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Presentation of DataModule 6Basic StatisticsSRSTHSMs. Pegollo

MCPegollo/Basic Statistics/SRSTHS

Presentation of Data

Objectives: At the end of the lesson, the students should be able to:

1. Prepare a stem-and-leaf plot2. Describe data in textual form3. Construct frequency distribution

table4. Create graphs5. Read and interpret graphs and

tables

Ungrouped vs. Grouped Data

Data can be classified as grouped or ungrouped.

Ungrouped data are data that are not organized, or if arranged, could only be from highest to lowest or lowest to highest.

Grouped data are data that are organized and arranged into different classes or categories.

Presentation of Data

Textual Method

• Rearrangement from lowest to highest

• Stem-and-leaf plot

Tabular Method

• Frequency distribution table (FDT)

• Relative FDT

• Cumulative FDT

• Contingency Table

Graphical Method

• Bar Chart• Histogram• Frequency

Polygon• Pie Chart• Less than,

greater than Ogive

Textual Presentation of Data

Data can be presented using paragraphs or sentences. It involves enumerating important characteristics, emphasizing significant figures and identifying important features of data.

Textual Presentation of DataExample. You are asked to present

the performance of your section in the Statistics test. The following are the test scores of your class:

34 42 20 50 17 9 34 43

50 18 35 43 50 23 23 35

37 38 38 39 39 38 38 39

24 29 25 26 28 27 44 44

49 48 46 45 45 46 45 46

SolutionFirst, arrange the data in order for you to

identify the important characteristics. This can be done in two ways: rearranging from lowest to highest or using the stem-and-leaf plot.

Below is the rearrangement of data from lowest to highest:

9 23 28 35 38 43 45 48

17 24 29 37 39 43 45 49

18 25 34 38 39 44 46 50

20 26 34 38 39 44 46 50

23 27 35 38 42 45 46 50

With the rearranged data, pertinent data worth mentioning can be easily recognized. The following is one way of presenting data in textual form.

In the Statistics class of 40 students, 3 obtained the perfect score of 50. Sixteen students got a score of 40 and above, while only 3 got 19 and below. Generally, the students performed well in the test with 23 or 70% getting a passing score of 38 and above.

Another way of rearranging data is by making use of the stem-and-leaf plot.

Stem-and-leaf Plot is a table which sorts data according to a certain pattern. It involves separating a number into two parts. In a two-digit number, the stem consists of the first digit, and the leaf consists of the second digit. While in a three-digit number, the stem consists of the first two digits, and the leaf consists of the last digit. In a one-digit number, the stem is zero.

What is a stem-and-leaf plot?

Below is the stem-and-leaf plot of the ungrouped data given in the example. Stem Leaves

2 0,3,3,4,5,6,7,8,9

3 4,4,5,5,7,8,8,8,8,9,9,9

4 2,3,3,4,4,5,5,5,6,6,6,8,9

5 0,0,0

Utilizing the stem-and-leaf plot, we can readily see the order of the data. Thus, we can say that the top ten got scores 50, 50, 50, 49, 48, 46, 46, 46,45, and 45 and the ten lowest scores are 9, 17, 18, 20, 23,23,24,25,26, and 27.

Exercise:

Prepare a stem-and-leaf plot and present in textual form.

The ages of 40 teachers in a public school23 27 28 36 35 38 39 40

32 42 44 54 56 48 55 48

30 31 35 36 47 48 43 38

34 26 28 29 45 34 45 44

36 38 39 38 36 35 40 40

Stem Leaf

2 3,6,7,8,8,9

3 0,1,2,4,4,5,5,5,6,6,6,6,8,8,8,8,9,9

4 0,0,0,2,3,4,4,5,5,7,8,8,8

5 4,5,6

Tabular Presentation of Data

http://www.sws.org.ph/youth.htm

Table Number

Table Title

Column Header

Row Classifier

Source Note

Below is a sample of a table with all of its parts indicated:

Frequency Distribution Table

A frequency distribution table is a table which shows the data arranged into different classes(or categories) and the number of cases(or frequencies) which fall into each class.

The following is an illustration of a frequency distribution table for ungrouped data:

Sample of a Frequency Distribution Table for Ungrouped Data

Table 1.1Frequency Distribution for the Ages of 50

Students Enrolled in StatisticsAge Frequency

N = 50

Sample of a Frequency Distribution Table for Grouped Data

Table 1.2Frequency Distribution Table for the Quiz

Scores of 50 Students in Geometry

0 - 2 1

3 - 5 2

6 - 8 13

9 - 11 15

12 - 14 19

Scores Frequency

Lower Class Limits are the smallest numbers that can actually belong

to different classes

0 - 2 1

3 - 5 2

6 - 8 13

9 - 11 15

12 - 14 19

Rating Frequency

Lower Class Limits

Lower ClassLimits

0 - 2 1

3 - 5 2

6 - 8 13

9 - 11 15

12 - 14 19

Rating Frequency

are the smallest numbers that can actually belong to different classes

Upper Class Limits

0 - 2 1

3 - 5 2

6 - 8 13

9 - 11 15

12 - 14 19

Rating Frequency

are the largest numbers that can actually belong to different classes

Upper Class Limits

Upper ClassLimits

0 - 2 1

3 - 5 2

6 - 8 13

9 - 11 15

12 - 14 19

Rating Frequency

are the largest numbers that can actually belong to different classes

are the numbers used to separate classes, but without the gaps created by class limits

Class Boundaries

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

number separating classes

Class Boundaries

ClassBoundaries

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

number separating classes

The Class Mark or Class Midpoint is the respective average of each

class limits

Class Midpoints

midpoints of the classesClass Midpoints

ClassMidpoints

0 - 1 2 20

3 - 4 5 14

6 - 7 8 15

9 - 10 11 2

12 - 13 14 1

Rating Frequency

Class Width

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

is the difference between two consecutive lower class limits or two consecutive class boundaries

Class Width

3 0 - 2 20

3 3 - 5 14

3 6 - 8 15

3 9 - 11 2

3 12 - 14 1

Rating Frequency

is the difference between two consecutive lower class limits or two consecutive class boundaries

1. Be sure that the classes are mutually exclusive.

2. Include all classes, even if the frequency is zero.

3. Try to use the same width for all classes.

4. Select convenient numbers for class limits.

5. Use between 5 and 20 classes.

6. The sum of the class frequencies must equal the number of original data values.

Guidelines For Frequency Tables

3. Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score.

4. Add the class width to the starting point to get the second lower class limit, add the width to the second lower limit to get the

third, and so on.

5. List the lower class limits in a vertical column and enter the upper class limits.

6. Represent each score by a tally mark in the appropriate class.

Total tally marks to find the total frequency for each class.

Constructing A Frequency Table1. Decide on the number of classes .

2. Determine the class width by dividing the range by the number of classes (range = highest score - lowest score) and round up.

class width round up of

number of classes

HomeworkGather data on the ages of your

classmates’ fathers, include your own. Construct a frequency distribution table

for the data gathered using grouped and ungrouped data.

What are the advantages and disadvantages of using ungrouped frequency distribution table?

What are the advantages and disadvantages of using grouped frequency distribution table?

Relative Frequency Table

relative frequency =class frequency

sum of all frequencies

Relative Frequency Table

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

0 - 2 38.5%

3 - 5 26.9%

6 - 8 28.8%

9 - 11 3.8%

12 - 14 1.9%

RatingRelativeFrequency

20/52 = 38.5%

14/52 = 26.9%

Table 2-5Total frequency = 52

Cumulative Frequency Table

CumulativeFrequencies

0 - 2 20 20 52

3 – 5 14 34 32

6 – 8 15 49 18

9 – 11 2 51 3

12 – 14 1 52 1

Rating <cf

Table 2-6

>cfFrequency

Frequency Tables

0 - 2 20

3 - 5 14

6 - 8 15

9 - 11 2

12 - 14 1

Rating Frequency

0 - 2 38.5%

3 - 5 26.9%

6 - 8 28.8%

9 - 11 3.8%

12 - 14 1.9%

RatingRelativeFrequency

0 – 2 20

3 – 5 34

6 – 8 49

9 – 11 51

12 – 14 52

RatingCumulative Frequency

Table 2-6Table 2-5Table 2-3

Complete FDT

A complete FDT has class mark or midpoint (x), class boundaries (c.b), relative frequency or percentage frequency, and the less than cumulative frequency (<cf) and the greater than cumulative frequency (>cf).

Complete Frequency Table

0 - 2 20 1 -0.5 – 2.5 38.5% 20 52

3 – 5 14 4 2.5 – 5.5 26.9% 34 32

6 – 8 15 7 5.5 – 8.5 28.8% 49 18

9 – 11 2 10 8.5 – 11.5 3.8% 51 3

12 – 14 1 13 11.5 – 14.5 1.9% 52 1

Class Intervals

(ci)<cf

Table 2-6

>cfFrequency

(f) Class

Mark (x)

Relative Frequency

Class Boundary (cb)

Grouped Frequency Distribution for the Test Scores of 52 Students in

Statistics

Exercise:For each of the following class intervals,

give the class width(i), class mark (x), and class boundary (cb)

Class interval (ci)

Class Width

Class Mark Class Boundary

a. 4 – 8

b. 35 – 44

c. 17 – 21

d. 53 – 57

e. 8 – 11

f. 108 – 119

g. 10 – 19

h. 2.5 – 2. 9

i. 1. 75 – 2. 25

Construct a complete FDT with 7 classes

The following are the IQ scores of 60 student applicants in a certain high school128 106 96 94 85 75

113 103 96 91 94 70

109 113 109 100 81 81

103 113 91 88 78 75

106 103 100 88 81 81

113 106 100 96 88 78

96 109 94 96 88 70

103 102 88 78 95 90

99 89 87 96 95 104

89 99 101 105 103 125

Contingency Table

This is a table which shows the data enumerated by cell. One type of such table is the “r by c” (r x c) where the columns refer to “c” samples and the rows refer to “r” choices or alternatives.

ExampleTable 1

The Contingency Table for the Opinion of Viewers on the TV program “Budoy”

Choice/Sample

Men Women Children

Like the Program

50 56 45 151

Indifferent 23 16 12 51

Do not like the program

43 55 40 138

Total 116 127 97 340

Give as many findings as you can, and draw as many conclusions from your findings. The next table can help you identify significant findings.

ExampleTable 1

The Contingency Table for the Opinion of Viewers on the TV program “Budoy”

Choice/Sample

Men Women Children Total

Like the Program

50 (33%)(43%)

56(37%)(44%)

45(30%)(46%)

151(44%)

Indifferent 23(45%)(20%)

16(31%)(13%)

12(24%)(12%)

51(15%)

Do not like the program

43(53%)(37%)

55(40%)(43%)

40(29%)(41%)

138(41%)

Total 116(34%)

127(37%)

97(28%)

340Do not use this table for presentation because the percentages might confuse the readers. Can you explain the percentages in each cell?

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