United International University, MBA Faculty: Rashed Mohammad Saadullah Assistant Professor School of Business and Economics

1. Answer to the Question No. 2
8. Population Interquartile Range (IQR):
Ascending Order:
21 22 22 22 22 23 27 28 29 33 36 36
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:
σ2
x =
2
Xi
− (26.75)= 2
=
8941
= 29.52
10. Population Standard Deviation:
σx= √휎2 = √29.52 = 5.43
11. Population Mean Absolute Deviation (MAD):
MAD =

x
i x The calculation for MAD are set out in the table:
Here,
Q1 = First Quartile
Q3 = Third Quartile
1 2
x
N
i
N


 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)²
12
715.56
12

σ = standard deviation, σ² = 29.52 [from (9)]
N
N
i

1
(  )
μ = Population Mean = 26.75 [from (1)], N = 12

2. ∴ MAD =
57
12
= 4.75
Xi Xi - μx = Xi – 26.75 (Xi - μx)
21 -5.75 5.75
22 -4.75 4.75
27 0.25 0.25
36 9.25 9.25
22 -4.75 4.75
29 2.25 2.25
22 -4.75 4.75
23 -3.75 3.75
22 -4.75 4.75
28 1.25 1.25
36 9.25 9.25
33 6.25 6.25
* Sums = 0 Sums = 57
12. Population Coefficient of Variation:
C. V. =
σx
μx
× 100 =
5.43
26.75
× 100 = 20.29%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge =
푄1+ 푄3
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
퐗̅
=
=
X
i 1
n
n
i
21 + 27 + 36 + 22 + 29 + 33
6
=
168
6
= 28
15. Sample Median: Arranging n=6, observations in ascending order, we have
Ascending order:
21 22 27 29 33 36
Q₁ = 22, Q₃ = 32 [from (8)]
21 27 36 22 29 33
C.V = coefficient of variation
σx = 5.43 [from (10)]
μx = 26.75 [from (1)]

3. Median =
th th
n n
 
 
6

56
2

= 28
 
 

6 2

16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange =
XS+ XL
2
=
21+36
2
=
57
2
= 28.5
18. Sample Harmonic Mean:
H. M. =
푛
1
푎1
1
푎2
+
+−−−−−−−−+
1
푎푛
= 6
1
21+ 1
27+ 1
36+ 1
22+ 1
29+ 1
33
=
6
.04+.03+.02+.04+.03+.03
3 4
= 31.57
19. Sample Geometric Mean:
n
G. M. = √푎1 × 푎2 × 푎3 × − − − × 푎푛
= √21 × 27 × 36 × 22 × 29 × 33 6
= 27.47
27 29
Measures of Dispersion for Sample
20. Sample Range:
Range = XL – XS = 36 – 21 = 15
2
2
2
2
2
2
2
2
2














rd th
th th
Here,
XS = Smallest observations
XL = Largest observations
Here,
XL = Largest observations
XS = Smallest observations

4. 21. Sample Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
21 22 27 29 33 36
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:
2 =
Sx
2− 푛푥² 푛푖
=1
Σ 푥푖
푛−1
=
=
(21)² + (27)² + (36)² + (22)² + (29)² + (33)² - 6(28)²
4880−4704
5
=
176
5
= 35.2
2 = √35.2 = 5.93
23. Sample Standard Deviation: Sx = √Sx
24. Sample Mean Absolute Deviation (MAD):
MAD =
The calculation for MAD are set out in the table:
∴ MAD =
28
6
= 4.66
Here,
Q1 = First Quartile
Q3 = Third Quartile
6-1
i 
x x
n
n
i

1
( )

5. 25. Sample Coefficient of Variation:
C. V. =
푆푥
푋
× 100 =
5.93
28
× 100 = 21.17%
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge =
푄1+ 푄3
2
=
21.75 + 33.75
2
= 27.75
Answer to the Question No. 3
Population:
3.6 3.1 3.9 3.7 3.5 3.7 3.4 3.0 3.6 3.4
Measures of Central Tendency for Population
1. Population Mean (Average): The population contains N=10 observations, so the Mean is
μx =
=
=

x
i 1
N
9. 34
= 3.49
2. Population Median: Arranging N=10 observations in ascending order, we have
Ascending order:
Median =
=
=
Q₁ = 21.5, Q₃ = 33.75 [from (21)]
N
i
10
 
 
2
2
2
2
th th
N N






 
 
10 2
2
2
10
2
th th






5 6 
3.5 3.6
2
2

 th th
Here,
μx = Population Mean
N = Number of observation
Xi = Observations
3.6 + 3.1 + 3.9 + 3.7 + 3.5 + 3.7 + 3.4 + 3.0 + 3.6 + 3.4
10
3.0 3.1 3.4 3.4 3.5 3.6 3.6 3.7 3.7 3.9

6. = 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+ XL
2
=
3.0+3.9
2
=
6.9
2
= 3.45
5. Population Harmonic Mean:
H. M. =
N
1
푎1
+ 1
푎2
+−−−−−−−−−−−−−−−−−−−+ 1
푎푁
=
10
1
3.6
+
1
3.1
1
3.9
+
+
1
3.7
+
1
3.5
+
1
3.7
1
3.4
+
+
1
3.0
+
1
3.6
1
3.4
+
=
10
.27+.32+.25+.27+.28+.27+.29+.33+.27+.29
= 3.52
6. Population Geometric Mean:
푁
G. M. = √푎1 × 푎2 × 푎3 × − − − − − − − × 푎푁
= √3.6 × 3.1 × 3.9 × 3.7 × 3.5 × 3.7 × 3.4 × 3.0 × 3.6 × 3.4 10
= 3.47
7. Population Range:
Range = XL – XS = 3.9 – 3.0 = 0.9
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 Quartile
Q3 = Third Quartile
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

7. 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:
σ2
x =
2
Xi
σx² = variance, μx = Population Mean = 3.49 [from (1)], N=10
(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)²
− = (3.49)2
=
122.49
= .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 =
2.12
10
= .212
1 2
x
N
i
N



12.18
10


x
i x N
N
i

1
(  )
10
σ = standard deviation, σ² = .069 [from (9)]
μ = Population Mean = 3.49 [from (1), N=10
Xi Xi - μx = Xi - 3.49 (Xi - μx)
3.6 0.11 0.11
3.1 -0.39 0.39
3.9 0.41 0.41
3.7 0.21 0.21
3.5 0.01 0.01
3.7 0.21 0.21
3.4 -0.09 0.09
3.0 -0.49 0.49
3.6 0.11 0.11
3.4 -0.09 0.09
* Sums = 0 Sums=2.12

8. 12. Population Coefficient of Variation:
C. V. =
σx
μx
× 100 =
.262
3.49
× 100 = 7.50%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge =
푄1+ 푄3
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
퐗̅
=
=

X
i 1
n
3.6 + 3.0 + 3.7 + 3.4 + 3.9
5
=
17.6
5
= 3.52
15. Sample Median: Arranging n=6, observations in ascending order, we have
Ascending order:
Median =
n
i
 1

  th
rd
th n
3
5 1
2
2




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)]

9. = 3.6
16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange =
XS+ XL
2
=
3.0+3.9
2
=
6.9
2
= 3.45
18. Sample Harmonic Mean:
H. M. =
푛
1
푎1
1
푎2
+
+−−−−−−−−+
1
푎푛
=
5
1
3.6
+
1
3.0
1
3.7
+
+
1
3.4
+
1
3.9
=
5
.27+.33+.27+.29+.25
= 3.54
19. Sample Geometric Mean:
푛
G. M. = √푎1 × 푎2 × 푎3 × − − − × 푎푛
= √3.6 × 3.0 × 3.7 × 3.4 × 3.9 5
= 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

10. 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 = 0.585
22. Sample Variance:
2 =
Sx
2− 푛푥² 푛푖
=1
Σ 푥푖
푛−1
=
=
62.42−61.95
4 =
.468
4
= .117
2 = √. 117 = 0.34
23. Sample Standard Deviation: Sx = √Sx
24. Sample Mean Absolute Deviation (MAD):
MAD =
i 
x x
The calculation for MAD are set out in the table:
∴ MAD =
1.28
5
= .256
Here,
Q1 = First Quartile
Q3 = Third Quartile
n
n
i

1
( )
3.0 3.4 3.6 3.7 3.9
(3.6)² + (3.0)² + (3.7)² + (3.4)²+ (3.9)² - 5(3.52)²
5 - 1
Sx = standard deviation, Sx² = .117 [from (22)]

11. 25. Sample Coefficient of Variation:
C. V. =
푆푥
푋
× 100 =
0.34
3.52
× 100 = 9.65%
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge =
푄1+ 푄3
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 =
=
=

x
i 1
N
25
= 3.125
2. Population Median: Arranging N=8 observations in ascending order, we have
Ascending order:
Median =
=
σ₁ = 3.215, σ₃ = 3.8 [from (21)]
Here,
Q1 = First Quartile
Q3 = Third Quartile
2 4 2 3 5 4 3 2
N
i
8
 
 
2
2
2
2
th th
N N




 
 
 
8 2
2
2
8
2
th th






Here,
μx = Population Mean
N = Number of observation
Xi = Observations
2 + 4 + 2 + 3 + 5 + 4 + 3 + 2
8
2 2 2 3 3 4 4 5

12. =
th th
5 4 
= 3
2

3. Population Mode:
3 3
2
The Mode of a set of observations is the value that occurs most frequently. So, Mode = 2
4. Population Midrange:
Midrange =
XS+ XL
2
=
2+5
2
=
7
2
= 3.5
5. Population Harmonic Mean:
H. M. =
N
1
푎1
+
1
푎2
+−−−−−−−−−−−−−−−−−−−+
1
푎푁
= 8
1
2+1
4+1
2+1
3+1
5+1
4+1
3+1
2
=
8
.5+.25+.5+.33+.2+.25+.33+.5
= 2.7
6. Population Geometric Mean:
N
G. M. = √푎1 × 푎2 × 푎3 × − − − − − − − × 푎푁
= √2 × 4 × 2 × 3 × 5 × 4 × 3 × 2 8
= 2.95
7. Population Range:
Range = XL – XS = 5 – 2 = 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 Quartile
Q3 = Third Quartile
Measures of Dispersion for Population
2 2 2 3 3 4 4 5

13. Q1 = .25(N+1)th = .25(8+1)th =2.25th
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:
σ2
x =
2
Xi
− (3.12 = 5 ) 2
=
87
= 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:
1 2
x
N
i
N



9.765
8


x
i x N
N
i

1
(  )
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)²
8
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 1.125
4 0.875 0.875
2 -1.125 1.125
3 -0.125 0.125
5 1.875 1.875
4 0.875 0.875
3 -0.125 0.125
2 -1.125 1.125
* Sums = 0 Sums=7.25

14. ∴ MAD =
2.25
8
= .90625
12. Population Coefficient of Variation:
C. V. =
σx
μx
× 100 =
1.05
3.125
× 100 = 33.6%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge =
푄1+ 푄3
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
X̅
=
=
X
i 1
n
n
i
2 + 4 + 3 + 5
4
=
14
4
= 3.5
15. Sample Median: Arranging n=4, observations in ascending order, we have
th th

n n
 
 

Ascending order:
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
3 4
2
2 3
2
 
2
4 2
2
4
2
2
2
2
2








 








nd rd
th th

15. Median =
= 3.5
16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange =
XS+ XL
2
=
2+5
2
=
7
2
= 3.5
18. Sample Harmonic Mean:
H. M. =
푛
1
푎1
1
푎2
+
+−−−−−−−−+
1
푎푛
=
4
1
2
+
1
3
1
4
+
+
1
5
=
4
.5+.33+.25+.2
= 3.125
19. Sample Geometric Mean:
푛
G. M. = √푎1 × 푎2 × − − − × 푎푛
= √2 × 4 × 3 × 5 4
= 3.309
Measures of Dispersion for Sample
20. Sample Range:
Range = XL – XS = 5 – 2 = 3
Here,
XS = Smallest observations
XL = Largest observations
Here,
XL = Largest observations
XS = Smallest observations

16. 21. Sample Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
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:
2 =
Sx
2− 푛푥² 푛푖
=1
Σ 푥푖
푛−1
=
=
54−49
3 =
5
3
= 1.6666
2 = √1.6666 = 1.29
23. Sample Standard Deviation: Sx = √Sx
24. Sample Mean Absolute Deviation (MAD):
MAD =
i 
x x
The calculation for MAD are set out in the table:
∴ MAD =
4
4
= 1
Here,
Q1 = First Quartile
Q3 = Third Quartile
n
n
i


1
( )
2 3 4 5
(2)²+ (4)²+ (3)²+ (5)² - 4(3.5)²
4 - 1
Sx = standard deviation, Sx² = 1.6666 [from (22)]

17. 25. Sample Coefficient of Variation:
C. V. =
푆푥
푋
× 100 =
1.29
3.5
× 100 = 36.85%
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge =
푄1+ 푄3
2
=
2.25 + 4.75
2
= 3.5
Answer to the Question No. 5
Population:
42 29 21 37 40 33 38 26 39 47
Measures of Central Tendency for Population
1. Population Mean (Average):
The population contains N=10 observations, so the Mean is
μx =
=
=

x
i 1
N
N
i
42 + 29 + 21 + 37 + 40 + 33 + 38 + 26 + 39 + 47
352
10
= 35.2
Q₁ = 2.25, Q₃ = 4.75 [from (21)]
Here,
μx = Population Mean
N = Number of observation
Xi = Observations
10
2. Population Median: Arranging N=10 observations in ascending order, we have
21 26 29 33 37 38 39 40 42 47

18. Ascending order:
Median =
=
=
N N
2
th th


 
 



10
2
th th


 
 



 th th
5 6 
2
= 37.5


2
10 2


2

3. Population Mode:
2
2



2



37 38
2
The Mode of a set of observations is the value that occurs most frequently. So, there is no Mode.
4. Population Midrange:
Midrange =
XS+ XL
2
=
21+47
2
=
68
2
= 37.5
5. Population Harmonic Mean:
H. M. =
푁
1
푎1
+
1
푎2
+−−−−−−−−−−−−−−−+
1
푎푁
= 10
1
42+ 1
29+ 1
21+ 1
37+ 1
40+ 1
33+ 1
38+ 1
26+ 1
39+ 1
47
=
10
.02+.03+.04+.02+.02+.03+.02+.03+.02+.02
= 40
6. Population Geometric Mean:
퐍
G. M. = √푎1 × 푎2 × 푎3 × − − − − − − − × 푎푁
= √42 × 29 × 21 × 37 × 40 × 33 × 38 × 26 × 39 × 47 ퟏퟎ
= 34.31
7. Population Range:
Measures of Dispersion for Population
Range = XL – XS = 47 – 21 = 26
Here,
XS = Smallest observations
XL = Largest observations
Here,
XL = Largest observations
XS = Smallest observations

19. 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).
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:
σ2
x =
2
Xi
− = (35.2)2
=
12954
= 56.36
10. Population Standard Deviation:
σx= √휎2 = √56.36 = 7.50
11. Population Mean Absolute Deviation (MAD):
MAD =

x
i x The calculation for MAD are set out in the table:
Here,
Q1 = First Quartile
Q3 = Third Quartile
1 2
x
N
i
N



1239.04
10

N
N
i

1
(  )
21 26 29 33 37 38 39 40 42 47
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)²
10
σ = standard deviation, σ² = 56.36 [from (9)]
μ = Population Mean = 35.2 [from (1), N=10

20. ∴ MAD =
63.6
10
= 6.36
Xi Xi - μx = Xi - 35.2 (Xi - μx)
42 6.8 6.8
29 -6.2 6.2
21 -14.2 14.2
37 1.8 1.8
40 4.8 4.8
33 -2.2 2.2
38 2.8 2.8
26 -9.2 9.2
39 3.8 3.8
47 11.8 11.8
* Sums = 0 Sums=63.6
12. Population Coefficient of Variation:
C. V. =
σx
μx
× 100 =
7.50
35.2
× 100 = 21.30%
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge =
푄1+ 푄3
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
퐗̅
=
=
X
i 
1
29 + 21 + 33 + 39 + 47
5
=
169
5
= 33.8
n
n
i
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. 15. Sample Median: Arranging n=5, observations in ascending order, we have
Ascending Order:
Median =
21 29 33 39 47
1
5 1
16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange =
XS+ XL
2
=
21+47
2
=
68
2
= 34
18. Sample Harmonic Mean:
H. M. =
푛
1
푎1
+
1
푎2
+−−−−−−−−+
1
푎푛
= 5
1
29+ 1
21+ 1
33+ 1
39+ 1
47
=
5
..03+.04+.03+.02+.02
= 35.71
19. Sample Geometric Mean:
퐧
G. M. = √푎1 × 푎2 × 푎3 × − − − × 푎푛
= √29 × 21 × 33 × 39 × 47 5
= 32.60
Here,
XS = Smallest observations
XL = Largest observations
Measures of Dispersion for Sample
 
 
3 33
2
2
 



rd
th
th n

22. 20. Sample Range:
Range = XL – XS = 47 – 21 = 26
21. Sample Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
Q1 = .25(n+1)th = .25(5+1)th =1.5th
Here,
XL = Largest observations
XS = Smallest observations
Here,
Q1 = First Quartile
Q3 = Third Quartile
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:
2 =
Sx
2− 푛푥² 푛푖
=1
Σ 푥푖
푛−1
=
=
6101−5712.2
4
=
388.8
4
= 97.2
2 = √97.2 = 9.85
23. Sample Standard Deviation: Sx = √푆푥
24. Sample Mean Absolute Deviation (MAD):
i 
x x
MAD =
The calculation for MAD are set out in the table:
n
n
i

1
( )
21 29 33 39 47
(29)²+ (21)²+ (33)²+ (39)²+ (47)² - 5(33.8)²
5 - 1
Sx = standard deviation, Sx² = 97.2 [from (22)]

23. ∴ MAD =
36.8
5
= 7.36
25. Sample Coefficient of Variation:
C. V. =
푆푥
푋
× 100 =
9.85
33.8
× 100 = 29.14%
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge =
푄1+ 푄3
2
=
25 + 43
2
= 34
Answer to the Question No. 6
Population:
10.2 3.1 5.9 7.0 3.7 2.9 6.8 7.3 8.2 4.3
Measures of Central Tendency for Population
1. Population Mean (Average):
The population contains N=10 observations, so the Mean is
μx =
=
=
x
i 1
N
59.4
= 5.94
Q₁ = 25, Q₃ = 43 [from (21)]
N
i
10
Here,
μx = Population Mean
N = Number of observation
Xi = Observations
10.2 + 3.1 + 5.9 + 7.0 + 3.7 + 2.9 + 6.8 + 7.3 + 8.2 + 4.3
10

24. 2. Population Median: Arranging N=10 observations in ascending order, we have
Ascending Order:
Median =
=
=
2.9 3.1 3.7 4.3 5.9 6.8 7.0 7.3 8.2 10.2


 
 
10


 
 
5 6 
= 6.35

10 2

5.9 6.8
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+ XL
2
=
2.9+10.2
2
=
13.1
2
= 6.55
5. Population Harmonic Mean:
H. M. =
N
1
푎1
+
1
푎2
+−−−−−−−−−−−−−−−+
1
푎푁
=
10
1
10.2
+
1
3.1
+
1
5.9
1
7.0
+
+
1
3.7
+
1
2.9
+
1
6.8
1
7.3
+
+
1
8.2
+
1
4.3
=
10
.09+.32+.16+.14+.27+.34+.14+.13+.12+.23
= 4.85
6. Population Geometric Mean:
퐍
G. M. = √푎1 × 푎2 × 푎3 × − − − − − − − × 푎푁
= √10.2 × 3.1 × 5.9 × 7.0 × 3.7 × 2.9 × 6.8 × 7.3 × 8.2 × 4.3 ퟏퟎ
= 5.48
2
2
2
2
th th
N N







2
2
2
th th







2
2

 th th
Here,
XS = Smallest observations
XL = Largest observations
Measures of Dispersion for Population

25. 7. Population Range:
Range = XL – XS = 10.2 – 2.9 = 7.3
8. Population Interquartile Range (IQR):
Ascending Order:
IQR = Q3 – Q1
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:
σ2
x =
2
Xi
− (5.94)= 2
=
404.82
= 5.20
10. Population Standard Deviation: σx= √σ2 = √5.20 = 2.28
11. Population Mean Absolute Deviation (MAD):
MAD =

x
i x The calculation for MAD are set out in the table:
Here,
XL = Largest observations
XS = Smallest observations
Here,
Q1 = First Quartile
Q3 = Third Quartile
1 2
x
N
i
N



35.28
10

N
N
i

1
(  )
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)²
10
σ = standard deviation, σ² = 5.20 [from (9)]
μ = Population Mean = 5.94 [from (1), N=10

26. ∴ MAD =
19.6
10
= 1.96
Xi Xi - μx = Xi - 5.94 (Xi - μx)
10.2 4.26 4.26
3.1 -2.84 2.84
5.9 -0.04 0.04
7.0 1.06 1.06
3.7 -2.24 2.24
2.9 -3.04 3.04
6.8 0.86 0.86
7.3 1.36 1.36
8.2 2.26 2.26
4.3 -1.64 1.64
* Sums = 0 Sums 19.6
12. Population Coefficient of Variation:
C. V. =
σx
μx
× 100 =
2.28
5.94
× 100 = 38.38%
C.V = coefficient of variation
σx = 2.28 [from (10)]
μx = 5.94 [from (1)]
Measures of Central Tendency for Population
13. Population Midhinge:
Midhinge =
푄1+ 푄3
2
=
3.55 + 7.525
2
= 5.5375
σ₁ = 3.215, σ₃ = 3.8 [from (8)]
Measures of Central Tendency for Sample
Sample:
3.1 5.9 7.0 4.3 8.2
14. Sample Mean: The sample contains n=5, observations, so the Mean is
퐗̅
=
=
X
i 1
n
n
i
3.1 + 5.9 + 7.0 + 4.3 + 8.2
5
28.5
5
= 5.7
=
Here,
Q1 = First Quartile
Q3 = Third Quartile
15. Sample Median: Arranging n=5, observations in ascending order, we have

27. Ascending order:
 n
1
 th  5 1
 th rd
Median = = 5.9
16. Sample Mode: There is no Mode.
17. Sample Midrange:
Midrange =
XS+ XL
2
=
3.1+8.2
2
=
11.3
2
= 5.65
18. Sample Harmonic Mean:
H. M. =
푛
1
푎1
1
푎2
+
+−−−−−−−−+
1
푎푛
=
5
1
3.1
+
1
5.9
1
7.0
+
+
1
4.3
+
1
8.2
=
6
.32+.17+.14+.23+.12
= 5.102
19. Sample Geometric Mean:
푛
G. M. = √푎1 × 푎2 × 푎3 × − − − × 푎푛
= √3.1 × 5.9 × 7.0 × 4.3 × 8.2 ퟔ
= 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
3
2
2




Here,
XS = Smallest observations
XL = Largest observations
Here,
XL = Largest observations
XS = Smallest observations
Here,
Q1 = First Quartile
Q3 = Third Quartile
3.1 4.3 5.9 7.0 8.2
3.1 4.3 5.9 7.0 8.2

28. 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
22. Sample Variance:
Sx
Σ 푥푖
2 =
2− 푛푥² 푛푖
=1
푛−1
=
=
(3.1)²+ (4.3)²+ (5.9)²+ (7.0)²+ (8.2)² - 5(5.7)²
16.7
4
= 4.175
2 = √4.175 = 2.04
5 - 1
23. Sample Standard Deviation: Sx = √Sx
24. Sample Mean Absolute Deviation (MAD):
MAD =
i 
x x
( )
n
n
i

1
The calculation for MAD are set out in the table:
∴ MAD =
8
5
= 1.6
25. Sample Coefficient of Variation:
Sx = standard deviation, Sx² = 4.175 [from (22)]

29. C. V. =
Sx
X
× 100 =
2.04
5.7
× 100 = 35.75%
Measures of Central Tendency for Sample
26. Sample Midhinge: Midhinge =
푄1+ 푄3
2
=
4 + 7.3
2
= 5.65
Answer to the Question No. 7
Population:
15.8 7.3 28.4 18.2 15.0 24.7 13.1 10.2 29.3 34.7 16.9 25.3
Measures of Central Tendency for Population
1.Population Mean (Average):
The population contains N=12 observations, so the Mean is
μx =
=
=

x
i 1
N
238.9
= 19.90
2. Population Median: Arranging N=12 observations in ascending order, we have
Ascending Order:
Median =
=
=
Q₁ = 4, Q₃ = 7.3 [from (21)]
N
i
12
 
 
2
2
2
2
th th
N N






 
 
12 2
2
2
12
2
th th







6 7 
16.9 18.2
2
2

 th th
Here,
μx = Population Mean
N = Number of observation
Xi = 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
12
7.3 10.2 13.1 15.0 15.8 16.9 18.2 24.7 25.3 28.4 29.3 34.7

30. = 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+ XL
2
=
7.3+34.7
2
=
42
2
= 21
5. Population Harmonic Mean:
H. M. =
푁
1
푎1
+
1
푎2
+−−−−−−−−−−−−−−−+
1
푎푁
=
12
1
15.8
+
1
7.3
+
1
28.4
+
1
18.2
+
1
15
+
1
24.7
+
1
13.1
1
10.2
+
1
29.3
+
1
34.7
+
+
1
16.9
+
1
25.3
=
12
.06+.13+.03+.05+.06+.04+.07+.09+.03+.02+.05+.03
= 18.18
6. Population Geometric Mean:
퐍
G. M. = √푎1 × 푎2 × 푎3 × − − − − − − − × 푎푁
= √15.8 × 7.3 × 28.4 × 18.2 × 15.0 × 24.7 × 13.1 × 10.2 × 29.3 × 34.7 × 16.9 × 25.3 ퟏퟐ
= 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
Here,
XS = Smallest observations
XL = Largest observations
Here,
XL = Largest observations
XS = Smallest observations
Here,
Q1 = First Quartile
Q3 = 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

31. 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:
σ2
x =
2
Xi
σx² = variance, μx = Population Mean = 19.90 [from (1)], N=12
(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)²
− (19.90)= 2
=
5539.75
= 65.63
10. Population Standard Deviation: σx= √σ2 = √65.63 = 8.10
11. Population Coefficient of Variation:
C. V. =
σx
μx
× 100 =
8.10
19.90
× 100 = 40.70%
Measures of Central Tendency for Population
12. Population Midhinge:
Midhinge =
푄1+ 푄3
2
=
13.58 + 27.63
2
= 20.61
Measures of Central Tendency for Sample
1 2
x
N
i
N
 


396.01
12

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)]
Here,
Q1 = First Quartile
Q3 = Third Quartile
15.8 7.3 24.7 29.3 34.7 25.3

32. Sample:
13. Sample Mean: The sample contains n=6, observations, so the Mean is
퐗̅
=
=

X
i 1
15.8 + 7.3 + 24.7 + 29.3 + 34.7+25.3
6
168
=
6
= 22.85
14. Sample Median: Arranging n=6, observations in ascending order, we have
Ascending order:
Median =
=



 
 
 
 
6 2
6



3 4 
24.7 25.3
= = 25
15. Sample Mode: There is no Mode.
16. Sample Midrange:
Midrange =
XS+ XL
2
=
7.3+34.7
2
=
42
2
= 21
17. Sample Harmonic Mean:
H. M. =
푛
1
푎1
1
푎2
+
+−−−−−−−−+
1
푎푛
=
6
1
15.8
+
1
7.3
+
1
24.7
+
1
29.3
+
1
34.7
+
1
25.3
n
n
i
Here,
XS = Smallest observations
XL = Largest observations
7.3 15.8 24.7 25.3 29.3 34.7
2
2
2
2
th th
n n







2
2
2
th th







2
2

 rd th

33. =
6
.06+.13+.04+.03+.02+.03
= 19.35
18. Sample Geometric Mean:
푛
G. M. = √푎1 × 푎2 × 푎3 × − − − × 푎푛
= √15.8 × 7.3 × 24.7 × 29.3 × 34.7 × 25.3 ퟔ
= 20.45
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:
2 =
Sx
2− 푛푥² 푛푖
=1
Σ 푥푖
푛−1
=
Here,
XL = Largest observations
XS = Smallest observations
Here,
Q1 = First Quartile
Q3 = Third Quartile
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)²
6 - 1

34. =
3615.69−3132.735
5
=
482.955
5
= 95.591
2 = √95.591 = 9.77
22. Sample Standard Deviation: Sx = √Sx
23. Sample Mean Absolute Deviation (MAD):
MAD =
i 
x x
The calculation for MAD are set out in the table:
∴ MAD =
45.2
6
= 7.53
24. Sample Coefficient of Variation:
C. V. =
Sx
X
× 100 =
9.77
22.85
× 100 = 42.75%
Measures of Central Tendency for Sample
25. Sample Midhinge: Midhinge =
푄1+ 푄3
2
=
13.675 + 30.65
2
= 22.16
Answer to the Question No. 16
Population:
Measures of Central Tendency for Population
n
n
i

1
( )
Sx = standard deviation, Sx² = 95.591 [from (21)]
Q₁ = 13.675, Q₃ = 30.65 [from (20)]
12 7 4 16 21 5 9 3 11 14 10 6

35. 1. Population Mean (Average):
The population contains N=12 observations, so the Mean is
μx =
=
=

x
i 1
N
12 + 7 + 4 + 16 + 21 + 5 + 9 + 3 + 11 + 14 + 10 + 6
118
= 9.83
2. Population Median: Arranging N=12 observations in ascending order, we have
Ascending Order:
Median =
=
=

12

6 7 
= 9.5



 
  
 
12 2


9 10
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+ XL
2
=
3+21
2
=
24
2
= 12
5. Population Harmonic Mean:
H. M. =
푁
1
푎1
+
1
푎2
+−−−−−−−−−−−−−−−+
1
푎푁
N
i
12
2
2
2
2
th th
N N







2
2
2
th th







2
2

 th th
Here,
μx = Population Mean
N = Number of observation
Xi = Observations
Here,
XS = Smallest observations
XL = Largest observations
12
3 4 5 6 7 9 10 11 12 14 16 21

36. =
12
1
12
+
1
7
+
1
4
1
16
+
1
21
+
+
1
5
1
9
+
+
1
3
+
1
11
+
1
14
+
1
10
+
1
6
=
12
.08+.14+.25+.06+.04+.2+.11+.13+.09+.07+.1+.16
= 7.40
6. Population Geometric Mean:
퐍
G. M. = √푎1 × 푎2 × 푎3 × − − − − − − − × 푎푁
= √12 × 7 × 4 × 16 × 21 × 5 × 9 × 3 × 11 × 14 × 10 × 6 ퟏퟐ
= 6.95
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:
σ2
x =
Here,
XL = Largest observations
XS = Smallest observations
Here,
Q1 = First Quartile
Q3 = Third Quartile
2
Xi
1 2
x
N
i
N


Measures of Dispersion for Population
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)²
12

37. = − (9.83)2
=
147 4
= 26.21
10. Population Standard Deviation: σx= √σ2 = √26.21 = 5.11
11. Population Coefficient of Variation:
C. V. =
σx
μx
× 100 =
5.11
9.83
× 100 = 51.98%
Measures of Central Tendency for Population
12. Population Midhinge:
Midhinge =
푄1+ 푄3
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
퐗̅
=
=
X
i 1
12 + 16 + 21 + 3 + 10+ 6
6
=
68
6
62. 96
12

n
n
i
σ = standard deviation, σ² = 26.21 [from (9)]
C.V = coefficient of variation
σx = 5.11 [from (10)]
μx = 9.83 [from (1)]
Q₁ = 5.25, Q₃ = 13.5 [from (8)]
12 16 21 3 10 6

38. = 11.33
14. Sample Median: Arranging n=6, observations in ascending order, we have
Ascending order:
Median =
=
=

6

3 6 10 12 16 21
3 4 
= 11

 
 

 
 
2 6


10 12
15. Sample Mode: There is no Mode.
16. Sample Midrange:
Midrange =
XS+ XL
2
=
3+21
2
=
24
2
= 12
17. Sample Harmonic Mean:
H. M. =
푛
1
푎1
1
푎2
+
+−−−−−−−−+
1
푎푛
=
6
1
12
+
1
16
+
1
3
+
1
3
1
10
+
+
1
6
=
6
.08+.06+.04+.33+.1+.16
= 8.10
18. Sample Geometric Mean:
Here,
XS = Smallest observations
XL = Largest observations
2
2
2
2
th th
n n







2
2
2
th th







2
2

 rd th

39. 푛
G. M. = √푎1 × 푎2 × 푎3 × − − − × 푎푛
= √12 × 16 × 21 × 3 × 10 × 6 ퟔ
= 9.47
Measures of Dispersion for Sample
19. Sample Range:
Range = XL – XS = 21 – 3 = 18
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 (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:
2 =
Sx
2− 푛푥² 푛푖=1
Σ 푥푖
푛−1
=
=
986−770.21
5
=
215.79
5
= 43.15
2 = √43.15 = 6.56
22. Sample Standard Deviation: Sx = √Sx
23. Sample Mean Absolute Deviation (MAD):
Here,
XL = Largest observations
XS = Smallest observations
Here,
Q1 = First Quartile
Q3 = Third Quartile
3 6 10 12 16 21
(12)² + (16)² + (21)² + (3)²+ (10)² + (6)² - 6(11.33)²
6 - 1
Sx = standard deviation, Sx² = 43.15 [from (21)]

40. MAD =
i 
x x
24. Sample Coefficient of Variation:
C. V. =
Sx
X
× 100 =
6.56
11.33
× 100 = 57.89%
Measures of Central Tendency for Sample
25. Sample Midhinge: Midhinge =
푄1+ 푄3
2
=
13.675 + 30.65
2
= 22.16
n
n
i

1
( )
Q₁ = 5.25, Q₃ = 17.25 [from (20)]