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Prelim (Pstat)

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    Definition of StatisticsDefinition of Statistics

    Introduction

    Statistics is a set of tools used to organize and analyze data. Data

    must either be numeric in origin or transformed by researchers into

    numbers. For instance, statistics could be used to analyze

    percentage scores English students receive on a grammar test: thepercentage scores ranging from 0 to 100 are already in numeric

    form. Statistics could also be used to analyze grades on an essay by

    assigning numeric values to the letter grades, e.g., A=4, B=3, C=2,

    D=1 and F=0.

    Employing statistics serves two purposes, (1) description and (2)

    prediction. Statistics are used to describe the characteristics of

    groups. These characteristics are referred to as variables..

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    STATISTICS

    Is the science of the collection, organization, and interpretation of

    data. It deals with all aspects of this, including the planning of data

    collection in terms of the design ofsurveys and experiments.

    A branch of applied mathematics concerned with the collection

    and interpretation of quantitative data and the use of probability

    theory to estimate population parameters

    Statistics is the science of making effective use of numerical data

    relating to groups of individuals or experiments

    Practice of collecting, organizing, describing, and analyzing data

    to draw conclusions from the data to apply to a cause.

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    Importance of StatisticsImportance of Statistics

    In Computer Engineering:

    To make systems(programs) with no flawsfor ISO certifications and process verifications

    It is used to study repetitive operations in order to setstandards.

    For data interpretation that is frequently seen in computer

    engineering.

    Example:They will make a program for data interpretation that

    is easy to use for people in the same time has no f laws.

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    InAstronomy:

    Astronomy is one of the oldest branch of statistical study, itdeals with the measurement of distance, sizes, masses anddensities of heavenly bodies by means of observations.During these measurements errors are unavoidable so most

    probable measurements are founded by using statisticalmethods.

    Example:An astronomer found 10 heavenly bodies that isthe same look. And he studied the 10 and 8 out of 10 of the

    heavenly bodies were of the same characteristics.

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    In Mathematics:

    Statistics is branch of applied mathematics. The largenumber of statistical methods like probability averages,dispersions, estimation etc. is used in mathematics anddifferent techniques of pure mathematics like integration,differentiation and algebra are used in statistics.

    Example: There are problems in math that needsanalization, interpretation, and explanation to solved theproblems.

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    In Economics:

    Statistics play an important role in economics. In economicsresearch statistical methods are used for collecting andanalysis the data and testing hypothesis. The relationshipbetween supply and demands is studies by statistical

    methods, the imports and exports, the inflation rate, the percapita income are the problems which require goodknowledge of statistics.

    Example:A sole proprietor wants to know what product is

    the most demanded today and the cost of that said product.

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    In Health and Medicine:

    edical Statistics deals with the application of biostatisticsto medicine and the health sciences, includingepidemiology, public health, forensic medicine, and clinicalresearch. Find data on indicators of the nations health, such

    as health inequalities, mobility rates, smoking drinking anddrug use, and abortion statistics.

    Example: In pharmacological research, statistics is used tosummarize (descriptive statistics) experimental data in

    terms of central tendency (mean or median) and variance(standard deviation, standard error of the mean, confidenceinterval or range) but more importantly it enables us toconduct hypothesis testing.

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    Basic Statistical TermsBasic Statistical Terms

    VARIABLE - is a quantity that may assume any of set ofvalues.Examples are monthly income, average grade, volume, priceand so forth.

    CONSTANT - is a quantity that does not change its value.Example: The mathematical symbol (Greek alphabet pi), isa constant because its value does not change which is alwaysequal to 3.1416... Likewise, the equivalence of 2.54

    centimeters to an inch is a constant.

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    TYPES OF DATA

    UNGROUPED (orRAW) DATA-which are not organized inany specific way. They are simply the collection of data asthey are gathered. While some computations may be madefor this kind of data, analysis and interpretation.

    INTERVALS NON-CUMUL. CUMULATIVELower Upper Abs. Rel. Abs. Rel.1.00 1.50 24 .169 24 .169

    1.50 2.00 24 .169 48 .3382.00 2.50 21 .148 69 .486

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    PRIMARYDATA- are measured and gathered by theresearcher that published it.

    SECONDARYDATA- are republished by another researcheror agency.

    POPULATION - is the entire collection of all possibleobservations of a particular characteristic of interest.

    Example: the population of the grades of all students who tookan entrance examination; the monthly incomes of all

    employees in the government.

    SAMPLE - is a representative set of observations that reflectthe characteristic of the whole, that is, the population fromwhich it is taken.

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    PARAMETER- is any statistical characteristic of a

    population.

    Example: the Mean and the Standard Deviation. Thus, we saythe population Mean is a parameter of the population.

    STATISTIC - is any statistical characteristic of a sample suchas the Mean and the Standard Deviation also. The SampleMean is a sample statistic.

    FREQUENCYDISTRIBUTION - is a tabulation of the valuesthat one or more variables take in a sample. Each entry in thetable contains the frequencyor count of the occurrences ofvalues within a particular group or interval, and in this waythe table summarizes the distribution of values in the

    sample.

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    For example, the heights of the students in a class could be

    organized into the following frequency table.

    Height range Number of student Cumulative number4.55.0 feet 25 255.05.5 feet 35 60

    5.56 feet 20 806.06.5 feet 20 100

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    LEVELS / SCALESOF MEASUREMENT

    Statistical operations on numerical values depend upon thenature of such values. Numerical values may be categorizedby levels of measurement namely, nominal, ordinal, intervalandratio (Seigel, Castillian, 1988).

    NOMINALLEVEL - is the crudest form of measurement. Thenumber of symbols is used for the purpose of categorizingforms into groups. The categories are mutually exclusive, that

    is being one category automatically excludes another.

    Example:Sex: M Male Faculty Tenure: 1 Tenured

    F - Female

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    ORDINALLEVEL - is a sort of improvement of nominal level.

    Data are ranked from "bottom to top" or "low to high" mannerstatements of the kind "greater than" or "less than" maybemade here

    Example:Class Standing: (Excellent, Good, Poor)Teacher's Evaluation: 1 - Poor

    2 - Fair3 - Good

    4 - Very Good

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    INTERVALLEVEL - possesses the properties of the nominal andordinal levels. The distances between any two numbers on the

    scale are known and it does not have a stable starting point.

    Example:Consider the I.Q. scores of four students, 90, 150, 85 and

    145. Here we can say that the difference between 90 and 150 is the

    same as the difference between 85 and 145, but we cannot claimthat the second student is twice as intelligent as the first.

    RATIOLEVEL - possesses all the properties of the nominal,ordinal and interval levels, in addition, this has an absolute zero

    point. Data can be classified and be placed in a proper order. Wecan compare the magnitudes of these data.

    Example: Age, income, exam scores, performance, ratings,grades of students and tuition fees as examples of ratio variables.

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    Sampling TechniquesSampling Techniques

    Sampling Technique or Sampling PlanIs the procedure of gathering sampling units from the

    population.

    Is the method of selecting a sample size (n) from auniverse (N) such that each member of the population hasan equal chance of being included in the sample and allpossible combinations of size (n) have an equal chance ofbeing selected as the sample.

    Technique of drawing sample from the population.Sampling is being applied once that the entire elements ofthe population is not available or the population size is too

    large.

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    A.ProbabilitySampling Technique is a sampling technique

    wherein each of the population unit has an equal chance ofbeing drawn or being selected as members of the sample.is a sampling technique in which the probability of getting anyparticular sample may be calculated

    1.Random SamplingIs a basic type of probability sampling. Using this

    technique, each individual in the population has an equalchance of being drawn into the sample.

    Is the method of selecting a sample size (n) from auniverse (N) such that each member of the population has anequal chance of being included in the sample and all possiblecombinations of size (n) have an equal chance of beingselected as the sample.

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    1.1.LotterySampling orRaffle Sampling

    Assigning numbers to each member of the populationusually carries out the lottery sampling method. the items areplaced in a container. All are thoroughly mixed, and elementsare drawn as needed.

    The lottery sampling method is usually carried out by

    assigning numbers to each member of the population.

    1.2.Table ofRandom NumberThe selection of each member of the population is left

    adequately to chance, and every member of the population hasan equal chance of being chosen.

    1.2.1 Direct Selection MethodIs used when there are only few sample units to be

    selected.

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    1.2.2.RemainderMethod

    Is used whenever the direct selection method cannotbe applied. There are two ways of conducting the remaindermethod:

    1. When the number taken from the Table of Random

    Numbers is subtracted from the upper limit within which thisnumber falls, the remainder is the sample unit.2. When the upper limit of the set is subtracted from thenumber taken from the Random Table and yields a numberequal or less than N, the remainder is the sample unit.

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    2.Systematic SamplingUsed to select the members of the sample from a large

    population.Picking every nth element of the population as a

    member of the sample when using this method.The most common form of systematic sampling is an

    equal-probability method, in which everykth

    element in theframe is selected, where k, the sampling interval (sometimesknown as the skip), is calculated as:

    where n is the sample size, andNis the population size.

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    3.StratifiedSampling

    In this technique, the set of interest is divided intogroups or aggregates from which the actual sampling is done.In this method, the population is subdivided into at

    least two different subpopulations (or strata) that share thesame characteristics and then the elements of the sample are

    drawn from its stratum proportionately.

    Determining sample sizeSLOVINs FORMULA

    n = ___N____1+Ne^2where:n = sample size N = populatione = margin of error

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    4.Cluster orArea Sampling

    Is a sampling wherein group or clusters instead ofindividuals are randomly chosen

    This involves dividing the population into non-overlapping clusters

    4.Cluster orArea SamplingIs a sampling wherein group or clusters instead of

    individuals are randomly chosen

    This involves dividing the population into non-overlapping clusters

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    B. Non-Probability Sampling TechniqueIs a sampling technique wherein the sample units do

    not have equal chances of being drawnis a sampling technique wherein members of the sample aredrawn from the population based on the judgment of theresearchers

    Examples of non-probability sampling include:a. Convenience or Haphazard- members of the population

    are chosen based on their relative ease of access. To samplefriends, co-workers, or shoppers at a single mall, are all

    examples of convenience.b. Snowball Sampling - The first respondent refers a friend.The friend also refers a friend, etc.

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    c. Judgmental sampling or Purposive samplingThe researcher chooses the sample based on who they

    think would be appropriate for the study. This is used primarilywhen there is a limited number of people that have expertise inthe area being researched.d. Deviant Case

    Get cases that substantially differ from the dominantpattern (a special type of purposive sample).e.Case study

    The research is limited to one group, often with a similarcharacteristic or of small size.

    f. ad hoc quotasA quota is established (say 65% women) and researchers

    are free to choose any respondent they wish as long as thequota is met.

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    Collection of DataCollection of Data

    Any statistical investigation must necessarily be basedon accurate data. In order to ensure the accuracy of data , onemust know the right sources and methods of collecting them.

    1.PrimarydataRefer to information which are gathered directly from

    an original source, or which are based on direct or first-handexperience.

    Examples:First-person accounts, autobiographies, and

    diaries.

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    2.SecondarydataRefer to information which are taken from published

    or unpublished data which were previously gathered byother individuals or agencies.Examples:

    Published books, newspapers, magazines,

    biographies, business reports, and the like.

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    Methods used in the collection of data

    The direct or interview methodThis is a method of person-to-person exchange

    between the interviewer and the interviewee. The interviewmethod provides consistent and more precise informationsince clarification may be given by the interviewee.

    The indirect or questionnaire methodWritten responses are given to prepared questions. A

    questionnaire is a list of questions which are intended to

    elicit answers to the problems of a study. This method isinexpensive and can cover a wide area in a shorter span oftime.

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    The registration methodThis method of gathering information is enforced by certain

    laws. Examples are the registration of births, deaths, motorvehicles, marriages, and licenses. The advantage of thismethod is that information is kept systematized.

    The observation method

    In this method, the investigator observes the behavior ofpersons or organizations and their outcomes. It is usuallyused when the subjects cannot talk or write

    The experiment methodThis method is used when the objective is to determine thecause and effect relationship of certain phenomena undercontrolled conditions. Scientific researchers usually use theexperiment method.

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    Types of questions

    Structured questionThis is a type of question that leaves only one way or

    few alternative ways of answering it. Here are some examplesof this type of question.

    Unstructured or open-ended questionsAs the name suggests, there are questions which can be

    answered in many ways. Probing questions or questions thatwant to elicit reasons are normally of this type.

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    Presenting a DataPresenting a Data

    Textual formPresenting a data in a paragraph form.

    Example:

    There are 100 students enrolled in the college of education.54 students came from the English major. 25 students camefrom mathematics major. And lastly 25 students came fromFilipino major.

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    Tabular fromThe data are places in a table.

    Example:Age Interval(yrs) Frequency15-19 13

    20-24 1525-29 2030-34 10

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    Graphical presentationThe data are presented in a diagrammatic form. The

    graphical representation of data makes the reading moreinteresting, less time-consuming and easily understandable.

    Kinds ofGraphical presentation:Pie Chart

    A pie chart is a way of summarizing a set of categoricaldata. It is a circle which is divided into segments. Eachsegment represents a particular category.

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    BarChartA bar chart is a way of summarizing a set of categorical

    data. It is often used in exploratory data analysis to illustratethe major features of the distribution of the data in aconvenient form. It displays the data using a number ofrectangles, of the same width, each of which represents a

    particular category. The length (and hence area) of eachrectangle is proportional to the number of cases in thecategory it represents, for example, age group, religiousaffiliation.

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    HistogramA histogram is a way of summarizing data that are

    measured on an interval scale (either discrete orcontinuous). It is often used in exploratory data analysis toillustrate the major features of the distribution of the data ina convenient form. It divides up the range of possible values

    in a data set into classes or groups.

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    FrequencydistributionFrequency distribution is a tabulation of the values

    that one or more variables take in a sample. Each entry in thetable contains the frequency or count of the occurrences ofvalues within a particular group or interval, and in this waythe table summarizes the distribution of values in the

    sample.

    Raw data is a term for data collected on source whichhas not been subjected to processing or any othermanipulation, it is also known as primary data.

    An array is a systematic arrangement of objects,usually in rows and columns.

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    The range of a sample (or a data set) is a measure ofthe spread or the dispersion of the observations. It is thedifference between the largest and the smallest observedvalue of some quantitative characteristic and is very easy tocalculate.

    The average of the values of the class limits for agiven class. A class mark is also called a mid-value or centralvalue and usually denoted by x.

    When data are continuous, class boundary are used.

    We can get the class boundary when we subtract .5 by thelover limit and adding .5 by the upper limit.

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    Frequency polygons are a graphical device forunderstanding the shapes of distributions. They serve thesame purpose as histograms, but are especially helpful incomparing sets of data. Frequency polygons are also a goodchoice for displaying cumulative frequency distributions. Tocreate a frequency polygon, start just as for histograms, by

    choosing a class interval. Then draw an X-axis representingthe values of the scores in your data. Mark the middle ofeach class interval with a tick mark, and label it with themiddle value represented by the class. Draw the Y-axis toindicate the frequency of each class. Place a point in the

    middle of each class interval at the height corresponding toits frequency. Finally, connect the points. You should includeone class interval below the lowest value in your data andone above the highest value.

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    Cumulative FrequencyCorresponding to a particular value is the sum of all

    the frequencies up to and including that value.

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    Ogive(a cumulative line graph) Is best used when you

    want to display the total at any given time. The relativeslopes from point to point will indicate greater or lesserincreases.

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    Summation NotationSummation Notation

    Is the operation of combining a sequence of numbers usingaddition; be summed may be integers, rational, number, realnumbers, or complex numbers. Besides numbers, othertypes of values can be added as well: vectors, matrices,

    polynomials, and in general elements of any additive group.

    EXAMPLES:4

    (2+i^2) = (2+1^2) + (2+2^2) + (2+3^2) +(2+4^2)i=1 = 3+6+11+18

    = 40

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    6

    (2) = 2+2+2+2+2+2i=1 =12

    5 (5-i) = (5-1) + (5-2) + (5-3) + (5-4) + (5-5)

    i=1 = 4 + 3 + 2 +1 + 0= 10

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