Organization of Data

8/8/2019 Organization of Data

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Chapter 6

Organization of Data

Chapter 6. Organization of Data

Definition of Raw Data (page 122)

Raw data or unclassified data is the setof data in its original form. It has not beenorganized in any manner and is recorded inthe order observed.


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Example of Raw Data (page 122)

Days-to-maturity for 40 short-terminvestments

70 64 99 55 64 89 87 65

62 38 67 70 60 69 78 39

75 36 71 51 99 68 95 86

57 53 47 50 55 81 80 89

51 36 63 66 85 79 83 70


NCR Region 5 Region 12

(National 1st District 120,663 (Bicol Albay 553,629(SOCCSKSARGEN) NCotabto 509,463

Capital 2nd District 229,301 Region)Camarines

Nort 301,147 Saranggi 223,279

Region)1 3rd District 292,611Camarines

Sur 765,373 SCot aba 469,874

4th District 206,387 Catanduanes 116,866 Sultan K 344,172

CAR Masbate 483 ,651 Cotab C 49,997

(Cordillera Abra 110,937 Sorsogon 319,952 Region 13

Administrative Apayao 28,770 Region 6 (Caraga) Agus dN 259,475

Region) Benguet 122,762 (Western Aklan 186,813 AgusdS 353,825

Ifugao 113,719 Visayas) Antique 208,169 Surig dN 232,065

Kalinga 83,844 Capiz 328,635 Suri dS 225,640

Mt. Prov 76,137 Guimaras 37,838 ARMM

Region1 Iloilo 690,639(AutonomousRegion Basilan 123,825

(Ilocos Ilocos N 115,116Negros

ccid ntal 1,312,961in MuslimMindanao) Lanao dS 432,307

Region) IlocosS 190,297 Region 7 Maguindanao 534,628

La Union 253,382 (Central Bohol 590,926 Sulu 397,119

Pangasinan 888,844 Visayas) Cebu 973,490 Tawi-tawi 160,562

Region2 NOriental 427,509

(Cagayan Batanes 2,535 Siquijor 25,237

Valley) Cagayan 251,222 Region 8

Isabela 424,580 (Eastern Biliran 58,135

N Vizcaya 82,895 Visayas) ESamar 202,680

Quirino 59,555 Leyte 680,536

Pages 123 -124

Region3 NSamar 240,228

(Central Aurora 59,985 SSamar 116,738

Luzon) Bataan 68,659 WSamar 348,054

Bu lacan 147,812 Region 9

N Ecija 532,961 (Zamb ZambdN 433,091

Pampanga 331,739 Peninsula) ZambdS 821,793

Tarlac 360,109 Zamb S

Zambales 193,962 Isabela C

Region 4a Region 10(CALABARZON) Batangas 440,603 (Northern Bukidnon 449,647

Cavite 244,712 Mindanao) Camiguin 41,017

Laguna 207,184 Lanao DN 424,819

Quezon 667,385 MisOci 260,764

Rizal 139,449 MisOr 404,002

Region 4b Region 11

(MIMAROPA) Marinduque 113,553(Davao

Re ion Davao dN 637 ,298

OccMindor 177,823 DavaldS 412,442

Or Mindoro 340,690 Davao O 172,627

Palawan 228,004 Compo V

Romblon 170,917


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Definition of Frequency Distribution Table

(page 125)

Frequency distribution table (fdt) is a summarytable that shows the number of observations thatbelong in the different classes.

Note: The classes may be distinct values/qualitativecategories or the classes may be intervals of valuesof the variable. If the classes are distinct values thenthe fdt is called single value grouping. If theclasses are intervals of values then the fdt is called

grouping by class intervals.


Example of Single Value Grouping(page 126)

We illustrate single value grouping. Suppose we have data on the number of children of 50currently married women using any modern contraceptive method. Constructa summary table for the data set below.

0 0 1 2 2 2 3 3 4 40 0 1 2 2 3 3 3 4 40 1 1 2 2 3 3 3 4 40 1 1 2 2 3 3 3 4 50 1 1 2 2 3 3 3 4 5

Single Value Grouping of Number of Children of Currently Married Women Using AnyModern Method of Contraceptive:

Number of Children No. of Married Women %

0 7 141 8 162 11 223 14 284 8 165 2 4

TOTAL 50 100


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Examples of Grouping by Class Intervals

(page 128)TABLE 4a TABLE 4b TABLE 4c

No. of Poor PeopleNo. of

Provinces No. of Poor PeopleNo. of

Provinces No. of Poor PeopleNo. ofProvinces

2,500 - 152,499 24 2,500 - 202,499 31 2,500 - 192,499 30

152,500 - 302,499 24 202,500 - 402,499 26 192,500 - 382,499 26

302,500 - 452,499 18 402,500 - 602,499 16 382,500 - 572,499 16

452,500 - 602,499 7 602,500 - 802,499 5 572,500 - 762,499 5

602,500 - 752,499 4 802,500 - 1,002,499 3 762,500 - 952,499 3

752,500 - 902,499 3 1,002,500- 1,202,499 0 952,500 - 1,142,499 1

902,500 - 1,052,499 1 1,202,500- 1,402,499 1 1,142,500- 1,332,499 1

1,052,500- 1,202,499 0 Total 82 Total 82

1,202,500- 1,352,499 1

Total 82


Definition of Terms (page 127)

Class interval is the rangeof values that belong in thecategory.

Class frequency is thenumber of observations thatbelong in a class interval.

Class limits are the end

numbers of a class interval.The lower class limit (LCL) isthe lower end of the classinterval and the upper classlimit (UCL) is the upper endof the class interval.

No. of Poor People No. of Provinces

2,500 - 202,499 31202,500 - 402,499 26402,500 - 602,499 16602,500 - 802,499 5802,500 - 1,002,499 3

1,002,500- 1,202,499 01,202,500- 1,402,499 1

Total 82


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Definition of Terms (contd) Class size is the size of the class

interval. It is the differencebetween the upper class limits ofthe class and the preceding class; orthe difference between the lowerclass limits of the next class and theclass.

Example:

First class: class size=11 -1=10

Second class: class size=51-11=40 orclass size=50-10=40

Fifth class: class size=500-200=300

Sales No. of(in thousands of pesos) Products

1 101911 5044

51 - 100 ..22

101- 200 ..13

201- 500 4


Using Excel in Tallying theNumber of Observations

1. Enter data, one column per variable(include column label).

2. In another column, enter the upper classlimits.

3. Click Tools/Data Analysis/Histogram/OK.4. Fill up dialogue box. Identify cells

containing the data in Input Range box.Identify cells containing the upper classlimits in Bin Range box. Click Labels thenOK.


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Steps in Constructing FDT

using Equal Class Sizes (page 127-128)

Step 1: Determine the adequate number of classes, K.

There must be an adequate number of classes to show the essentialcharacteristics of the data. The larger the number of classes in a frequencydistribution, the more detail is shown. If the number of classes is too large,though, the table loses its effectiveness in summarizing the data. Too fewclasses, on the other hand, condense the information so much as to leavelittle insight into the pattern of the distribution.

There are no precise rules concerning the optimal number of classesbut the following formula can be used as a first approximation:

Sturges formula: K = 1 + 3.322 log n

where K = number of classesn = number of observations


Example

Exercise no. 4 (page 135)

n=30

Sturges formula: K = 1 + (3.322)(log(30))=5.9.

We consider using 6 classes.

The following data represent the weight of 30 children:

39.12 61.74 37.29 44.35 57.2964.1 48.25 67.25 58.95 39.9538.42 55.8 44.35 38.75 63.9151.5 40.15 60.29 41.26 49.3236.07 46.01 41.13 67.29 45.6863.55 62.12 36.85 45.97 42.89


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Step 2: Determine the range, R.

R = Maximum Minimum

Example: R=67.29 36.07 = 31.22

The following data represent the weight of 30 children:

39.12 61.74 37.29 44.35 57.2964.1 48.25 67.25 58.95 39.9538.42 55.8 44.35 38.75 63.9151.5 40.15 60.29 41.26 49.32

36.07 46.01 41.13 67.29 45.6863.55 62.12 36.85 45.97 42.89


Steps in Constructing FDTusing Equal Class Sizes (page 127-128)

Step 3: Calculate the approximate classsize, C.

C = R/K

Example: C = 31.22/6 =5.2


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Step 4: Determine the class size, C, byrounding off C to a number that is easyto work with. We recommend classsizes of multiples of 5, 10, 15, 20, etc.

Example: Consider using C=5.



Step 5: List the required number (K) of class intervals.

Choose the lower class limit (LCL) of the first class.

Important Pointers:1. The number of significant digits of LCL must be the same as the values in the

data.2. The LCL of first class should be less than or equal to the minimum value of the

data set.3. Often times, the LCL is selected so that we end up with numbers that are easy

to work with and read such as multiples of 5s, 10s and so on.4. If the observations tend to be concentrated at specific values throughout the

range of data for example, prices are often multiples of 50 or 100 or 1000 one may have to experiment on choosing the LCL so that the midpoint of theclass interval will be at these values.

5. Whenever appropriate, choose the LCL so that the groupings will bemeaningful. For example, when the values are grades then choose the LCL ofthe first class so that the passing grade will be the LCL of one of the classes.


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Example: Choose the lower class limit of the first class.

The number must have 4 significant digits. Its value should not be higher than 36.07. The observations do not tend to concentrate at any

specific value. A meaningful grouping should separate underweight,

normal and overweight kids. In our case, well just choose a limit that is easy to

work with. Let us use 35.00 as the lower limit of thefirst class.



After choosing the LCL of the first class, identify the LCL of the succeedingclasses by successively adding the class size, C. Stop this process once thecomputed LCL is larger than the largest observation.

Example: We have selected C=5 and the LCL of the first class=35.00. The largestobservation is 67.29. Thus, the lower class limits are:

LCL35.00

35.00 + 5 = 40.00 40.0040.00 + 5 = 45.00 45.0045.00 + 5 = 50.00 50.00

50.00 + 5 = 55.00 55.0055.00 + 5 = 60.00 60.0060.00 + 5 = 65.00 65.0065.00 + 5 = 70.00 STOP


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After determining the LCLs, obtain the UCLs by identifying thenumber with the same number of significant digits that precedethe LCLs.

Example:LCL UCL35.00 39.9940.00 44.9945.00 49.9950.00 54.9955.00 59.9960.00 64.9965.00 69.99



Step 6: Tally the frequency for each class interval.Step 7: Sum the frequency column and check against the total number of

observations.

Example:LCL UCL f 35.00 39.99 740.00 44.99 645.00 49.99 550.00 54.99 155.00 59.99 360.00 64.99 665.00 69.99 2

30


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Additional Pointers (page 129) When we present the frequency distribution in tabular form, always put the

appropriate column labels to describe the data. Thus, instead of using lowerand upper class limits and f, we use the variable of interest and identify theunits being counted.

Frequency Distribution of Weights of Children(in pounds)

Weight No. of Children

35.00 - 39.99. 740.00 - 44.99. 6

45.00 - 49.99. 550.00 - 54.99. 1

55.00 - 59.99..360.00 - 64.99..665.00 - 69.99..2

Total30


Additional Pointers

Whenever possible, all classes should be of the same size. If the classes sizesare not equal, it becomes difficult to tell whether the differences in classfrequencies result mainly from differences in the concentration of items or fromdifferences in the class sizes. However, there are certain instances when it isnot practical to use equal class sizes. One such case is when the distribution isbadly skewed.

Example: Suppose the salaries range from 20,000 to 1,000,000 but 95% ofobservations are less than 80,000. If all class sizes are equal to 100,000 thenthe fdt is almost useless. There will be around 10 classes but 95% belong inthe first class. No information would be provided about the distribution ofsalaries of the 95% with salaries from 1 100,000.

For such distributions, unequal class intervals are generally used. For instance,equal class sizes of say 10,000 might be used for the range wherein most of thesalaries fall, after which the size might increase to say 100,000. Another optionis to use an open class interval to account for the remainder of salaries that arenot included in the classification.


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Definition of Open Class Interval

(page 127) Open class interval is a class interval with either no lower class limit or no upper class

limit.

Example:Frequency Distribution of Magnitude of Poor Population

of the Provinces in the Phil ippines

No. of Poor People No. of Provinces

0 9 9,9 99 13100 ,000 199,999 ... 18200 ,000 299 ,999 16300 ,000 399,999 10400,000 499,999 .. 11500,000 599,999 .. 5

600,000 699,999 .. 4700,000 and above 5


Definition of Class Boundaries (page 127)

Class boundaries are the true class limits. Forrounded figures, the lower class boundary (LCB) isdefined as halfway between the lower class limit ofthe class and the upper class limit of the precedingclass while the upper class boundary (UCB) is definedas halfway between the upper class limit of the classand the lower class limit of the next class.

Rationale: If observation is rounded to nearest tenththen an observation of 10.0 is actually any measurein the interval [9.95, 10.05).


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Example of Class Boundaries Correction in page 129. The class limits are

the class boundaries themselves since theobservations are not rounded figures.TABLE 6. Frequency Distribution with Class Boundaries

and Class Marks

Class Limits Class Boundaries

LCL UCL LCB UCB Class Mark f

2,500 - 192,499 2,500 - 192,499 97,500 30

192,500 - 382,499 192,500 - 382,499 287,500 26

382,500 - 572,499 382,500 - 572,499 477,500 16

572,500 - 762,499 572,500 - 762,499 667,500 5

762,500 - 952,499 762,500 - 952,499 857,500 3

952,500 - 1,142,499 952,500 - 1,142,499 1,047,500 1

1,142,500 - 1,332,499 1,142,500 - 1,332,499 1,237,500 1

82


Example of Class Boundaries Using the fdt of weight of children,

First class: UCB =(39.99+40.00)/2 = 39.995

LCB = (34.99+35.00)/2 = 34.995

Note: For rounded figures, there are no gaps in the class boundaries. Thenumber of decimal place is one more than the number of decimal place ofthe class limits.

LCL UCL LCB UCB

35. 00 39.99 34. 995 39. 995

40. 00 44.99 39. 995 44. 995

45. 00 49.99 44. 995 49. 995

50. 00 54.99 49. 995 54. 995

55. 00 59.99 54. 995 59. 995

60. 00 64.99 59. 995 64. 995

65. 00 69.99 64. 995 69. 995


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Definition of Class Mark(page 127)

Class markis the midpoint of a class interval. It is the average of thelower class limit and the upper class limit or the average of the lowerclass boundary and upper class boundary of a class interval.

Example: Using the fdt of weight of children,First class: CM = (35.00+39.99)/2 = 37.495Second class: CM = (40.00+44.99)/2 = 42.495

LCL UCL CM

35.00 39.99 37.495

40.00 44.99 42.495

45.00 49.99 47.495

50.00 54.99 52.49555.00 59.99 57.495

60.00 64.99 62.495

65.00 69.99 67.495


Graphical Presentation of FDT (page 131)

The frequency histogram is the barchart of the fdt and presents the shapeof the distribution of the data set. Theheight of the bar represents thefrequency of the class interval. We plot

the sides of the bars at the classboundaries.


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Example of Frequency Histogram

LCB UCB f

34.995 39.995 7

39.995 44.995 6

44.995 49.995 5

49.995 54.995 1

54.995 59.995 3

59.995 64.995 6

64.995 69.995 2

Frequency Histogram of Weight of Children

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9

Weight (in lbs)

No.ofChildren

34.995 39.995 44.995 49.995 54.995 59.995 64.995 69.995


Frequency Histogram for FDTwith Unequal Class Sizes (not in notes)

When the class intervals of the frequencydistribution are not equal, the heights of thehistogram rectangles must be adjusted tomake the areas proportional to the class size.

Step 1: Select a unit class size, c.Step 2: Adjust class size of ith class, ci

*= ci/c.

Step 3: Adjust frequency of ith class, fi*= fi/ci

*.


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The table below shows the frequency distribution of income for policy-making purposes

regarding tax exemptions for taxpayers with annual incomes of P200,000 or below.

Choose unit class size, c =10Adjusted Adjusted

No.of size freq.Taxpayers ci

*= ci /10 fi*= fi/ci

*

ci Income fi

10 51 - 60 6 1 6

10 61 - 70 102 1 10210 71 - 80 134 1 134

20 81 - 100 293 2 146.550 101 - 150 364 5 72.8

50 151 - 200 101 5 20.2

Frequency Histogram of Annual Income of Taxpayers

0

20

40

60

80

100

120

140

160

45.5 65.5 85.5 105.5 125.5 145.5 165.5 185.5 205.5

Annual Income (in thousands)

NumberofTaxpayer


Frequency Polygon (page 132)

The frequency polygon is the line graph ofthe frequency distribution table. We plot thefrequencies against the corresponding classmarks then connect the points by straightlines. Since this is a polygon, we need toclose the chart by putting an additional classmark at both ends of the horizontal axis and

bring down the line to the horizontal axis atthe midpoints of the additional class marks.


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Example of Frequency Polygon

CM No. of children

32.495 0

37.495 7

42.495 6

47.495 5

52.495 1

57.495 3

62.495 6

67.495 272.495 0

Frequency Polygon of Weight of Children

0

1

2

3

4

5

6

7

8

32.495 37.495 42.495 47.495 52.495 57.495 62.495 67.495 72.495

Weight (in l bs)

No.ofChildren


Other Remarks (page 132)

The frequency polygon also shows the shape of the datadistribution. The advantage of the frequency polygon over thefrequency histogram is that we may draw and compare two ormore frequency distributions. The advantage of the frequencyhistogram is that it is easier to interpret since the boundaries ofthe class intervals are clearly displayed as the boundaries of thebars.

From the frequency histogram, we can easily construct thefrequency polygon by connecting the midpoints of the adjoining

bars and then putting two additional classes at both ends toclose it. The area under the frequency polygon is the same asthe area under the frequency histogram. Both areas representthe total number of observations.


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Variations of FDT:Relative Frequency and

Relative Frequency Percentage (page 129 130)

We find the relative frequency for each class interval by dividing the class frequency of aclass interval to the number of observations. The sum of the relative frequency column isone. On the other hand, we derive the relative frequency percentage from the relativefrequency. We simply multiply the relative frequency by 100% to get the relative frequencypercentage. The sum of the relative frequency percentage column is one hundred percent.

TABLE 7. Frequency Distribution with Relative Frequencyand Relative Frequency Percentage Columns

RelativeClass Limits Relative Frequency

LCL UCL f Frequency Percentage2,500 - 192,499 30 0.366 36.6

192,500 - 382,499 26 0.317 31.7

382,500 - 572,499 16 0.195 19.5

572,500 - 762,499 5 0.061 6.1

762,500 - 952,499 3 0.037 3.7

952,500 -1,142,499 1 0.012 1.2

1,142,500 -1,332,499 1 0.012 1.282 1.000 100.0


Variations of FDT:

Cumulative Frequency Distribution(page 130)

Less than cumulative frequencydistribution (< CFD) shows the number ofobservations with values smaller than theupper class boundary.

Greater than cumulative frequencydistribution (>CFD) shows the number ofobservations with values larger than thelower class boundary.


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How to Construct the < CFD (page 130)

For the first class interval, simply copy theclass frequency.

For the successive class intervals, add theclass frequency and the less thancumulative frequency of the precedingclass.

The < cumulative frequency of the last

class interval should be the same as thetotal number of observations.


Example

W eight Frequenc y


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How to Construct >CFD (page 130)

Begin with the last class interval and copythe class frequency.

For the preceding class intervals, add theclass frequency and the greater cumulativefrequency of the next class.

The > cumulative frequency of the firstclass interval should be the same as the

total number of observations.


Example

Weight Frequency >CFD Computation

35.00 - 39.99 7 30 23 + 7

40.00 - 44.99 6 23 17 + 6

45.00 - 49.99 5 17 12 + 5

50.00 - 54.99 1 12 11 + 1

55.00 - 59.99 3 11 8 + 360.00 - 64.99 6 8 2 + 6

65.00 - 69.99 2 2 2


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Ogive (page 133)

The ogives are the line charts of the cumulativefrequency distribution.

For the less than ogive, we plot the less thancumulative frequencies against the correspondingupper class boundaries.

For the greater than ogive, we plot the greater thancumulative frequencies against the correspondinglower class boundaries.

If we superimpose the less than ogive and thegreater than ogive, the intersection is the median(the value that divides the array into 2 equal parts).


Example of Less than Ogive

Class Boundaries


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Example of Greater than Ogive

Class Boundaries >CFD

34.995 - 39.995 30

39.995 - 44.995 23

44.995 - 49.995 17

49.995 - 54.995 12

54.995 - 59.995 11

59.995 - 64.995 8

64.995 - 69.995 2

Greater than Ogive of Weight of Children

0

5

10

15

20

25

30

35

34.995 39.995 44.995 49.995 54.995 59.995 64.995

Weight (in lbs)

GreaterthanCumulativeFrequency


Assignment

Use data in Exercise no. 5, page 135.1. Construct a frequency distribution and compute

for the relative frequency percentages. Presentthe frequency distribution and the relativefrequency percentage distribution in a formalstatistical table.

2. Present the frequency histogram of fdt in no. 1.3. Present the frequency polygon of fdt in no. 1.4. Construct the CF.5. Present the ogives of the cumulative frequencies

in no. 4.

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Organization of Data