MAT 254 Probability and Statistics Sprin

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MAT 254- Probability and Statistics

Spring 2015

LECTURE 2

DATA COLLECTION AND

PRESENTATION (Charts, graphs, etc)

25 30 35 40 45 50 55 60 65 70 75 80 85 90 9 8 7 6 5 4 3 2 1 0 f midpoint 29.5 - UCB

27- midpoint

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Types of Data

Quantitative data are measurements that are recorded

on a naturally occurring numerical scale.

Exp. Height in cm. ,weight in kg. ,blood pressure

(mm/Hg)

Qualitative data are measurements that cannot be

measured on a natural numerical scale; they can only be

classified into one of a group of categories.

Exp . Sex, tall or short, blood group

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SAMPLING TECHNIQUES

3/11/2015 [email protected]

Sampling techniques

are used to economize (on

the part of the researcher)

the following:



Time



Effort

(4)

POPULATION

SAMPLE

(5)

Sampling techniques

are classified

into:

•

probability sampling

•

non-probability sampling

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PROBABILITY SAMPLING



It is a method of selecting a sample (n)

from a universe (N) such that each

member of the population has an equal

chance of being included in the sample

and all possible combinations of size (n)

have an equal chance of being chosen

as the sample.

(7)

NON-PROBABILTY SAMPLING

It is a method wherein the

manner of selecting a sample

(n) from a universe (N)

depends on some inclusion

rule as specified by the

researcher.

(8)

PROBABILITY SAMPLING

TECHNIQUES

•

Simple Random (Lottery) Sampling

•

Systematic Sampling

•

Stratified Sampling

•

Cluster or Area Sampling

(9)

SRS or Lottery Sampling



It is done by simply assigning number

to each member of the population in a

piece of paper, placing them in a

container and drawing the desired

number of samples from it.



This applies to a

not-so-large population

when listing is still

possible.

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SYSTEMATIC SAMPLING

Ex: N = 100, n = 25

N/n = 100/25

= 4

•

This means every

4

th

element in a series should be

taken as a sample.



This method still

uses the concept of

random sampling

and involves the

selection of the n

th

element of a series

representing the

(11)

Note:

All numbers in yellow color are the desired

samples.

1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92

3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96

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STRATIFIED SAMPLING

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MULTI-STAGE SAMPLING

Ex: Region

–

1

st

level

Province

–

2

nd

Level

City

–

3

rd

Level

Barangay

–

4

th

Level

A technique that considers

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MULTI-STAGE SAMPLING

Regions

Divisions

School Districts

Schools

Divisions

Schools

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NON-PROBABILITY SAMPLING

TECHNIQUES

•

Purposive Sampling, based on a criteria

or qualifications given by the researcher.

Those who will satisfy the criteria are

included.

•

Quota Sampling

It is quick and cheap

since the interviewer is given a definite

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Presentation of Data

Objectives: At the end of the lesson, the

students should be able to:

1. Prepare a stem-and-leaf plot

2. Describe data in textual form

3. Construct frequency distribution table

4. Create graphs

5. Read and interpret graphs and tables

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Presentation of Data

Textual

Method

•

Rearrangeme

nt 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

(18)

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.

(19)

Solution

First, 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

(20)

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.

(21)

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 two-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.

MCPegollo/Basic Statistics/SRSTHS

(22)

Below is the stem-and-leaf plot of the

ungrouped data given in the example.

Stem Leaves

0 9

1 7,8

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.

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Tabular Presentation of Data

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

Table Number Table Title

Column Header

Row Classifier Body

Source Note

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Sample of a Frequency Distribution

Table for Grouped Data

Table 1.2

Frequency 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

(25)

Lower Class Limits

Lower Class

Limits

0 - 2 1 3 - 5 2 6 - 8 13 9 - 11 15

12 - 14 19

Rating Frequency

(26)

Upper Class Limits

0 - 2 1 3 - 5 2 6 - 8 13 9 - 11 15 12 - 14 19

Rating Frequency

(27)

Upper Class Limits

Upper Class

Limits

0 - 2 1 3 - 5 2 6 - 8 13 9 - 11 15 12 - 14 19

Rating Frequency

(28)

Class Boundaries

Class

Boundaries

0 - 2

20

3 - 5

14

6 - 8

15

9 - 11

2

12 - 14

1

Rating Frequency

- 0.5 2.5 5.5 8.5 11.5 14.5

(29)

midpoints

of

the classes

Class Midpoints

Class

Midpoints

0 - 1 2 20 3 - 4 5 14 6 - 7 8 15 9 - 10 11 2 12 - 13 14 1

(30)

Class Width

3 0 - 2 20

3 3 - 5 14

3 6 - 8 15

3 9 - 11 2

3 12 - 14 1

Rating Frequency

(31)

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 Table

1. 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

(32)

Relative Frequency Table

relative frequency =

class frequency

(33)

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%

Rating

Relative

Frequency

20/52 = 38.5%

14/52 = 26.9%

etc.

Table 2-5

(34)

Cumulative Frequency Table

Cumulative

Frequencies

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

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Ahmed-Refat-ZU

Graphical Presentation

The diagram should be:

•

Simple

•

Easy to understand

•

Save a lot of words

•

Self explanatory

•

Has a clear title indicating its content

•

Fully labeled

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Ahmed-Refat-ZU

Graphical Presentation

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The Line diagram

37

Example:

(38)

38

Cumulative Frequency Graph

A cumulative frequency graph or ogive, is a line graph that

displays the cumulative frequency of each class at its upper class

boundary.

17.5

Age (in years)

Ages of Students

24 18 12 6 30 0 C u m u la ti ve f re q u e n cy (p o rt io n o f st u d e n ts )

25.5 33.5 41.5 49.5 57.5

The graph ends at the upper

(39)

Ahmed-Refat-ZU

Graphical Presentation

Bar chart

•

It is used for presenting

discrete or qualitative

data.

•

It represent the measured value (or %) by separated

rectangles

of constant width and its lengths

proportional to the frequency

•

Type:

•

>>>Simple ,

•

>>> Multiple,

(40)

Bar diagram

(41)

Ahmed-Refat-ZU

Pie diagram:

Percentage of causes of child death in Egypt

diarrhea 50% chest infection

30%

congenital 10%

(42)

Ahmed-Refat-ZU

Histogram:

•

It is very similar to the bar chart with the difference

that the rectangles or bars are

adherent (without

gaps).

•

It is used for presenting class frequency table

(continuous data).

(43)

Ahmed-Refat-ZU

Graphical Presentation

Frequency Polygon

•

Derived from a histogram by connecting the

mid

points of the tops of the rectangles in the histogram.

•

The line connecting the centers of histogram

rectangles is called frequency polygon.

•

We can draw polygon without rectangles so we will

get simpler form of line graph.

•

A special type of frequency polygon is the

Normal

(44)

44

The Frequency Polygon

• Examples:

Age in Years Sex Mid-point of interval

Males Females

20-30 3 2 (20+30)/2=25 30-40 5 5 (30+40)/2=35 40-50 7 8 (40+50)/2=45 50-60 4 3 (50+60)/2=55 60-70 2 4 (60+70)/2=65

(45)

The Frequency Polygon

• Example:

Figure : Distribution of a group of subjects by age and sex

(46)

Ahmed-Refat-ZU

Graphical Presentation

Scatter diagram

(47)

Ahmed-Refat-ZU

This scatter diagram showed a positive or direct

relationship between NAG and

albumin/creatinine among diabetic patients

Correlation between NAG and albumin creatinine ratio in group of early diabetics

0 5 10 15 20 25 30 35

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

albumin creatinine ratio

N

A

(48)

Graphical Presentation

Box Plots

Box Plots are another way of representing all the same information that can be found on a Cumulative Frequency graph.

!

Lowest value

Upper Quartile

Highest value Lower Quartile

Median

Inter-Quartile Range

Range