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 15^{th} and 16^{th} 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 
AB 
Square (AB) 
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 (AB) = 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 
(AB)/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 Zscore 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 
XMean 
Square(XMean) 
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 3^{rd} day. But the steps again reduced on 15tg, 17^{th} and 18^{th} 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. 1336). Research Institute of Organic Agriculture FiBL and IFOAM Organics International.
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