Determining The Level Of Measure

Determining The Level Of Measure

Question one:

In statistics there are various levels of measurement, these levels include nominal variables, ordinal variables, interval variables and ratio variables. Given a survey undertaken in the United States we can determine the level of measure for the question.

Given the question “the most important problem facing the United States” we can determine the level of measure; these variables are given in percentage of respondents regarding the most important problem, this level of measure can be termed as nominal because the objects measured could be assigned numerical values and then classified.

Given the question “how often the United States use of preemptive force can be justified” we can still determine the level of measure, the answers for this question included often, sometimes, rarely and never. This requires respondents to rank their views regarding the issue, for this reason therefore we conclude that this is ordinary level of measure because it required ranking the responses.

Question two:

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The mean:

The mean is determined by getting the total of the values and then dividing by the sample size, given the following data we add up the values and then divide the total by the size or number of data:

Country

2005

1

Albania

6

2

Algeria

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6

3

Angola

15

4

Antigua and Barbuda

11

5

Argentina

34

6

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Australia

25

7

Austria

34

8

Azerbaijan

11

9

Bangladesh

2

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10

Barbados

13

11

Belarus

29

12

Belize

7

13

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Benin

7

14

Bhutan

9

total

209

mean

14.92857

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mode

6

median

11

The total for the sample that contains 14 countries is 209; we now determine the mean by dividing 209 by 14 as follows:

209/14 = 14.92857

Therefore our mean value is 14.92857.

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Median:

The median is also a measure of central tendency of data, to obtain the median we arrange the given data in an ascending order, when this is done we determine the value that half of the observations lie above and half of the observations lie below this value.

In our case we arrange our observations in an ascending order and the result is as follows:

Bangladesh

2

x

Albania

6

x

Algeria

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6

x

Belize

7

x

Benin

7

x

Bhutan

9

x

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Antigua and Barbuda

11

Azerbaijan

11

x

Barbados

13

Angola

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15

x

Australia

25

x

Belarus

29

x

Argentina

34

x

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Austria

34

x

Having arranged the observations in an ascending order we determine which value is at the center or that value which divides the observations into two half, from the above table it is evident that our median is 11.

Median is therefore 11

Mode:

The mode is that observation value that has the highest occurrence in data, for example is 2 is repeated many times that any other variable then we take it to be the mode. In our case we first determine the variables represented in the data and record their frequency, the variable with the highest frequency is the mode, we do this as follows:

value

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frequency

2

1

6

2

7

2

9

1

11

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2

13

1

15

1

25

1

29

1

34

2

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From the above table it is clear that 6,7,11 and 34 have a frequency of 2 each which is the highest in the data, for this reason therefore we conclude that we do not have a mode for this data set given because we do not have an observation with high frequency.

Question three:

Variance and standard deviation:

Variance:

Variance is a measure of dispersion from the mean of given data, the first step is to determine the mean, the mean is determined by dividing the total by the number of observation, in our case the number of observations (n) is 20, the total is 99.93, therefore our mean will be given by 99.93/20 = 4.9965.

After determining our mean which we denoted as x” and the observations as x, our next step will be to determine the difference of each observation and the mean, this is determined by subtracting the mean from each observation which is denoted as (x-x”). We do this for each observation.

Our next step is to square the values derived from (x-x”), this is denoted in the table below as (x-x")2, after determining these values we get the total for this values derived in the table which can be denoted as ∑(x-x")2

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We now determine our variance using the formula

Var = (∑ (x-x") 2)/n

Country

Share of Personnel for  Multinational Peace Operations – 2003 (%)

x – x"

(x-x")2

1

Australia

1.15

-3.8465

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14.79556

2

Belgium

0.76

-4.2365

17.94793

3

Canada

4.71

-0.2865

0.082082

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4

Czech Republic

1.34

-3.6565

13.36999

5

Denmark

1.42

-3.5765

12.79135

6

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France

13.45

8.4535

71.46166

7

Germany

13.97

8.9735

80.5237

8

Greece

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2.27

-2.7265

7.433802

9

Hungary

2.23

-2.7665

7.653522

10

Italy

14.14

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9.1435

83.60359

11

Japan

1.34

-3.6565

13.36999

12

Netherlands

3.99

-1.0065

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1.013042

13

Norway

1.51

-3.4865

12.15568

14

Spain

5.65

0.6535

0.427062

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15

Poland

4.71

-0.2865

0.082082

16

Portugal

2.68

-2.3165

5.366172

17

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Republic of Korea

0.16

-4.8365

23.39173

18

Turkey

3.02

-1.9765

3.906552

19

United Kingdom

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6.71

1.7135

2.936082

20

United States

14.72

9.7235

94.54645

total

99.93

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466.8581

mean

4.9965

Given that the value of ∑ (x-x") 2 and n = 20 our variance value will be derived from

466.8581 / 20 = 23.3429

Therefore our variance = 23.3429

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Standard deviation:

Standard deviation is also a measure of dispersion from the mean of given data, the standard deviation is usually the square root of the variance, it is also determined using the following formula:

STD dev = [(∑ (x-x") 2) / n] ½

Therefore our standard deviation will be derived from the following:

= [466.8581 / 20] ½

= 4.831449

Therefore our standard deviation value is 4.831449

Question 4:

Yule Q measures the strength of a relationship in a 2 by 2 table, it is a symmetric measure that is derived from the difference between Discordant (Q) and concordant (P) pairs of data, and given the following 2 by 2 table we derive the Q as follows:

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a

b

c

d

Q = (a X d) – (c X b) / (a X d) + (c X b)

Having our table we derive our results as follows:

above mean

below mean

against

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12

11

for

7

14

(ad – bc)

91

(ad + bc)

245

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Q

0.371428571

Therefore the Yule’s Q is equal to 0.3714 which is represented as a percentage and therefore it is equal to 37.14%.

If we interchanged rows in such a way that those who voted against were placed in the second and those who voted for were placed in the first row the following would be the result:

The two by two table is as follows:

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above mean

below mean

for

7

14

against

12

11

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(ad – bc)

-91

(ad + bc)

245

Q

-0.371428571

Therefore the Yule’s Q is equal to -0.3714 which is represented as a percentage and therefore it is equal to -37.14%.

For this reason therefore it is clear that the value is negative and not positive as derived from the above calculation, therefore the difference is that this value has a negative sign, both values shows the same strength of relationship between the variables because their values are equal

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and the only difference is the sign.

Making one of the values to be equal to zero then there will be different values of the Yule’s Q, if one value in the table was zero then the results would be either 1 or negative 1, the value of 1 shows perfect association between variables, when we get this result then this would be a good measure of association between variables because the value derived shows perfect association.

We take the example of the following table:

above mean

below mean

against

12

11

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for

0

14

The results will be as follows:

(ad – bc)

168

(ad + bc)

168

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Q

1

Q = 1 showing that there is perfect association.

Question 5:

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Ordinary association gamma is also a measure of association and it gives the strength of relationships that exist between variable, we have a 3 by three matrix as follows:

a

b

c

d

e

f

g

h

i

To derive gamma from the above we have to consider the S and P as follows

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S = A (e + f + h+i) + D (h+i) + B (f+i) + E (i)

P = C(d +e + g + h) + I (g + h) + b (d +g) + e ( g)

Therefore Gamma = (S – P)/ (S+P)

In our case we have the following table:

Time spent watching  television news (minutes)

Years of Education

12-Sep

13-16

17-20

0-10

15

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9

3

20-Nov

8

10

4

21 or more

4

5

6

S = 15 (10+4+ 5+6i) + 8 (5+6) + 9(4+6) + 10(6)

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S= 613

P=3(8+10+4+5)+6(4+5)+9(8+4)+10(4)

P=283

Gamma = (S – P)/ (S+P)

Gamma = (613 – 283)/ (613+283)

Gamma =0.368304

Therefore gamma = 0.368304, this value of gamma depicts a weak relationship between the variables.

Question 6:

The Somers d value is calculated as follows given a 3 by 3 table:

a

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b

c

d

e

f

g

h

i

S = A(e + f + h+i) + D(h+i) + B(f+i) + E(i)

P = C(d +e + g + h) + I (g + h) + b (d +g) + e ( g)

Y0 = a (b+c) + bc + d (e+f) + ef +g (h+i) + hi

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Therefore Somers d = (S – P)/ (S+P + y0)

Given the following data we can calculate the Somers d as follows:

Time spent watching  television news (minutes)

Years of Education

12-Sep

13-16

17-20

0-10

15

9

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3

20-Nov

8

10

4

21 or more

4

5

6

S = 15 (10+4+ 5+6i) + 8 (5+6) + 9(4+6) + 10(6)

S= 613

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P=3(8+10+4+5)+6(4+5)+9(8+4)+10(4)

P=283

Y0 = 15(9+3) + (9) (3) + 8(10+4) + (10) (4) +4(5+6) + (5) (6)

Y0 = 433

Somers d = (613– 283)/ (613+283 + 433)

Somers D = 0.248307

Question 7:

Scatter diagram:

Slope:

To determine the slope of the regression model we use the classical regression model to estimate the model using the data provided:

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The classical linear regression model is stated as follows:

b = ∑xiyi/∑xi2

given our data as follows we determine the independent and dependent variable, the independent variable is denoted by X and the dependent is represented by Y, for this reason therefore GDP is represented by Y and export days required is represented by X. when we have done this we calculate the values of X squares, Y squared and Y multiplied by X for each set of data.

We get their totals for each value in the table and also determine the mean for Y and X, the following table summarizes the calculation process:

Country

Number of days required for  export

Gross national income per  capita

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X

Y

x

y

xy

x2

Albania

37

21

9.1

-12.1

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-110.11

82.81

Brazil

39

31

11.1

-2.1

-23.31

123.21

El Salvador

43

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24

15.1

-9.1

-137.41

228.01

Jamaica

20

29

-7.9

-4.1

32.39

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62.41

Jordan

28

22

0.1

-11.1

-1.11

0.01

Namibia

32

24

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4.1

-9.1

-37.31

16.81

Chile

23

49

-4.9

15.9

-77.91

24.01

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Lithuania

6

57

-21.9

23.9

-523.41

479.61

South Africa

31

36

3.1

2.9

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8.99

9.61

Turkey

20

38

-7.9

4.9

-38.71

62.41

total

279

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331

1.42E-14

-1.4E-14

-907.9

1088.9

mean

27.9

33.1

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b = ∑XiYi/∑xi2

∑xiyi =-907.9

∑xi2 = 1088.9

B = -907.9 / 1088.9

Therefore our slope is equal to -0.833777206

Interpretation:

To interpret the slope value we state that if all the other factors were held constant and we increase the number of days for exports by one unit then the GDP per capita would decline by 0.833777

Autonomous value (a)

Given a model of the form Y = a + bX we can determine the value of a as follows:

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A = y’ – bx’

Where y’ is the mean of Y and x’ is the mean of x

Therefore our a is determined as follows

= 33.1 – (27.9) (- 0.833777) Therefore a = 56.36238406

Interpretation:

The value of the autonomous value depict that if the number of days for exports is zero then the value of GDP per capita would be equal to 56.3623

Pearson correlation coefficient:

The Pearson correlation coefficient value depict the strength of the relationship that exist between two variables, using excel the calculation is summarized below:

r = n (∑XY – ∑X∑Y)/ (n∑x2 – (∑x) 2) (n∑y2 – (∑y) 2)½

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r = n(∑XY – ∑X∑Y)/ (n∑x2 –  (∑x)2) (n∑y2 – (∑y)2)½

N

10

∑XY

8327

∑X

279

∑Y

331

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∑x2

8873

∑y2

12269

n(∑XY – ∑X∑Y)

-9079

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n∑x2 – (∑x)2

10889

n∑y2 – (∑y)2

13129

(n∑x2 – (∑x)2) (n∑y2 – (∑y)2)

142961681

(n∑x2 – (∑x)2) (n∑y2 – (∑y)2)½

11956.65844

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r

-0.759325864

From the calculation our correlation coefficient is -0.7593, this value shows that there is a strong relationship between the number of days for export and GDP per capita, the negative value shows that there is an inverse relationship between the two variables, this means that when one variable increase the other variable decreases.

Correlations of determination r squared:

R squared = b2 ∑x2/ ∑y2

correlation of determination

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R2 = b2 ∑x2/ ∑y2

∑x2

1088.9

∑y2

1312.9

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b

-0.833777206

b2

0.69518443

b2 ∑x2

756.9863256

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b2 ∑x2/ ∑y2

0.576575768

The correlation of determination value is 0.5766, this means that there is a moderately strong relationship between the two variables; it also means that 57.66% of deviations in GDP per capita is explained by the number of days for export.

References:

Louise White (1998) Political Analysis: Techniques and Practice, McGraw Hill Press, New York

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