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Categorical Data

Data is the small fragments of raw information collected for study and analysis for making it useful in form an informed conclusion. Data is of three types namely, Categorical, discrete numerical and continuous numerical. Categorical data is also understood as the qualitative data which represents various characteristics like gender, marital status, city, etc. Such data may have a numerical value but which don’t have any mathematical meaning. 

On the other hand, numerical data, as the name itself suggest, is with quantitative characteristic of measurement, like height, weight, etc. Further dividing into two as Discrete data which can be counted and have possible values which is either fixed or in a range going on to infinity; and Continuous data represent measurements that can be described using intervals.

As per the views of Cressie (2015), mmeasurement scales are of three types namely, Nominal, ordinal, interval and ratio scale. The nominal scale measures variable with a descriptive category, but have no Natural Numerical Value. The ordinal scale has both identity and magnitude property. The interval scale has identity, magnitude and equal intervals as its properties (Willer and Lernoud, 2016). The ratio scale has all four properties of measurement namely, identity, magnitude, equal intervals and minimum value of zero. 

What is your gender? (Male = 0, Female = 1)

Data Type: Categorical data with qualitative characteristic of a gender.

Measurement Level: Nominal scale of measurement as satisfies only identity property.

What is your approximate undergraduate college GPA? (1.0 to 4.0)

Data Type: Discrete data with finite possible values.

Measurement Level: Ordinal scale of measurement satisfying both identity and magnitude as property.

About how many hours per week do you expect to work at an outside job this semester?

Data Type: Discrete date with infinite possible values.

Measurement Level: Ordinal scale of measurement satisfying both identity and magnitude as property.

What do you think is the ideal number of children for a married couple?

Data Type: Discrete data with finite possible values.

Measurement Level: Interval scale of measurement with identity, magnitude and equal intervals.

On a 1 to 5 scale, which best describes your parents? (1 = Mother clearly dominant ? 5 = Father clearly dominant)

Data Type: Discrete data

Measurement Level: Ordinal scale of measurement satisfying both identity and magnitude as property.

No. of Students (N): 30

Monthly Rent paid: 730           730    730      930      700      570

690      1,030   740      620      720      670

560       740    650      660      850      930

Discrete Data

600       620    760      690      710      500

730       800    820      840      720      700

(a)

Total of values = 18850

Mean = x = Σx / n

            = 18850/30 = 628.33

Median = [(n/2)+(n/2+1)] / 2

            = [(30/2)+(30/2+1)] / 2 = (15+16) / 2

            = 15.5 i.e. average of the 15th and 16th value = 820+930/2 = 875

Mode = The values occurring more than once therefore it’s a multimodal data, thus grouping will give the more appropriate mode which is 730 occurring 4 times. 

(b) Agreement of the measure of Central Tendency:

Since the values of mean, median and mode are not very close to each other, the measures of central tendency are more scattered. Since mean takes into account all the values; median calculates the mid value and mode analyses value that occurs more frequently, any value over 730 would be more favorable and agreeable situation. 

(c) Calculation of Standard Deviation =

Value

Mean

A-B

Square (A-B)

A

B

C

560

628.33

-68.33

4668.989

600

628.33

-28.33

802.5889

690

628.33

61.67

3803.189

730

628.33

101.67

10336.79

730

628.33

101.67

10336.79

 730

628.33

101.67

10336.79

1030

628.33

401.67

161338.8

 620

628.33

-8.33

69.3889

 740

628.33

111.67

12470.19

 800

628.33

171.67

29470.59

730

628.33

101.67

10336.79

740

628.33

111.67

12470.19

650

628.33

21.67

469.5889

760

628.33

131.67

17336.99

820

628.33

191.67

36737.39

930

628.33

301.67

91004.79

620

628.33

-8.33

69.3889

660

628.33

31.67

1002.989

690

628.33

61.67

3803.189

840

628.33

211.67

44804.19

700

628.33

71.67

5136.589

720

628.33

91.67

8403.389

850

628.33

221.67

49137.59

710

628.33

81.67

6669.989

720

628.33

91.67

8403.389

570

628.33

-58.33

3402.389

670

628.33

41.67

1736.389

930

628.33

301.67

91004.79

500

628.33

-128.33

16468.59

700

628.33

71.67

5136.589

Sum of Square of (A-B) = C = 657169.267  

Mean of C = D = 21905.64223         

Square Root of D = 148.0055404      

(d) Sort and standardize the data. 

Standardized value = X – μ / σ 

Where:

X is the value

μ is the mean

σ is the standard deviation 

Data is sorted from smallest to largest in the following table with their standard values:

Value

Mean

SD

Standardized Data

A

B

C

(A-B)/C

500

628.33

148.0055

-0.86706

560

628.33

148.0055

-0.46167

570

628.33

148.0055

-0.39411

600

628.33

148.0055

-0.19141

620

628.33

148.0055

-0.05628

620

628.33

148.0055

-0.05628

650

628.33

148.0055

0.146413

660

628.33

148.0055

0.213979

670

628.33

148.0055

0.281544

690

628.33

148.0055

0.416674

690

628.33

148.0055

0.416674

700

628.33

148.0055

0.484239

700

628.33

148.0055

0.484239

710

628.33

148.0055

0.551804

720

628.33

148.0055

0.619369

720

628.33

148.0055

0.619369

730

628.33

148.0055

0.686934

730

628.33

148.0055

0.686934

730

628.33

148.0055

0.686934

730

628.33

148.0055

0.686934

740

628.33

148.0055

0.754499

740

628.33

148.0055

0.754499

760

628.33

148.0055

0.889629

800

628.33

148.0055

1.159889

820

628.33

148.0055

1.295019

840

628.33

148.0055

1.43015

850

628.33

148.0055

1.497715

930

628.33

148.0055

2.038235

930

628.33

148.0055

2.038235

1030

628.33

148.0055

2.713886

(e) Are there outliers or unusual data values?

Values which have a standardized value or Z-score of over 2 are the unusual value like 930 and 1030. 

(f) Using the Empirical Rule, do you think the data could be from a normal population?

For Normal Distribution, the Empirical rule is defined as values that fall in 1 Standard Deviation of the mean is 68%, 95% falls in 2 standard deviation of the mean and 99.73% in 4 standard deviation of the mean. That means

% of Values falling in range

Higher Value

Lower Value

68%

mean ± sd

776.3355

480.3245

95%

mean ± 2 sd

924.341

332.319

99.73%

mean ± 3 sd

1220.352

36.308

Considering the Empirical Rule data is from the normal population apparently.

Find the mean, median, and mode for each quiz.

I

II

III

IV

60

65

66

10

60

65

67

49

60

65

70

70

60

65

71

80

71

70

72

85

73

74

72

88

74

79

74

90

75

79

74

93

88

79

95

97

99

79

99

98

Mean

72

72

76

76

Median

72

72

72

86.5

Mode

60

65

72, 74

 Do these measures of center agree? Explain.

Yes the measures of centre agree as the mean and median and mode are closely related. 

For each data set, note strengths or weaknesses of each statistic of center.

Quiz

Strength

Weakness

I

Mean: Very useful measure results into average score of class.

Median: The mid value derived minimize the error in skewed distribution

Mode: Easily markable.

Mean: Unusually high scores affect the average score

Median: Insensitive to extreme values of the sample.

Mode: Least useful information scope.

II

Mean: Symmetric average score.

Median: Minimized error in skewed distribution

Mode: Easily spotted.

Mean: Minimal difference affect the average score.

Median: High sensitivity to fresh additions.

Mode: Two common scores create multimodal result..

III

Mean: Very useful central tendency result.

Median: Minimized skewed distribution errors

Mode: Mode easily spot able.

Mean: Competitive scores affect the average score.

Median: Close scattering of scores.

Mode: Small sample of frequency.

IV

Mean: Blend for average score of class.

Median: The mid value minimize the error in skewed distribution

Mode: Couldnot be spotted.

Mean: Very scattered scores does not portray correct potential of class.

Median: Mid values too high than least values.

Mode: No frequency of two similar scores could be spotted

Are the data symmetric or skewed? If skewed, which direction?

Data of quiz II, III and I is symmetric in order of its mention with the long tail of skewness extending to right. However the scores of quiz IV is very asymmetric with higher levels of variation and differences and also median is higher than mean, resulting into data skewed to left.

Continuous Data

Briefly describe and compare student performance on each quiz. 

Student performance in Quiz II is symmetrical in order with a very low difference in the minimum and maximum score. On the other hand performance in Quiz I and II is more competitive with higher differences in the least and the maximum score. Lastly Quiz IV results show that many students were confident and well prepared than a few who turned out to be a low performing in this case. 

Total Probability (one of the alternator fail or both fail or none fails) = 1

P (alternator 1 or 2 fail) = P(1 fails) or P(2 fails) = 0.02

P (Alternator 1 or 2 works well) = P(1 works) or P(2 works) = 1 - 0.02 = 0.98

Probability that both alternator fails = P(1 fails) * P(2 fails)  (Anderberg, 2014)

= 0.02*0.02

= 0.0004

Probability that neither of Alternators fail = P(1 works) * P(2 works)

= 0.98 * 0.98

= 0.9604 

Probability that one or the other alternator will fail

= P (1 fails) * P(2 works) OR P(2 fails) * P(1 works)

= 0.02 * 0.98

= 0.0196 

Mean = x = Σx / n

            = 59017/18

            = 3278.722

X

Mean

X-Mean

Square(X-Mean)

3450

3278.722

171.278

29336.15

3363

3278.722

84.278

7102.781

3228

3278.722

-50.722

2572.721

3360

3278.722

81.278

6606.113

3304

3278.722

25.278

638.9773

3407

3278.722

128.278

16455.25

3324

3278.722

45.278

2050.097

3365

3278.722

86.278

7443.893

3290

3278.722

11.278

127.1933

3289

3278.722

10.278

105.6373

3346

3278.722

67.278

4526.329

3252

3278.722

-26.722

714.0653

3237

3278.722

-41.722

1740.725

3210

3278.722

-68.722

4722.713

3140

3278.722

-138.722

19243.79

3220

3278.722

-58.722

3448.273

3103

3278.722

-175.722

30878.22

3129

3278.722

-149.722

22416.68

59017

160129.6

 Standard deviation = ( Rohatgi and Saleh,  2015)

                                    = Square Root [160129.6/18]

                                    = 94.31908

Standard error = Standard deviation / SQRT of no. of observation (Allen,  2014)

= 94.31908 / SQRT 18

 = 22.23122

E = 22.23122 * 1.96 = 43.57319

95% confidence interval =  (3278.722 – 43.57319) to (3278.722 + 43.57319)

= 3235.149 to 3322.295 steps

 Sample size to obtain an error of ± 20 steps with 95 percent confidence

= [(1.96 * Standard deviation)/20]^2

= 85.43804

Line chart of the data

The chart chart shows that the No. Of steps taken by Dave while jogging has gone down from the first day. But he picked up gradually after the 3rd day. But the steps again reduced on 15tg, 17th and 18th day.

References

Books and Journal 

Allen, A.O., 2014. Probability, statistics, and queueing theory. Academic Press.

Anderberg, M.R., 2014. Cluster analysis for applications: probability and mathematical statistics: a series of monographs and textbooks (Vol. 19). Academic press.

Cressie, N., 2015. Statistics for spatial data. John Wiley & Sons.

Rohatgi, V.K. and Saleh, A.M.E., 2015. An introduction to probability and statistics. John Wiley & Sons.

Willer, H. and Lernoud, J., 2016. The world of organic agriculture. Statistics and emerging trends 2016 (Pp. 1-336). Research Institute of Organic Agriculture FiBL and IFOAM Organics International.

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