III. Measures of Central Tendency (ppt) PDF

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Summary

Measures of Central Tendency: mean, median, and mode....

Description

DATA PRESENTATION

REVIEW

OBJECTIVES

•

Distinguish the three main forms of data presentation

•

Describe a data in textual form

•

Organize data in tables

•

Read and interpret tables and graphs

•

Choose appropriate diagrams/graphs to present a given set of data

•

Realize the importance of the measure of central tendency in describing the characteristics of a set of data

TEXTUAL PRESENTATION •

Keep your text/paragraphs simple and short with additional explanations about the relevance of the figures and its implications

•

Include important characteristics, and identify important features of your data

GRAPHICAL PRESENTATION

Bar Graph -To compare different categories (group sizes, ratings, inventories or survey responses) -Use to organize discrete data Histogram -Similar to bar graph but use to present categories of data (continuous data) Pie Chart -To show the proportion/percentage of each part or category to the whole Line Graph -To show trends in data over a period of time -Use to organize continuous data

BAR GRAPH

HISTOGRAM

PIE CHART

LINE GRAPH

GRAPH

TABULAR PRESENTATION

•to represent individual information and represents both quantitative and qualitative information

FREQUENCY DISTRIBUTION TABLE

Ungrouped Data -> raw data that has not been placed in any group or category after collection

Grouped Data -> data which is classified into groups after collection ->frequency distribution for grouped data: observations are sorted into classes of more than one value

FREQUENCY DISTRIBUTION TABLE

Table 1. Frequency Distribution of the blood types of BS Biology Students

Table 2. Frequency Distribution of the mass of the students Class limits 45 – 49

Class Boundaries 44.5 – 49.5

Frequency 2

50 – 54

49.5 – 54.5

4

55 – 59

54.5 – 59.5

7

60 – 64

59.5 – 64.5

10

65 – 69

64.5 – 69.5

4

70 – 74

69.5 – 74.5

6

75 – 79

74.5 – 79.5

7

RELATIVE FREQUENCY DISTRIBUTION TABLE Relative Frequency = proportion of each frequency to the total frequency

for decimal value: RF = 𝒇/𝑵 for percentage value: % RF = 𝒇/𝑵 𝒙 𝟏𝟎𝟎 where f=frequency, and N=total frequency

Table 3. Relative Frequency Distribution of the mass of the students

Class limits

Relative Relative Class Frequency Frequency Frequency Boundaries (Percentage) (Decimal)

45 – 49

44.5 – 49.5

2

5%

0.05

50 – 54

49.5 – 54.5

4

10%

0.1

55 – 59

54.5 – 59.5

7

17.5%

0.175

60 – 64

59.5 – 64.5

10

25%

0.25

65 – 69

64.5 – 69.5

4

10%

0.1

70 – 74

69.5 – 74.5

6

15%

0.15

75 – 79

74.5 – 79.5

7

17.5%

0.175

40

100%

1.00

Total

MEASURES OF CENTRAL TENDENCY

MEASURE OF CENTRAL TENDENCY

•

measures that represent a set of data

•

the value around which the data tends to be centered 3 primary measures of the central tendency:  Mean  Median  Mode

MEAN •

nominal average

•

the sum of the data values divided by the number of data values.

Advantages of Mean: ✦ Simple to understand and easy to calculate. ✦ It is rigidly defined. ✦ It is least affected fluctuation of sampling. ✦ It takes into account all the values

UNGROUPED DATA Sample Mean Population Mean

∑ 𝑥 𝑥 = 𝑛 𝜇=

∑  𝑥 𝑁

where: 𝑥 = 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠/𝑖𝑡ℎ observation 𝑛 = total no. of sample observations N = total no. of observations

GROUPED DATA weighted mean Sample Mean Population Mean

𝑥 =

∑ 𝑓𝑥 ∑  𝑓

∑  𝑓𝑥 𝜇= ∑  𝑓

where: 𝑥 = 𝑑𝑎𝑡𝑎 𝑣𝑎𝑙𝑢𝑒𝑠 𝑛 = no. of sample observations N = no. of observations 𝑓= frequency

MEAN

EXAMPLE Consider the following grades in five quizzes in Statistics of the two students:

Student A: 80, 60, 88, 75, 90 Student B: 70, 100, 85, 92, 86

Student A’s mean: Student B’s mean:

∑  







∑  







The means suggest the average grade of the two students in five quizzes in Statistics: 78.6 for Student A while 86.6 for Student B.

EXAMPLE Consider the following grades of a student last semester: Subject

Weighted mean/average:

Grade (x)

No. of units (f)

A

1.0

3

B

1.75

3

C

2.0

3

D

1.5

5

E

1.0

5

𝑥 =

∑ 𝑓𝑥 ∑ 𝑓

EXAMPLE Consider the table:

Weighted mean/average:

𝑥 =

∑  𝑓𝑥 ∑ 𝑓

EXAMPLE Weighted mean/average:

Class limits 45 – 49 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74 75 – 79

𝑥 =

∑ 𝑓𝑥 ∑ 𝑓

Class Boundaries 44.5 – 49.5 49.5 – 54.5 54.5 – 59.5 59.5 – 64.5 64.5 – 69.5 69.5 – 74.5 74.5 – 79.5

Frequency 2 4 7 10 4 6 7

middle value of class interval

x 47 52 57 62 67 72 77

EXAMPLE Weighted mean/average:

Class limits 45 – 49 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74 75 – 79

𝑥 =

∑ 𝑓𝑥 ∑ 𝑓

Class Boundaries 44.5 – 49.5 49.5 – 54.5 54.5 – 59.5 59.5 – 64.5 64.5 – 69.5 69.5 – 74.5 74.5 – 79.5

Frequency 2 4 7 10 4 6 7 ∑𝑓 = 40

x

fx 47 52 57 62 67 72 77

94 208 399 620 268 432 539

EXAMPLE Class limits 45 – 49 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74 75 – 79

Class Boundaries 44.5 – 49.5 49.5 – 54.5 54.5 – 59.5 59.5 – 64.5 64.5 – 69.5 69.5 – 74.5 74.5 – 79.5

Frequency 2 4 7 10 4 6 7 ∑𝑓 = 40

𝑥 =

∑ 𝑓𝑥 2560 = 64 =  40 ∑ 𝑓 The average mass of the students is 64 kg.

x

fx 47 52 57 62 67 72 77

94 208 399 620 268 432 539 ∑𝑓x = 2560

MEDIAN • •

positional average “middle observation” when the data set is sorted (in either increasing or decreasing order).

Advantages of Median: ✦ The median is not affected by the size of extreme values but by the number of observations. ✦ The median can be calculated even when the frequency distribution contains “openended” intervals. ✦ It can also be used to define the middle of a number of objects, properties, or quantities which are not really quantitative in a nature. ✦ It can be easily interpreted

MEDIAN Md, For Ungrouped Data: •

If the number of data is odd, the median is the centermost score.

•

If the number of data is even, the median is found by computing the average of the two middle numbers

MEDIAN For Grouped Data: 𝑥 = 𝐿 +

𝑛 2 − 𝑐𝑓 𝑐𝑤 𝑓

where 𝐿 = lower limit boundary of the median class 𝑛 = total frequency 𝑐𝑓 = cumulative frequency before median class 𝑓 = frequency of median class 𝑐𝑤 = class width

The median class is the class interval with the  ( )th item. 

EXAMPLE Consider the following grades in five quizzes in Statistics of the two students:

Student A: 80, 60, 88, 75, 90 Student B: 70, 100, 85, 92, 86  

The median of Student A’s grades is 80 while the median of Student B’s grades is 86.

EXAMPLE Consider the following scores in an activity in Statistics:

75.5, 97, 95.5, 100, 93.3, 66.7, 92, 75.5, 73.3, 85.6 Arranging from lowest to highest: 66.7, 73.3, 75.5, 75.5, 85.6, 92, 93.3, 95.5, 97, 100

The median of the scores in an activity in Statistics is 88.8.

EXAMPLE Consider the table:

𝑥 = 𝐿 +

𝑛 2 − 𝑐𝑓 𝑐𝑤 𝑓

EXAMPLE

Class limits 45 – 49 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74 75 – 79

Class Boundaries 44.5 – 49.5 49.5 – 54.5 54.5 – 59.5 59.5 – 64.5 64.5 – 69.5 69.5 – 74.5 74.5 – 79.5

Frequency 2 4 7 10 4 6 7

x 47 52 57 62 67 72 77

EXAMPLE Cumulative frequency

Median Class

Class limits 45 – 49 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74 75 – 79

Class Boundaries 44.5 – 49.5 49.5 – 54.5 54.5 – 59.5 59.5 – 64.5 64.5 – 69.5 69.5 – 74.5 74.5 – 79.5

Frequency 2 4 7 10 4 6 7

Median for grouped data:

n= 40

𝑥 = 𝐿 +

 = 

𝑛 − 𝑐𝑓 2 𝑐𝑤 𝑓

20

x

cf 47 52 57 62 67 72 77 𝐿 = 59.5 𝑓 = 10 𝑐𝑓 = 13

The median class is the class containing the 20th item.

2 6 13 23 27 33 40

EXAMPLE Given: 𝐿 = 59.5  = 20  𝑐𝑓 = 13 𝑓 = 10 𝑐𝑤 = 5

Median for grouped data: 𝑥 = 𝐿 +

𝑛 − 𝑐𝑓 2 𝑐𝑤 = 59.5 + 𝑓

20 − 13 5 = 𝟔𝟑 10

The median of the mass of the students is 63 kg.

MODE •

nominal average

•

It is the most frequently occurring value in a list of data.

•

It is an appropriate measure of average for data using the nominal scale of measurement.

•

It is the only measure of central tendency used in both quantitative and qualitative data

MODE Advantages: ✦ The mode is easy to understand. ✦ Like the median, it is not greatly affected by extreme values. ✦ Like the median, it can be computed even when the frequency distribution contains “open-ended” intervals.

For ungrouped data: The mode is the most frequently occurring data (if there is one). For grouped data: 𝑥 = 𝐿 +

𝑑 𝑐𝑤 𝑑 + 𝑑

where 𝑳𝒃 = lower limit boundary of the modal class 𝒅𝟏 = difference between the highest frequency and the frequency just below it 𝒅𝟐 = difference between the highest frequency and the frequency just above it 𝒄𝒘 = class width

The modal class is the class interval with the highest frequency. If there are two class interval that contains the highest frequency, always choose the highest class interval.

MODE

EXAMPLE Class limits 45 – 49 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74 75 – 79

Class Boundaries 44.5 – 49.5 49.5 – 54.5 54.5 – 59.5 59.5 – 64.5 64.5 – 69.5 69.5 – 74.5 74.5 – 79.5

The modal class is the class interval with the highest frequency. The modal class is 60 - 64

Frequency 2 4 7 10 4 6 7

x

cf 47 52 57 62 67 72 77

2 6 13 23 27 33 40

EXAMPLE

𝒅𝟏 = 10 – 7 = 3 𝒅𝟐 = 10 – 4 = 6 𝑳𝒃 = 59.5

𝑥 = 𝐿 +

Class limits 45 – 49 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74 75 – 79

Class Boundaries 44.5 – 49.5 49.5 – 54.5 54.5 – 59.5 59.5 – 64.5 64.5 – 69.5 69.5 – 74.5 74.5 – 79.5

Frequency 2 4 7 10 4 6 7

x

𝑑 3 5 = 𝟔𝟏. 𝟐 𝑐𝑤 = 59.5 + 3+6 𝑑 + 𝑑 The mode of the mass of the students is 61.2 kg.

cf 47 52 57 62 67 72 77

2 6 13 23 27 33 40...