,XS = Smallest observations
XL = Largest observations
Here, = Sample Meann = Number of observationXi = Observations
C.V = coefficient of variation
σx = 2.28 [from (10)]
µx = 5.94 [from (1)]
3.1 5.9 7.0 4.3 8.2
3.1 4.3 5.9 7.0 8.2
σ₁ = 3.215, σ₃ = 3.8 [from (8)]
∑i=1
n
X i
n
[n+1 ]th
2=
[5+1 ]th
2=3rd
= 3.1+8.22
=11.32 = 5.65
18. Sample Harmonic Mean:
H. M. =n
1a1
+1a2
±−−−−−−∓1
an
¿ 513.1
+15.9
+17.0
+14.3
+18.2
=6
.32+ .17+.14+.23+ .12 = 5.102
19. Sample Geometric Mean:
G. M. = n√a1×a2×a3×−−−×an
= 6√3.1×5.9×7.0×4.3×8.2 = 335.94
Measures of Dispersion for Sample
20. Sample Range:
Range = XL – XS = 8.2 – 3.1 = 5.1
21. Sample Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(n+1)th = .25(5+1)th =1.5th
So, the first quartile is three quarters of the way from the 1st observation (3.1) to the 2nd (4.3).
Q1 = 3.1 + .75 (4.3 – 3.1) = 3.1 + .90 = 4
Again, Q3 = .75(n+1)th = .75(5+1)th = 4.5th
So, the third quartile is one quarter of the way from the 4th observation (7.0) to the 5th (8.2).
Q3 = 7.0 + .25(8.2 – 7.0) = 7.0 + .30 = 7.3
IQR = Q3 – Q1 = 7.3 – 4 = 3.3
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
3.1 4.3 5.9 7.0 8.2
22. Sample Variance:
Sx2 =∑i=1
n
x i2−n x ²
n−1
=
= 16.74 = 4.175
23. Sample Standard Deviation: Sx = √S x2 = √4.175 = 2.04
24. Sample Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD = 85
= 1.6
25. Sample Coefficient of Variation:
C. V. = SxX
× 100 = 2.045.7
× 100 = 35.75%
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge = Q1+Q 3
2=4+7.3
2
= 5.65
Answer to the Question No. 7
Sx² = variance, = sample mean = 5.7 [from (14)], n=5
(3.1)²+ (4.3)²+ (5.9)²+ (7.0)²+ (8.2)² - 5(5.7)²5 - 1
Sx = standard deviation, Sx² = 4.175 [from (22)]
Xi Xi - ̅� = Xi - 5.7 (Xi - )3.1 -2.6 2.65.9 0.2 0.27.0 1.3 1.34.3 -1.4 1.48.2 2.5 2.5
* Sums = 0 Sums=8
Here,C.V. = coefficient of variationSx = 2.04 [from (23)]
= 5.7 [from (14)]
Q₁ = 4, Q₃ = 7.3 [from (21)]
∑i=1
n
( x i−x )
n
Population:
Measures of Central Tendency for Population
1.Population Mean (Average): The population contains N=12 observations, so the Mean is
µx =
=
=
= 19.90
2. Population Median: Arranging N=12 observations in ascending order, we have
Ascending Order:
Median =
=
=
= 17.55
3. Population Mode: The Mode of a set of observations is the value that occurs most frequently. So, there is no Mode.
4. Population Midrange:
Midrange = XS+XL2
= 7.3+34.72=422 = 21
5. Population Harmonic Mean:
Here,µx = Population MeanN = Number of observationXi = Observations
Here,XS = Smallest observations
XL = Largest observations
15.8 7.3 28.4 18.2 15.0 24.7 13.1 10.2 29.3 34.7 16.9 25.3
15.8 + 7.3 + 28.4 + 18.2 + 15.0 + 24.7 + 13.1 + 10.2 + 29.3 + 34.7 + 16.9 + 25.312
7.3 10.2 13.1 15.0 15.8 16.9 18.2 24.7 25.3 28.4 29.3 34.7
∑i=1
N
x i
N
238 .912
[ N2 ]th
+[ N+22 ]
th
2
[122 ]th
+[12+22 ]th
2
6th+7th
2=16 .9+18 .2
2
H. M. =N
1a1
+1a2
±−−−−−−−−−−−−−∓1
aN
¿ 12115.8
+17.3
+128.4
+118.2
+115
+124.7
+113.1
+110.2
+129.3
+134.7
+116.9
+125.3
=12
.06+.13+.03+.05+.06+.04+.07+.09+.03+.02+.05+.03 = 18.18
6. Population Geometric Mean: G. M. = N√a1×a2×a3×−−−−−−−×aN
= 12√15.8×7.3×28.4×18.2×15.0×24.7×13.1×10.2×29.3×34.7×16.9×25.3 = 18.15
7. Population Range:
Range = XL – XS = 34.7 – 7.3 = 27.4
8. Population Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(N+1)th = .25(12+1)th =3.25th
So, the first quartile is one quarter of the way from the 3rd observation (13.1) to the 4th (15.0).
Q1 = 13.1 + .25(15 – 13.1) = 13.1+.48 = 13.58
Again, Q3 = .75(N+1)th = .75(12+1)th = 9.75th
So, the third quartile is three quarters of the way from the 9th observation (25.3) to the 10th (28.4).
Q3 = 25.3 + .75(28.4 – 25.3) = 25.3 + 2.325 = 27.63
IQR = Q3 – Q1 = 27.625 – 13.5 = 14.13
9. Population Variance:
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
Measures of Dispersion for Population
7.3 10.2 13.1 15.0 15.8 16.9 18.2 24.7 25.3 28.4 29.3 34.7
∑i=1
N
Xi2
N−μ
x2
σ2x =
= − (19.90)2
= = 65.63
10. Population Standard Deviation: σx= √σ2 = √65.63 = 8.10
11. Population Coefficient of Variation:
C. V. = σ xμx
× 100 = 8.1019.90
× 100 = 40.70%
Measures of Central Tendency for Population
12. Population Midhinge:
Midhinge = Q1+Q 3
2=13.58+27.63
2 = 20.61
Measures of Central Tendency for Sample
Sample:
13. Sample Mean: The sample contains n=6, observations, so the Mean is
X=¿
= 15.8+7.3+24.7+29.3+34.7+25.3
6
= 1686
= 22.85
Here, = Sample Meann = Number of observationXi = Observations
(15.8)² + (7.3)² + (28.4)² + (18.2)² + (15.0)² + (24.7)² + (13.1)²+ (10.2)² + (29.3)² + (34.7)² (16.9)² + (25.3)²12
σx² = variance, µx = Population Mean = 19.90 [from (1)], N=12
σ = standard deviation, σ² = 65.63 [from (9)]
C.V = coefficient of variation
σx = 8.10 [from (10)]
µx = 19.90 [from (1)]
σ₁ = 13.58, σ₃ = 27.63 [from (8)]
15.8 7.3 24.7 29.3 34.7 25.3
5539 .7512
−396 .01
∑i=1
n
X i
n
14. Sample Median: Arranging n=6, observations in ascending order, we have
Ascending order:
Median =
=
= = 25
15. Sample Mode: There is no Mode.
16. Sample Midrange:
Midrange =XS+XL2
= 7.3+34.7
2=422 = 21
17. Sample Harmonic Mean:
H. M. =n
1a1
+1a2
±−−−−−−∓1
an
¿ 6115.8
+17.3
+124.7
+129.3
+134.7
+125.3
=6
.06+.13+.04+ .03+.02+.03 = 19.35
18. Sample Geometric Mean:
G. M. = n√a1×a2×a3×−−−×an
= 6√15.8×7.3×24.7×29.3×34.7×25.3 = 20.45
Here,XS = Smallest observations
XL = Largest observations
7.3 15.8 24.7 25.3 29.3 34.7
[ n2 ]th
+[ n+22 ]th
2
[ 62 ]th
+[ 6+22 ]th
2
3rd+4 th
2=24 .7+25 .3
2
Measures of Dispersion for Sample
19. Sample Range:
Range = XL – XS = 34.7 – 7.3 = 27.4
20. Sample Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(n+1)th = .25(6+1)th =1.75th
So, the first quartile is three quarters of the way from the 1st observation (7.3) to the 2nd (15 Q1 = 7.3 + .75 (15.8 – 7.3) = 7.3 + 6.375 = 13.675
Again, Q3 = .75(n+1)th = .75(6+1)th = 5.25th
So, the third quartile is one quarter of the way from the 5th observation (29.3) to the 6th (34.7).
Q3 = 29.3 + .25(34.7 – 29.3) = 29.3 + 1.35 = 30.65
IQR = Q3 – Q1 = 30.65 – 13.675 = 16.975
21. Sample Variance:
Sx2 =∑i=1
n
x i2−n x ²
n−1
=
= 3615.69−3132.7355=482.955
5 = 95.591
22. Sample Standard Deviation: Sx = √S x2 = √95.591 = 9.77
23. Sample Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
7.3 15.8 24.7 25.3 29.3 34.7
(15.8)²+ (7.3)²+ (24.7)²+ (29.3)²+ (34.7)² + (25.3)² - 6(22.85)²6 - 1
Sx² = variance, = sample mean = 22.85 [from (13)], n=6
Sx = standard deviation, Sx² = 95.591 [from (21)]
Xi Xi - ̅� = Xi - 22.85 (Xi - )15.8 -7.05 7.057.3 -15.55 15.55
24.7 1.85 1.8529.3 6.45 6.4534.7 11.85 11.8525.3 2.45 2.45* Sums = 0 Sums = 45.2
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
∑i=1
n
( x i−x )
n
∴ MAD = 45.26
= 7.53
24. Sample Coefficient of Variation:
C. V. = SxX
× 100 = 9.7722.85
× 100 = 42.75%
Measures of Central Tendency for Sample
25. Sample Midhinge: Midhinge = Q1+Q 3
2=13.675+30.65
2
= 22.16
Answer to the Question No. 16
Population:
Measures of Central Tendency for Population
1. Population Mean (Average):
The population contains N=12 observations, so the Mean is
µx =
=
=
= 9.83
Here,C.V. = coefficient of variationSx = 9.77 [from (22)]
= 22.85 [from (13)]
Q₁ = 13.675, Q₃ = 30.65 [from (20)]
Here,µx = Population MeanN = Number of observationXi = Observations
12 7 4 16 21 5 9 3 11 14 10 6
12 + 7 + 4 + 16 + 21 + 5 + 9 + 3 + 11 + 14 + 10 + 612
∑i=1
N
x i
N
11812
2. Population Median: Arranging N=12 observations in ascending order, we have
Ascending Order:
Median =
=
=
= 9.5
3. Population Mode: The Mode of a set of observations is the value that occurs most frequently. So, there is no Mode.
4. Population Midrange:
Midrange = XS+XL2
= 3+212 =242 = 12
5. Population Harmonic Mean:
H. M. =
N1a1
+1a2
±−−−−−−−−−−−−−∓1
aN
¿ 12112
+17+14+116
+121
+15+19+13+111
+114
+110
+16
=12
.08+.14+.25+ .06+.04+.2+.11+.13+.09+.07+.1+.16 = 7.40
6. Population Geometric Mean: G. M. = N√a1×a2×a3×−−−−−−−×aN
= 12√12×7×4×16×21×5×9×3×11×14×10×6 = 6.95
Here,XS = Smallest observations
XL = Largest observations
Measures of Dispersion for Population
3 4 5 6 7 9 10 11 12 14 16 21
[ N2 ]th
+[ N+22 ]
th
2[122 ]th
+[12+22 ]th
2
6th+7th
2=9+10
2
7. Population Range:
Range = XL – XS = 21 – 3 = 18
8. Population Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(N+1)th = .25(12+1)th =3.25th
So, the first quartile is one quarter of the way from the 3rd observation (5) to the 4th (6).
Q1 = 5 + .25(6 – 5) = 5 +.25 = 5.25
Again, Q3 = .75(N+1)th = .75(12+1)th = 9.75th
So, the third quartile is three quarters of the way from the 9th observation (12) to the 10th (14).
Q3 = 12 + .75(14 – 12) = 12 + 1.5 = 13.5
IQR = Q3 – Q1 = 13.5 – 5.25 = 8.25
9. Population Variance:
σ2x =
= −
(9.83)2
= = 26.21
10. Population Standard Deviation: σx= √σ2 = √26.21 = 5.11
11. Population Coefficient of Variation:
C. V. = σ xμx
× 100 = 5.119.83
× 100 = 51.98%
Here,XL = Largest observations
XS = Smallest observations
Here,Q1 = First QuartileQ3 = Third Quartile
3 4 5 6 7 9 10 11 12 14 16 21
σx² = variance, µx = Population Mean = 9.83 [from (1)], N=12
(12)² + (7)² + (4)² + (16)² + (21)² + (5)² + (9)²+ (3)² + (11)² + (14)² + (10)² + (6)²12
σ = standard deviation, σ² = 26.21 [from (9)]
C.V = coefficient of variation
σx = 5.11 [from (10)]
µx = 9.83 [from (1)]
∑i=1
N
Xi2
N−μ
x2
147412
−96 .62
Measures of Central Tendency for Population
12. Population Midhinge:
Midhinge = Q1+Q 3
2=5.25+13.5
2 = 9.375
Measures of Central Tendency for Sample
Sample:
13. Sample Mean: The sample contains n=6, observations, so the Mean is
X=¿
= 12+16+21+3+10+6
6
= 686
= 11.33
14. Sample Median: Arranging n=6, observations in ascending order, we have
Ascending order:
Median =
=
Here, = Sample Meann = Number of observationXi = Observations
Q₁ = 5.25, Q₃ = 13.5 [from (8)]
12 16 21 3 10 6
3 6 10 12 16 21
∑i=1
n
X i
n
[ n2 ]th
+[ n+22 ]th
2
[ 62 ]th
+[ 6+22 ]th
2
3rd+4 th
2=10+12
2
=
= 11
15. Sample Mode: There is no Mode.
16. Sample Midrange:
Midrange =XS+XL2
= 3+212
=242 = 12
17. Sample Harmonic Mean:
H. M. =n
1a1
+1a2
±−−−−−−∓1
an
¿ 6112
+116
+13+13+110
+16
=6
.08+.06+.04+ .33+.1+.16 = 8.10
18. Sample Geometric Mean:
G. M. = n√a1×a2×a3×−−−×an
= 6√12×16×21×3×10×6 = 9.47
Measures of Dispersion for Sample
19. Sample Range:
Range = XL – XS = 21 – 3 = 18
20. Sample Interquartile Range (IQR):
Ascending Order:
Here,XS = Smallest observations
XL = Largest observations
Here,XL = Largest observations
XS = Smallest observations
3 6 10 12 16 21
IQR = Q3 – Q1
Q1 = .25(n+1)th = .25(6+1)th =1.75th
So, the first quartile is three quarters of the way from the 1st observation (3) to the 2nd (6).
Q1 = 3 + .75 (6 – 3) = 3 + 2.25 = 5.25
Again, Q3 = .75(n+1)th = .75(6+1)th = 5.25th
So, the third quartile is one quarter of the way from the 5th observation (16) to the 6th (21).
Q3 = 16 + .25(21 – 16) = 16 + 1.25 = 17.25 IQR = Q3 – Q1 = 17.25 – 5.25 = 12
21. Sample Variance:
Sx2 =∑i=1
n
x i2−n x ²
n−1
=
= 986−770.215=215.79
5 = 43.15
22. Sample Standard Deviation: Sx = √S x2 = √43.15 = 6.56
23. Sample Mean Absolute Deviation (MAD):
MAD =
24. Sample Coefficient of Variation:
C. V. = SxX
× 100 = 6.5611.33
× 100 = 57.89%
Measures of Central Tendency for Sample
Here,Q1 = First QuartileQ3 = Third Quartile
Sx² = variance, = sample mean = 11.33 [from (13)], n=6
(12)² + (16)² + (21)² + (3)²+ (10)² + (6)² - 6(11.33)²6 - 1
Sx = standard deviation, Sx² = 43.15 [from (21)]
Here,C.V. = coefficient of variationSx = 6.56 [from (22)]
= 11.33 [from (13)]
Q₁ = 5.25, Q₃ = 17.25 [from (20)]
∑i=1
n
( x i−x )
n
25. Sample Midhinge: Midhinge = Q1+Q 3
2=13.675+30.65
2
= 22.16