Types Of Data And Measurement Scales

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	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 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. 1-336). Research Institute of Organic Agriculture FiBL and IFOAM Organics International.