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 =



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%


26. Sample Midhinge: Midhinge = Q1+Q 3

2=21.75+33.75

2

= 27.75


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


Ascending Order:





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


σ2x =

= − (3.49)2

=

= .069


σx= √σ2 = √ .069 = .262


MAD =


∴ MAD = 2.1210

= .212


(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


C. V. = σ xμx

× 100 = .2623.49

× 100 = 7.50%



Midhinge = Q1+Q 3

2=3.325+3.7

2 = 3.51


Sample:


X=¿

= 3.6+3.0+3.7+3.4+3.9

5

= 17.65 = 3.52


Ascending order:

Median =


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


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



Midrange = XS+XL2

= 3.0+3.92

=6.92 = 3.45


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


G. M. = n√a1×a2×a3×−−−×an

= 5√3.6×3.0×3.7×3.4×3.9

= 3.50


20. Sample Range:

Range = XL – XS = 3.9 – 3.0 = 0.9






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


MAD =


∴ MAD = 1.285

= .256


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


C. V. = SxX

× 100 = 0.343.52

× 100 = 9.65%



2=3.215+3.8

2

= 3.50

Answer to the Question No – 4

Population:


1. Population Mean (Average): The population contains N=10 observations, so the Mean is

µx =

=

=

= 3.125


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


Midrange = XS+XL2

= 2+52 =7

2 = 3.5


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


G. M. = N√a1×a2×a3×−−−−−−−×aN

= 8√2×4×2×3×5×4×3×2

= 2.95


Range = XL – XS = 5 – 2 = 3


Ascending Order:

IQR = Q3 – Q1

Q1 = .25(N+1)th = .25(8+1)th =2.25th







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


Q3 = 4 + .75(4 – 4) = 4 + 0 = 4

IQR = Q3 – Q1 = 4 – 2 = 2


σ2x =

= − (3.125)2

=

= 1.11


σx= √σ2 = √1.11 = 1.05


MAD =


∴ MAD = 2.258

= .90625


(2)²+ (4)²+ (2)²+ (3)²+ (5)²+ (4)²+ (3)²+ (2)²8


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


C. V. = σ xμx

× 100 = 1.053.125

× 100 = 33.6%



Midhinge = Q1+Q 3

2=2+4

2 = 3


Sample:


X=¿

= 2+4+3+5

4

= 144 = 3.5


Ascending order:

Median =

= 3.5



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



Midrange = XS+XL2

= 2+52

=72 = 3.5


H. M. =n

1a1

+1a2

±−−−−−−∓1

an

¿ 412+13+14+15

=4

.5+.33+.25+.2 = 3.125


G. M. = n√a1×a2×−−−×an

= 4√2×4×3×5

= 3.309


20. Sample Range:

Range = XL – XS = 5 – 2 = 3


Ascending Order:

IQR = Q3 – Q1

2 3 4 5






Q1 = .25(n+1)th = .25(4+1)th =1.25th


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


Sx2 = ∑i=1

n

x i2−n x ²

n−1

=

= 54−493

= 53

= 1.6666



MAD =


∴ MAD = 44

= 1


C. V. = SxX

× 100 = 1.293.5

× 100 = 36.85%


(2)²+ (4)²+ (3)²+ (5)² - 4(3.5)²4 - 1

Sx = standard deviation, Sx² = 1.6666 [from (22)]


= 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



2=2.25+4.75

2

= 3.5


Population:


1. Population Mean (Average):

The population contains N=10 observations, so the Mean is

µx =

=

=

= 35.2


Ascending order:

Median =

=

=

Q₁ = 2.25, Q₃ = 4.75 [from (21)]


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.


Midrange = XS+XL2

= 21+472 =682 = 37.5


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


G. M. = N√a1×a2×a3×−−−−−−−×aN

= 10√42×29×21×37×40×33×38×26×39×47

= 34.31


Range = XL – XS = 47 – 21 = 26


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).







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


Q3 = 40 + .25(42 – 40) = 40 + 0.5 = 40.5

IQR = Q3 – Q1 = 40.5 – 28.5 = 12


σ2x =

= − (35.2)2

=

= 56.36


σx= √σ2 = √56.36 = 7.50


MAD =


∴ MAD = 63.610

= 6.36


(42)² + (29)² + (21)² + (37)² + (40)² + (33)² + (38)² + (26)² + (39)² + (47)²10



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


∑i=1

N

Xi2

N−μ

x2

1295410

−1239 .04

∑i=1

N

( x i−μx )

N


C. V. = σ xμx

× 100 = 7.5035.2

× 100 = 21.30%



Midhinge = Q1+Q 3

2=28.5+40.5

2 = 34.5


Sample:


X=¿

= 29+21+33+39+47

5

= 1695 = 33.8


Ascending Order:

Median =




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


Midrange =XS+XL2

= 21+472

=682 = 34


H. M. =n

1a1

+1a2

±−−−−−−∓1

an

¿ 5129

+121

+133

+139

+147

=5

..03+.04+.03+ .02+.02 = 35.71


G. M. = n√a1×a2×a3×−−−×an

= 5√29×21×33×39×47

= 32.60


20. Sample Range:

Range = XL – XS = 47 – 21 = 26


Ascending Order:

IQR = Q3 – Q1

21 29 33 39 47






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


Sx2 =∑i=1

n

x i2−n x ²

n−1

=

= 6101−5712.2

4 = 388.8

4 = 97.2



MAD =


∴ MAD = 36.85

= 7.36


C. V. = SxX

× 100 = 9.8533.8

× 100 = 29.14%


(29)²+ (21)²+ (33)²+ (39)²+ (47)² - 5(33.8)²5 - 1


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



= 33.8 [from (14)]

∑i=1

n

( x i−x )

n



2=25+43

2

= 34


Population:


1. Population Mean (Average) : The population contains N=10 observations, so the Mean is

µx =

=

=

= 5.94


Ascending Order:

Median =

=

=

Q₁ = 25, Q₃ = 43 [from (21)]


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.


Midrange = XS+XL2

= 2.9+10.22=13.12 = 6.55


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


Range = XL – XS = 10.2 – 2.9 = 7.3


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


Q3 = 7.3 + .25(8.2 – 7.3) = 7.3 + .225 = 7.525

IQR = Q3 – Q1 = 7.525 – 3.55 = 3.975


σ2x =

= − (5.94)2

= = 5.20

10. Population Standard Deviation: σx= √σ2 = √5.20 = 2.28


MAD =


∴ MAD = 19.610

= 1.96


(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



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


C. V. = σ xμx

× 100 = 2.285.94

× 100 = 38.38%



Midhinge = Q1+Q 3

2=3.55+7.525

2 = 5.5375


Sample:


X=¿

= 3.1+5.9+7.0+4.3+8.2

5

= 28.55

= 5.7


Ascending order:

Median = = 5.9



Midrange =XS+XL2





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


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


G. M. = n√a1×a2×a3×−−−×an

= 6√3.1×5.9×7.0×4.3×8.2 = 335.94


20. Sample Range:

Range = XL – XS = 8.2 – 3.1 = 5.1


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


Q3 = 7.0 + .25(8.2 – 7.0) = 7.0 + .30 = 7.3

IQR = Q3 – Q1 = 7.3 – 4 = 3.3




3.1 4.3 5.9 7.0 8.2


Sx2 =∑i=1

n

x i2−n x ²

n−1

=

= 16.74 = 4.175



MAD =


∴ MAD = 85

= 1.6


C. V. = SxX

× 100 = 2.045.7

× 100 = 35.75%



2=4+7.3

2

= 5.65



(3.1)²+ (4.3)²+ (5.9)²+ (7.0)²+ (8.2)² - 5(5.7)²5 - 1


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


= 5.7 [from (14)]

Q₁ = 4, Q₃ = 7.3 [from (21)]

∑i=1

n

( x i−x )

n

Population:


1.Population Mean (Average): The population contains N=12 observations, so the Mean is

µx =

=

=

= 19.90


Ascending Order:

Median =

=

=

= 17.55



Midrange = XS+XL2

= 7.3+34.72=422 = 21





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


= 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


Range = XL – XS = 34.7 – 7.3 = 27.4


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






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



C. V. = σ xμx

× 100 = 8.1019.90

× 100 = 40.70%



Midhinge = Q1+Q 3

2=13.58+27.63

2 = 20.61


Sample:


X=¿

= 15.8+7.3+24.7+29.3+34.7+25.3

6

= 1686

= 22.85


(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




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


Ascending order:

Median =

=

= = 25



Midrange =XS+XL2

= 7.3+34.7

2=422 = 21


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


G. M. = n√a1×a2×a3×−−−×an

= 6√15.8×7.3×24.7×29.3×34.7×25.3 = 20.45



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


19. Sample Range:

Range = XL – XS = 34.7 – 7.3 = 27.4


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


Q3 = 29.3 + .25(34.7 – 29.3) = 29.3 + 1.35 = 30.65

IQR = Q3 – Q1 = 30.65 – 13.675 = 16.975


Sx2 =∑i=1

n

x i2−n x ²

n−1

=

= 3615.69−3132.7355=482.955

5 = 95.591



MAD =


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



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




∑i=1

n

( x i−x )

n

∴ MAD = 45.26

= 7.53


C. V. = SxX

× 100 = 9.7722.85

× 100 = 42.75%



2=13.675+30.65

2

= 22.16


Population:


1. Population Mean (Average):

The population contains N=12 observations, so the Mean is

µx =

=

=

= 9.83


= 22.85 [from (13)]

Q₁ = 13.675, Q₃ = 30.65 [from (20)]


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


Ascending Order:

Median =

=

=

= 9.5



Midrange = XS+XL2

= 3+212 =242 = 12


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


= 12√12×7×4×16×21×5×9×3×11×14×10×6 = 6.95




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


Range = XL – XS = 21 – 3 = 18


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


Q3 = 12 + .75(14 – 12) = 12 + 1.5 = 13.5

IQR = Q3 – Q1 = 13.5 – 5.25 = 8.25


σ2x =

= −

(9.83)2

= = 26.21



C. V. = σ xμx

× 100 = 5.119.83

× 100 = 51.98%




3 4 5 6 7 9 10 11 12 14 16 21


(12)² + (7)² + (4)² + (16)² + (21)² + (5)² + (9)²+ (3)² + (11)² + (14)² + (10)² + (6)²12



σx = 5.11 [from (10)]

µx = 9.83 [from (1)]

∑i=1

N

Xi2

N−μ

x2

147412

−96 .62



Midhinge = Q1+Q 3

2=5.25+13.5

2 = 9.375


Sample:


X=¿

= 12+16+21+3+10+6

6

= 686

= 11.33


Ascending order:

Median =

=


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



Midrange =XS+XL2

= 3+212

=242 = 12


H. M. =n

1a1

+1a2

±−−−−−−∓1

an

¿ 6112

+116

+13+13+110

+16

=6

.08+.06+.04+ .33+.1+.16 = 8.10


G. M. = n√a1×a2×a3×−−−×an

= 6√12×16×21×3×10×6 = 9.47


19. Sample Range:

Range = XL – XS = 21 – 3 = 18


Ascending Order:





3 6 10 12 16 21

IQR = Q3 – Q1

Q1 = .25(n+1)th = .25(6+1)th =1.75th


Q1 = 3 + .75 (6 – 3) = 3 + 2.25 = 5.25

Again, Q3 = .75(n+1)th = .75(6+1)th = 5.25th


Q3 = 16 + .25(21 – 16) = 16 + 1.25 = 17.25 IQR = Q3 – Q1 = 17.25 – 5.25 = 12


Sx2 =∑i=1

n

x i2−n x ²

n−1

=

= 986−770.215=215.79

5 = 43.15



MAD =


C. V. = SxX

× 100 = 6.5611.33

× 100 = 57.89%




(12)² + (16)² + (21)² + (3)²+ (10)² + (6)² - 6(11.33)²6 - 1



= 11.33 [from (13)]

Q₁ = 5.25, Q₃ = 17.25 [from (20)]

∑i=1

n

( x i−x )

n


2=13.675+30.65

2

= 22.16

statistics assignment 2

Education

population range

population variance

population mode

sample mean

population midrange

population median

population midhinge

population harmonic