GENERAL GENERAL STATISTICSSTATISTICS

TABLE OF CONTENTSTABLE OF CONTENTS

Chapter 1.Chapter 1. Preliminary ConceptsPreliminary Concepts

1.11.1 Introduction and Basic ConceptsIntroduction and Basic Concepts

1.21.2 Variables and DataVariables and Data

1.31.3 SummationSummation

Chapter 2.Chapter 2. Data Collection and PresentationData Collection and Presentation

2.12.1 Data CollectionData Collection

2.22.2 Data PresentationData Presentation

2.32.3 Graphical Representation of Graphical Representation of Frequency DistributionFrequency Distribution

Chapter 3.Chapter 3. Measure of Central TendencyMeasure of Central Tendency

3.13.1 The MeanThe Mean

3.23.2 Median and ModeMedian and Mode

3.33.3 Percentiles, Deciles, and Percentiles, Deciles, and QuartilesQuartiles

Chapter 4.Chapter 4. Measure of Disfersion and Measure of Disfersion and SkewnessSkewness

4.14.1 Measure of VariabilityMeasure of Variability

4.24.2 Coefficient of VariationCoefficient of Variation

Chapter 5.Chapter 5. Permutations and CombinationsPermutations and Combinations

5.15.1 Principle of CountingPrinciple of Counting

5.25.2 PermutationsPermutations

5.35.3 CombinationsCombinations

Chapter 1Chapter 1

1.1 Introduction and 1.1 Introduction and Basic ConceptsBasic Concepts

This section aims to:This section aims to: discuss the background and the discuss the background and the

development of statistics;development of statistics; Define and differentiate the two branches Define and differentiate the two branches

of statistics; andof statistics; and Differentiate population from sample.Differentiate population from sample.

Statistical Statistical information and development information and development can be traced back from ancient times. can be traced back from ancient times. People compiled statistical data with regard People compiled statistical data with regard to all sorts of things such as agricultural to all sorts of things such as agricultural crops, athletic events, commerce and trade crops, athletic events, commerce and trade and so on. As time went by, statistical work and so on. As time went by, statistical work has continued to have a marked influence has continued to have a marked influence on the activities of mankind in a wider on the activities of mankind in a wider scope from describing important features of scope from describing important features of the data and analyzing them.the data and analyzing them.

StatisticsStatistics A science of conducting studies to collect, A science of conducting studies to collect,

organize, summarize, analyze, and draw organize, summarize, analyze, and draw conclusion from data; interpreting and conclusion from data; interpreting and presenting numerical data.presenting numerical data.

Can refer to the mere tabulation of Can refer to the mere tabulation of numeric information as in reports of stock, numeric information as in reports of stock, market, transactions, or to the body of market, transactions, or to the body of techniques used in processing or analyzing techniques used in processing or analyzing data.data.

DataData

Data are the raw material which the Data are the raw material which the statistician works. Data can be found statistician works. Data can be found through surveys, experiments, numerical through surveys, experiments, numerical records, and other modes of research.records, and other modes of research.

StatisticianStatistician

Statistician is also used in several ways. It Statistician is also used in several ways. It can be a person who simply collects can be a person who simply collects information or one who prepares analysis information or one who prepares analysis or interpretations. It may mean a scholar or interpretations. It may mean a scholar who develops a mathematical theory on who develops a mathematical theory on which the science of statistics is based.which the science of statistics is based.

Two Branches of Two Branches of StatisticsStatistics

Statistics can be organized into Statistics can be organized into descriptive statisticsdescriptive statistics and and inferential inferential statisticsstatistics..

Descriptive StatisticsDescriptive Statistics

Concerned with collecting, Concerned with collecting, organizing, presenting, and analyzing organizing, presenting, and analyzing numerical data.numerical data.

Inferential StatisticsInferential Statistics

Its main concern is to analyze the Its main concern is to analyze the organized data leading to prediction organized data leading to prediction or inferences.or inferences.

The word “population” and “sample” are the The word “population” and “sample” are the most commonly used words associated most commonly used words associated with statistics.with statistics.

PopulationPopulation

Refers to the groups or aggregates of Refers to the groups or aggregates of people, objects, materials, events or people, objects, materials, events or thing of any form.thing of any form.

SampleSample

Consist of few or more members of Consist of few or more members of the population.the population.

1.2 Variables and Data1.2 Variables and Data

This section aims to:This section aims to: Differentiate the two types of variables;Differentiate the two types of variables; Identify and illustrate the two areas of Identify and illustrate the two areas of

quantitative variables;quantitative variables; Enumerate the classifications of data; andEnumerate the classifications of data; and Apply the types of variables in various Apply the types of variables in various

fields of applications.fields of applications.

Statistical data or information can be Statistical data or information can be gathered through different ways such as gathered through different ways such as interviewing people, observing or interviewing people, observing or inspecting items, using questionnaires and inspecting items, using questionnaires and checklists. The characteristic that is being checklists. The characteristic that is being studied is called a variable. It varies from studied is called a variable. It varies from one person or thing to another. one person or thing to another.

Examples of variables for people are Examples of variables for people are heightheight, , weightweight, , ageage, , sexsex, , marital statusmarital status, , eye coloreye color, etc. , etc. The first three of the given The first three of the given variables yield numerical values and are variables yield numerical values and are examplesexamples of of quantitative variablesquantitative variables. . The last three yield non-numerical values The last three yield non-numerical values or attributes are or attributes are examplesexamples of of qualitative qualitative variables.variables.

Qualitative Variables Qualitative Variables are further classified are further classified as either as either discretediscrete or or continuouscontinuous. A . A discrete variable is a variable whose values discrete variable is a variable whose values can be counted using integral values such as can be counted using integral values such as the the number of enrolleesnumber of enrollees, , drop-outsdrop-outs, , graduates in a certain collegegraduates in a certain college, , deathsdeaths, and , and number of employeesnumber of employees. A . A continuous continuous variable variable is a variable that can assume any is a variable that can assume any numerical value over an interval or intervals. numerical value over an interval or intervals. HeightHeight, , weightweight, , temperaturetemperature, and , and timetime are are examplesexamples of of continuous variablescontinuous variables..

A variable can be A variable can be dependent dependent or or independentindependent depending on its use. To predict the value of depending on its use. To predict the value of variable on the other, variable on the other, independent variable independent variable is the predictor while the is the predictor while the dependent variable dependent variable is the variable whose value is being predictedis the variable whose value is being predicted. .

For exampleFor example, to predict the value of sunlight , to predict the value of sunlight on the growth of a certain plants, the on the growth of a certain plants, the dependent variable is the growth of the plant dependent variable is the growth of the plant while the independent variable is the amount while the independent variable is the amount of sunlight exposed to the plant.of sunlight exposed to the plant.

Scales of Scales of Measurement of Measurement of

DataData

Nominal DataNominal Data

Use numbers for the purpose of identifying Use numbers for the purpose of identifying name or membership in a group or name or membership in a group or category.category.

Ordinal Data Ordinal Data

Connote ranking or inequalities in this type Connote ranking or inequalities in this type of data, numbers represents “greater of data, numbers represents “greater than” or “less than” measurement, such as than” or “less than” measurement, such as preferences or rankings.preferences or rankings.

Interval Data Interval Data

Indicate an actual amount and there is Indicate an actual amount and there is equal unit of measurement separating equal unit of measurement separating each score, specifically equal intervals. each score, specifically equal intervals. The true zero is present.The true zero is present.

Ratio DataRatio Data

Similar to interval data but has an Similar to interval data but has an absolute zero and multiples are absolute zero and multiples are meaningful. It include all the usual meaningful. It include all the usual measurement of length, height, weight, measurement of length, height, weight, area, volume, density, velocity, money and area, volume, density, velocity, money and duration.duration.

1.3 Summation1.3 Summation

This section aims to:This section aims to: Introduce a special notation that will work Introduce a special notation that will work

as a shortcut for expressing sum of terms as a shortcut for expressing sum of terms and thereby appreciate mathematics as a and thereby appreciate mathematics as a tool of symbols; andtool of symbols; and

State and analyze the properties of State and analyze the properties of summation.summation.

When dealing with a sum of terms, we shall have When dealing with a sum of terms, we shall have occasions to use an abbreviated form. This special occasions to use an abbreviated form. This special symbol for writing of sums is called symbol for writing of sums is called summation.summation.

SummationSummation is denoted by ∑, is defined as is denoted by ∑, is defined as nn

∑∑ xxii=x=x11+x+x22+…+x+…+xnn

i=1i=1

ss

Where 1 and n are called the lower and Where 1 and n are called the lower and upper limits respectively. We note that x1, upper limits respectively. We note that x1, is read as “x sub 1”is read as “x sub 1”

Chapter 2Chapter 2

2.1 Data Collection2.1 Data Collection

This section aims to:This section aims to: Identify, compare and contrast the Identify, compare and contrast the

different types of data;different types of data; List and explain the various techniques of List and explain the various techniques of

selecting a sample; andselecting a sample; and Enumerate and illustrate the different Enumerate and illustrate the different

sampling techniques.sampling techniques.

Types of DataTypes of Data

Primary DataPrimary Data - data collected directly by - data collected directly by the researcher himself. These are first-the researcher himself. These are first-hand or original sources.hand or original sources.

They can be collected through the ff:They can be collected through the ff:1.1. Direct observation or measurement Direct observation or measurement

(primary source of info).(primary source of info).2.2. By interview (questionnaires or rating By interview (questionnaires or rating

scales).scales).3.3. By mail of recording or of recording forms.By mail of recording or of recording forms.4.4. Experimentation.Experimentation.

Secondary DataSecondary Data

Are information taken from published Are information taken from published or unpublished materials previously or unpublished materials previously gathered by other researchers or gathered by other researchers or agencies such as book, newspapers, agencies such as book, newspapers, magazines; journals, published and magazines; journals, published and unpublished thesis and dissertations.unpublished thesis and dissertations.

Two types of Sampling Two types of Sampling Technique:Technique:

Probability Sampling -Probability Sampling - every unit every unit has a chance of being selected and has a chance of being selected and that chance can be qualified.that chance can be qualified.

Non-Probability SamplingNon-Probability Sampling - every - every item in a population does not have item in a population does not have an equal chance of being selected.an equal chance of being selected.

Sampling TechniqueSampling Technique

Procedure in selecting the numbers of Procedure in selecting the numbers of samples from the entire population.samples from the entire population.

Different Types Different Types of Sampling of Sampling TechniquesTechniques

Simple Random SamplingSimple Random Sampling

It is recommended to prevent the It is recommended to prevent the possibility of a bias or erroneous inference. possibility of a bias or erroneous inference. Under the concept of randomness, each Under the concept of randomness, each member of the population has an equal member of the population has an equal chance to be included in the sample chance to be included in the sample gathered.gathered.

Systematic Random SamplingSystematic Random Sampling

The items or individuals are arranged The items or individuals are arranged in some way perhaps alphabetically in some way perhaps alphabetically or other sort.or other sort.

Stratified Random Stratified Random Sampling Sampling

In this type of planning a population is first In this type of planning a population is first divided into subsets based on homogenity divided into subsets based on homogenity called Strata. The Strata are internally called Strata. The Strata are internally homogenous as possible and at the same homogenous as possible and at the same time each stratum is different from one time each stratum is different from one another as much as possible.another as much as possible.

Cluster SamplingCluster Sampling

Can be done by subdividing the population Can be done by subdividing the population into smaller units and then selecting only into smaller units and then selecting only a random some primary units where the a random some primary units where the study would then be concentrated if study would then be concentrated if sometimes referred are sampling because sometimes referred are sampling because it is frequently applied on a geographical it is frequently applied on a geographical basisbasis

2.2 Data Presentation2.2 Data Presentation

This section aims to:This section aims to: Summarize and present data in different Summarize and present data in different

forms;forms; Arrange and organize the raw data into a n Arrange and organize the raw data into a n

array and construct the frequency array and construct the frequency distribution, stem and lead diagram; anddistribution, stem and lead diagram; and

Define, illustrate, and solve for the class Define, illustrate, and solve for the class limits, class boundaries and class marks.limits, class boundaries and class marks.

Methods in Presenting Methods in Presenting DataData

Textual Form Textual Form - data in paragraph form.- data in paragraph form.

Tabular FormTabular Form - systematic arrangement of - systematic arrangement of data in rows and columns.data in rows and columns.

Graphical FormGraphical Form - a graph or chart is a - a graph or chart is a device for showing numerical values in device for showing numerical values in pictorial form.pictorial form.

Semi Tabular/Semi Tabular FormSemi Tabular/Semi Tabular Form - the - the combination of Textual and Tabular Form.combination of Textual and Tabular Form.

Stem and Leaf DiagramStem and Leaf Diagram

Raw data Raw data are data collected in an are data collected in an investigation and they are not organized investigation and they are not organized systematically. systematically. Raw data Raw data that are that are presented in the form of a frequency presented in the form of a frequency distribution are called distribution are called grouped datagrouped data..

There are There are two methods of organizing two methods of organizing the raw datathe raw data – setting up an array and – setting up an array and stem-and leaf diagram.stem-and leaf diagram.

For For exampleexample, a nationwide travel agency , a nationwide travel agency offers special rates for package tours during offers special rates for package tours during summer. To economize spending for the summer. To economize spending for the advertisement only certain age group of advertisement only certain age group of people will be sent brochures for attraction. people will be sent brochures for attraction. The agency gets to previous passenger The agency gets to previous passenger customers from its files and groups them customers from its files and groups them according to ages. Only those age groups according to ages. Only those age groups with least people are sent brochures. The with least people are sent brochures. The following are the ages of the previous following are the ages of the previous customers:customers:

Example:Example:

5959 50 50 5252 3838 8080 6262 7777 5656

60 6160 61 5858 6262 5151 3636 5454 1818

7171 54 54 4444 5252 2626 6363 5858 5656

41 3441 34 6161 5050 6060 5353 6262 6262

5353 43 43 6363 7171 6565 7979 4545 6666

I. Setting up an array from I. Setting up an array from the largest to the smallestthe largest to the smallest

8080 7979 7777 7171 7171 6666 6666 6666

6363 6363 6262 5252 5252 5252 6161 6161

6060 6060 5959 5858 5858 5555 5454 5454

5353 5353 5252 5252 5050 5050 5050 4545

4444 4343 4141 3838 3636 3434 2626 1818

II. An II. An array array from the from the smallest to the largestsmallest to the largest

1818 2626 3434 3636 3838 4141 4343 4444

4545 5050 5050 5151 5252 5252 5353 5353

5454 5454 5555 5858 5858 5959 6060 6060

6161 6161 6262 6262 6262 6262 5353 5353

6666 6666 6666 7171 7171 7777 7979 8080

III. Setting up into III. Setting up into stem-and-leaf diagramstem-and-leaf diagram

11 88

22 66

33 44 66 88

44 11 33 44 55

55 00 00 11 2 22 2 3 43 4 4 5 8 84 5 8 8

66 00 00 11 1 21 2 2 22 2 3 3 6 63 3 6 6

77 11 11 77 99

88 00

Tally MethodTally Method

CLASS LIMITCLASS LIMIT TALLYTALLY ff CLASS CLASS BOUNDARYBOUNDARY

80-8980-89 II 11 79.5-89.579.5-89.5

70-7970-79 IIIIIIII 44 69.5-79.569.5-79.5

60-6960-69 IIIII-IIIII-IIIIIIII-IIIII-III 1313 59.5-69.559.5-69.5

50-5950-59 IIIII-IIIII-IIIIIIII-IIIII-III 1313 49.5-59.549.5-59.5

40-4940-49 IIIIIIII 44 39.5-49.539.5-49.5

30-3930-39 IIIIII 33 29.5-39.529.5-39.5

20-2920-29 II 11 19.5-29.519.5-29.5

10-1910-19 II 11 9.5-19.59.5-19.5

n=40n=40

2.3 Graphical 2.3 Graphical Representation of Representation of

Frequency DistributionFrequency Distribution

This section aims to:This section aims to: Define and illustrate histograms, Define and illustrate histograms,

frequency polygon, ogives and pie graphs;frequency polygon, ogives and pie graphs; Portray and apply the distribution of data Portray and apply the distribution of data

in various graphs such as histogram, in various graphs such as histogram, frequency polygon, and a cumulative frequency polygon, and a cumulative frequency polygon.frequency polygon.

Graphical forms Graphical forms of presenting of presenting information is often more helpful in information is often more helpful in making a stronger impact. There are making a stronger impact. There are some some featuresfeatures in tabular form, which can’t in tabular form, which can’t be discerned simply by looking at raw be discerned simply by looking at raw data.data.

Graphical Representation Graphical Representation of Frequency of Frequency DistributionDistribution

Frequency HistogramFrequency Histogram

It is a bar graph that displays the It is a bar graph that displays the classes or the horizontal axis and the classes or the horizontal axis and the frequency of the classes on the frequency of the classes on the vertical axis.vertical axis.

Frequency HistogramFrequency Histogram

0

2

4

6

8

10

12

14

14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5

Series1

Frequency PolygonFrequency Polygon

It is a line chart that is constructed It is a line chart that is constructed by plotting the frequencies and class by plotting the frequencies and class mark and connecting the plotted mark and connecting the plotted pointed by means of a straight line; pointed by means of a straight line; the polygon us closed by considering the polygon us closed by considering an additional class at each end and an additional class at each end and each end of the lines are brought each end of the lines are brought down to the horizontal axis at the down to the horizontal axis at the mid point of the additional classes.mid point of the additional classes.

Frequency PolygonFrequency Polygon

0

2

4

6

8

10

12

14

14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5

Series1

OgiveOgive

It is a graph of a cumulative It is a graph of a cumulative frequency distribution and frequency distribution and sometimes called a cumulative sometimes called a cumulative frequency distribution graph.frequency distribution graph.

OgiveOgive

0

5

10

15

20

25

30

35

40

45

9.5 19.5 29.5 39.5 49.5 59.5 69.5 79.5

Series1

Series2

Pie ChartPie Chart

It is a graphical presentation that It is a graphical presentation that uses circle or pie.uses circle or pie.

Pie ChartPie Chart

3636

9

9

117

117

2736

36

9

9

117

117

27

Chapter 3Chapter 3

3.1 The Mean3.1 The Mean

This section aims to:This section aims to: State and illustrate the definition of the State and illustrate the definition of the

mean both for grouped and raw data mean both for grouped and raw data (ungrouped);(ungrouped);

Apply the shortcut formula for calculating Apply the shortcut formula for calculating the mean.the mean.

The most commonly used measure of The most commonly used measure of central tendency is the central tendency is the meanmean. When . When taking an average, it is the taking an average, it is the meanmean that is that is often referring to.often referring to.

This section is divided into two: This section is divided into two: the mean the mean for ungrouped data for ungrouped data andand the mean for the mean for grouped datagrouped data

MEASURES OF CENTRAL MEASURES OF CENTRAL TENDENCYTENDENCY

-single number represent the given data.-single number represent the given data.

1.1.MeanMean – average value of the given data. – average value of the given data.

- not appropriate measures of - not appropriate measures of central tendency if there is central tendency if there is outerouter..

2. 2. MedianMedian – divide the distribution into two – divide the distribution into two equal parts (upper 50% and the lower equal parts (upper 50% and the lower 50%)50%)

3. 3. ModeMode – the most frequent occuring data. – the most frequent occuring data.

- nominal value/part.- nominal value/part.

UNGROUPED DATAUNGROUPED DATA

25 32 41 58 78 9 5 105 110 112 112 25 32 41 58 78 9 5 105 110 112 112 115115

Mean = Mean = ∑ X∑ X

nn

= = 883883

1111

= 80.2727= 80.2727

GROUPED DATAGROUPED DATA

Short MethodShort Method

Mean = AM (∑fd/n) iMean = AM (∑fd/n) i

Long MethodLong Method

Mean = Mean = ∑fx∑fx

nn = = 22502250

4040

Mean = 56.25Mean = 56.25

3.2 Median and Mode3.2 Median and Mode

This section aims to:This section aims to: Differentiate the three principal Differentiate the three principal

measurements of central tendency;measurements of central tendency; Apply the computations of the median and Apply the computations of the median and

mode in various sets of datamode in various sets of data

B. B. MedianMedian - is the middle measure in - is the middle measure in a set of measures arranged in order a set of measures arranged in order magnitude. If the total number of magnitude. If the total number of measure is given by the average of measure is given by the average of two middle measures. Thus, in the two middle measures. Thus, in the median, half the distribution lies median, half the distribution lies above it.above it.

Mode = 112Mode = 112 if in case of two mode, it is called if in case of two mode, it is called

bimodal.bimodal. if no mode, there is no pair of data.if no mode, there is no pair of data.

C. C. ModeMode - is the item or measure which - is the item or measure which occurs most often. It has the highest occurs most often. It has the highest number of frequency.number of frequency.

ASSUMED MEANASSUMED MEAN

Mean = AM+ (∑fd/n)iMean = AM+ (∑fd/n)i

= 64.5+ (-33/40)10= 64.5+ (-33/40)10

= 64.5-8.25= 64.5-8.25

Mean = 56.25Mean = 56.25

MedianMedian

Median = LL+Median = LL+(n/2-<cf)(n/2-<cf) i i

ff = LL + (20-9/13)10= LL + (20-9/13)10

= 49.5 + (11/13) 10= 49.5 + (11/13) 10

= 49.5 + 8.4615= 49.5 + 8.4615

Median =57.9615Median =57.9615

ModeMode

Mode= LL + (∆Mode= LL + (∆11/ ∆/ ∆11+ ∆+ ∆22) i) i

*where ∆1 = difference between the *where ∆1 = difference between the modal class and the next lower modal class and the next lower score. score.

∆ ∆2 = difference between the 2 = difference between the modal class and the next upper modal class and the next upper score.score.

3.3 Percentiles, Deciles, 3.3 Percentiles, Deciles, and Quartilesand Quartiles

This section aims to:This section aims to: Define, illustrate, and distinguish Define, illustrate, and distinguish

percentiles, deciles, and quartiles; andpercentiles, deciles, and quartiles; and Discuss the formulas of percentiles, Discuss the formulas of percentiles,

deciles, and quartiles.deciles, and quartiles.

Measure of LocationMeasure of Location

Position/LocationPosition/Location

QUARTILE (Q)QUARTILE (Q)

QQ11- 25%- 25% 1/21/2

QQ22- 50%- 50% 1/21/2

QQ33- 75%- 75% 3/43/4

DECILE (D)DECILE (D)

DD11 -10% -10% DD22– 20%– 20% DD33– 30%– 30% DD44- 40%- 40% DD55- 50%- 50% DD66- 60%- 60% DD77- 70%- 70% DD88- 80%- 80% DD99- 90%- 90%

PERCENTILE (P)PERCENTILE (P)

PP11- 1/100- 1/100PP22- 2/100- 2/100PP33- 3/100- 3/100PP44- 4/100- 4/100PP55- 5/100- 5/100

… …....PP9999- 99/100- 99/100

UNGROUPED DATAUNGROUPED DATA

85 92 105 118 126 149 165 189 205 210 22085 92 105 118 126 149 165 189 205 210 220

QQ11 : 0.25n = 0.25(11) = 2.75 : 0.25n = 0.25(11) = 2.75

d = 105-92 = 13d = 105-92 = 13

c = 13(0.75) = 9.75c = 13(0.75) = 9.75

QQ11= 92=9.75= 92=9.75

QQ11= 101.75= 101.75

QQ33 : 0.75n = 0.75(11) = 8.25 : 0.75n = 0.75(11) = 8.25

d = 205-189 = 16d = 205-189 = 16

c = 16(0.25) = 4c = 16(0.25) = 4

QQ33= 189+4= 189+4

QQ33= 193= 193

PP33 : 0.3n = 0.3(11) = 8.25 : 0.3n = 0.3(11) = 8.25

d = 118-105= 13d = 118-105= 13

c = 13(0.3) = 3.9c = 13(0.3) = 3.9

PP33= 105+3.9= 105+3.9

PP33= 108.9= 108.9

GROUPED DATAGROUPED DATA

C.I.C.I. ff <cf<cf XX <cf<cf f/nf/n sectorsector

80-8980-89 11 4040 84.584.5 11 0.00250.0025 9percen9percentt

70-7970-79 44 3939 74.574.5 55 0.10000.1000 3636

60-6960-69 1313 3535 64.564.5 1818 0.32500.3250 117117

50-5950-59 1313 2222 54.554.5 3131 0.32500.3250 117117

40-4940-49 44 99 44.544.5 3535 0.10000.1000 3636

30-3930-39 33 55 34.534.5 3838 0.07500.0750 2727

20-2920-29 11 22 24.524.5 3939 0.02500.0250 99

10-1910-19 11 11 14.514.5 4040 0.02500.0250 99

n=40n=40 ∑∑rf=1rf=1

QQ11 = LL + n/4 - <cf i = LL + n/4 - <cf i

ff

= 49.5 + 10-9 10= 49.5 + 10-9 10

1313

= 4905 + (1/13) 10= 4905 + (1/13) 10

= 4905 + 0.7692= 4905 + 0.7692

QQ11 = 50.2692 = 50.2692

QQ33 = LL + 3n/4 - <cf i = LL + 3n/4 - <cf i

ff

= 59.5 + 30-22 10= 59.5 + 30-22 10

1313

= 59.5 + (80/13) 10= 59.5 + (80/13) 10

= 59.5 + 6.1538= 59.5 + 6.1538

QQ33 = 65.6538 = 65.6538

DD22 = LL + 0.2n - <cf i = LL + 0.2n - <cf i

ff

= 39.5 + 8-5 10= 39.5 + 8-5 10

44

= 39.5 + (30/4) 10= 39.5 + (30/4) 10

= 39.5 + 7.5= 39.5 + 7.5

DD2 2 = 47 = 47

PP2323 = LL + 0.23n - <cf i = LL + 0.23n - <cf i ff = 49.5 + 92.2 - 9 10= 49.5 + 92.2 - 9 10

1313 = 49.5 + (0.2/13) 10= 49.5 + (0.2/13) 10

= 49.5 + (0.0153)10= 49.5 + (0.0153)10

= 49.5 + 0.1538= 49.5 + 0.1538

PP2323 = 49.6538 = 49.6538

QQ33 = LL + 3n/4 - <cf i = LL + 3n/4 - <cf i

ff

= 59.5 + 30-22 10= 59.5 + 30-22 10

1313

= 59.5 + (80/13) 10= 59.5 + (80/13) 10

= 59.5 + 6.1538= 59.5 + 6.1538

QQ33 = 65.6538 = 65.6538

MEASURE OF VARIABILITY OR MEASURE OF VARIABILITY OR DISPERSIONDISPERSION

Measure of the scatteredness of a Measure of the scatteredness of a particular data in a given data set.particular data in a given data set.

Average of distanceAverage of distance

1. Range = H.S. – L.S.1. Range = H.S. – L.S. C.L. RangeC.L. Range 80-89 8905-9.5 = 8080-89 8905-9.5 = 80

2. Mean Average Deviation2. Mean Average Deviation - takes into account all the variables in a - takes into account all the variables in a

given distribution.given distribution.

FORMULA FOR FINDING MEAN FORMULA FOR FINDING MEAN AVERAGE DEVIATION:AVERAGE DEVIATION:

MAD = ∑|x-x|MAD = ∑|x-x|

nn

3. Standard Deviation 3. Standard Deviation - the most commonly used in measures of variability- the most commonly used in measures of variability

UNGROUPED DATAUNGROUPED DATASample SDSample SD1. SD= ∑(x-x)1. SD= ∑(x-x)2 2. 2 2. SD= ∑xSD= ∑x22 – (x) – (x)2 2

n-1n-1 n-1 n-1

Population SDPopulation SD = ∑x-m)= ∑x-m)22

NN

GROUPED DATA:GROUPED DATA:

SD = i ∑f(dSD = i ∑f(d11))22 - ∑fd - ∑fd1 21 2

n nn n

4. 4. Quartile DeviationQuartile Deviation

- semi- center quartile range.- semi- center quartile range.

- represent mid-point of middle part - represent mid-point of middle part of a distribution. of a distribution.

FORMULA:FORMULA:

UNGROUPED DATA:UNGROUPED DATA:

QD = Q3 – Q1QD = Q3 – Q1

2 2

Chapter 4Chapter 4

MEASURE OF VARIABILITYMEASURE OF VARIABILITY

COEFFICIENT OF VARIATIONCOEFFICIENT OF VARIATION Coefficient of Variation denoted by CV Coefficient of Variation denoted by CV

allows the variability of scores in 2 sets of allows the variability of scores in 2 sets of data that do not necessarily measures the data that do not necessarily measures the same thing.same thing.

The one who got highest scores is the one The one who got highest scores is the one who needs improvement.who needs improvement.

FORMULA:

CV SD x 100%x

1 2 3 4 5 6 7 8 9 10

Coke 8 10 2 8 9 5 8 6 8 10

Pepsi 9 8 1 10 9 3 7 8 8 10

Example:

10- Highest1- Lowest

x= 7.4 (coke)x= 7.4 (coke)xx xx x-xx-x |x-x| |x-x| (x-x)(x-x)22 (x)(x)22

88 7.47.4 .6.6 .6.6 .36.36 6464

1010 7.47.4 2.62.6 2.62.6 6.766.76 100100

22 7.47.4 5.45.4 5.45.4 29.1629.16 44

88 7.47.4 .6.6 .6.6 .36.36 6464

99 7.47.4 1.61.6 1.61.6 2.562.56 8181

55 7.47.4 -2.4-2.4 2.42.4 5.765.76 2525

88 7.47.4 .6.6 .6.6 .36.36 6464

66 7.47.4 -1.4-1.4 1.41.4 1.961.96 3636

88 7.47.4 .6.6 .6.6 .36.36 6464

1010 7.47.4 2.62.6 2.62.6 6.766.76 100100

∑∑|x-x|=18.4|x-x|=18.4 ∑∑(x-(x-x)2=54.4x)2=54.4

∑∑(x)(x)22=60=6022

x= 7.3 (pepsi)x= 7.3 (pepsi)

xx xx x-xx-x |x-x| |x-x| (x-x)(x-x)22 (x)(x)22

99 7.37.3 1.71.7 1.71.7 2.872.87 8181

88 7.37.3 0.70.7 0.70.7 0.490.49 6464

11 7.37.3 -6.3-6.3 6.36.3 39.6939.69 11

1010 7.37.3 2.72.7 2.72.7 7.297.29 100100

99 7.37.3 1.71.7 1.71.7 2.872.87 8181

33 7.37.3 -4.3-4.3 4.34.3 18.4718.47 99

77 7.37.3 -0.3-0.3 0.30.3 0.090.09 4949

88 7.37.3 0.70.7 0.70.7 0.490.49 6464

88 7.37.3 0.70.7 0.70.7 0.490.49 6464

1010 7.37.3 2.72.7 2.72.7 7.297.29 100100

∑∑|x-x|=|x-x|= ∑∑(x-x)(x-x)22=80.1=80.1

Coke

CV= 204585 x 100% 7.4

CV = SD x 100% x

SD= ∑(x-x)2

n-1

= 54.4 9

= √6.04

SD= 2.4585CV= 33.2229%

SD= ∑(x-x)2

n-1

Pepsi

= 80.1

10-1

= 80.1

9

= √ 8.9

SD = 2.9833

SD2= 8.9

DECISION:

Pepsi needs more

improvement than coke in terms of

taste

A distribution of 2 different units is given to compare in dispersion of heights versus in dispersion of weights. The mean height is 5.70 feet with SD = 0.9 ft. The mean weight is 72.5 kg with SD = 801 kg. Compare the dispersion in heights and in weights.

CV = SD x 100% x

= 0.9 x 100% 5.7

= 0.15789

= 15.7985%

CV = SD x 100% x

= 8.1 x 10% 72.5

= 0.111724

= 11.1724%

HEIGHTS WEIGHTS

MEASURE OF SKEWNESSMEASURE OF SKEWNESS

Degree of symmetry or departure from symmetry.Degree of symmetry or departure from symmetry.

FORMULA:FORMULA:

1. SK1 = x-x 2. SK2 = 3(x-x)1. SK1 = x-x 2. SK2 = 3(x-x)

SDSD SD SD

3.SK3 = Q3 – 2Q2 + Q1 Q3 – Q1

4. SK4 = P90 – 2P50 + P10 P90 – P10

5. SK5 = ∑f(x-x)3 n(SD)3

UNGROUPED DATA

GROUPED DATA

SK5 = ∑f(x-x)3 n(SD)3

Negatively Skewed Distribution (all negative)

Positively Skewed Distribution (all positive)

Normal Distribution = 0

Measure of KurtosisMeasure of Kurtosis

It is the degree of peakednessIt is the degree of peakedness

FORMULA:FORMULA:

Ungrouped DataUngrouped Data GroupedGrouped

K = ∑(x-x)4K = ∑(x-x)4 K= ∑f(x-x)4 K= ∑f(x-x)4

n(SD)4n(SD)4 n(SD)4 n(SD)4

Leptokurtic Distribution

Mesokurtic Distribution

Platykurtic DistributionPlatykurtic Distribution

Chapter 5Chapter 5

5.1 PRINCIPLE OF COUNTING5.1 PRINCIPLE OF COUNTING

This section aims to:This section aims to:State and illustrate the principle of State and illustrate the principle of

counting;counting;Diagram the computations involving Diagram the computations involving

the principle of counting; andthe principle of counting; andApply the principle of counting in Apply the principle of counting in

various area of problem solving.various area of problem solving.

Principle of CountingPrinciple of Counting

If a choice of 2 steps of which the If a choice of 2 steps of which the first can be made in n1 ways and a first can be made in n1 ways and a second can be made in n2 ways, then second can be made in n2 ways, then the whole choice can be made by n1 the whole choice can be made by n1 n2 ways.n2 ways.

1. In a class of 20 the # of ways selecting president, Vice-President, Secretary, treasurer is 20 . 19 . 18 . 17 = 116280

2.Certain government employees are classified into 2 categoriesSex: (male, female) Marital Status : (single. Married, widow, separated)2 . 4 = 8

EXAMPLES:

GENERALIZATION OF GENERALIZATION OF PRINCIPLE OF COUNTINGPRINCIPLE OF COUNTING

If a choice has k steps of which the If a choice has k steps of which the first can be made ian N1 ways, of first can be made ian N1 ways, of which each of these 2which each of these 2ndnd can be made can be made in n2 ways…. 3in n2 ways…. 3rdrd of which of these of which of these kth can be made… in nk which then kth can be made… in nk which then the whole choice can be made by n1 the whole choice can be made by n1 . n2 . nk(ways). n2 . nk(ways)

EXAMPLES:

1. A test is compose of a 10 multiple question with each having four(4) possible answers.

4 . 4 . 4 . 4 . 4 . 4 . 4 . 4 . 4 . 4 = 1,048,576

2. How many nos, of five(5) digits each can be made from the digit 1-9 if:

a. No. must be oddb. The last two(2) digit each no. are even number.

• Repetition is not allowed

1 ,2 ,3 , 4 , 5

5.2 Permutations5.2 Permutations

This section aims to:This section aims to: Define and illustrate permutationsDefine and illustrate permutations Apply permutations in various situational Apply permutations in various situational

conditions; andconditions; and State and illustrate the circular State and illustrate the circular

permutation.permutation.

PERMUTATIONSPERMUTATIONS

Arrangement of group of things in a Arrangement of group of things in a definite order that is, there is a 1definite order that is, there is a 1stst element, 2element, 2ndnd element, 3 element, 3rdrd element etc. In element etc. In other words, the order of arrangement of other words, the order of arrangement of an element is important.an element is important.

EXAMPLES:

1. In how many ways can the five(5) starting position on the PBA team with 12 mean who can play any of the position.

12P5 = 12! = 12! = 12 .11 .10 .9 . 8 . 7 . 6 . 5 .4 .3 .2 .1 = 95,040

(12-5) 7! 7! 2. How many permutation can be made from the letter

of q word Sunday?a. If the four(4) letters are use at a time. 6P4 = 6! = 6! = 360

(6-4)! 2! b. All letters are used

6P6 = 6! = 720 6!

FORMULAS:

1ST Formula:• the number if permutation n things taken n at a time is nPn=n!

CIRCULAR PERMUTATIONCIRCULAR PERMUTATION

The permutation that occur by arranging The permutation that occur by arranging objects in a circle are called circular objects in a circle are called circular permutation…permutation…

P = (n-1)!P = (n-1)!

COMBINATIONCOMBINATION

A combination also concerns arrangement A combination also concerns arrangement but without regards to order. This means but without regards to order. This means that the order or arrangement in which the that the order or arrangement in which the element are taken is not important.element are taken is not important.

nCr = n!nCr = n!

r!(n-1)!r!(n-1)!

The EndThe End

Thank You!Thank You!

Presenters:

Mary Ann FrogosaMary Ann MosquerraBOA IV-1