Statistics (Chapter - 1)

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Statistics

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A science which deals with the collection ; tabulation or presentation; analysis; and interpretation of data

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-Defining the Problem-Collecting and Assembling Relevant Information-Conducting an Original investigation of the problem-Classifying the data-presenting the data and analyzing -interpreting the results

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6 Steos in Statistical Analysis

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3800 BC

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Babylonia

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3000

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China

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1491 BC

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Moses

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Achenwall

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-1719 - 1772

-first introduced the word statistics

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Zimmermann and Sinclair

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popularized the name statistics in their books

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16th Century

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European became interested in coolecting information

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Girotamo Cardano

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wrote the first known study of the principles of probability

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Mathematicians requested by gamblers

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-Pascal

-Leibnitz

-Fermat

-Bernoulli

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17th Century

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companies begin to thrive and were already compiling

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De Moivre

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discovered the equation for the normal distribution upon which many of the theories of inferential statistics have been based

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Adolf Quetelet

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-Belgian

-Father of Modern Statistics

-applied the theory of probability to anthropological measurements and expanded the same principle to the field of psychology and education

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Sir Francis Galton

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developed the use of percentiles and contributed the application if statistics to heredity

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Karl Pearson

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worked with Galton to develop the theory of regression and correlation

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William S. Gosset

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develop methods for decision-making derived from smaller sets of data

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Sir Ronald Fisher

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Fisher's Test

-used in the analysis of variance in inferential statisticd

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Division of Statistics

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-Descriptive Statistics

-Inferential Statistics

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Descriptive Statistics

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concerned with the gathering, classification and presentation of data and the collection of summarizing values to describe group characteristics

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Tpes of Sumamrizing Values Most COmmonly USed

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-Measure of Central Tendency or POsition or Location

-Measure of Variability

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Measure of Central Tendency or POsition or Location

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a single figure which is representative of the general level of magnitudes or values of the items in a set of dta

-a measure thst determines where the group tends to cluster or to the center

-a single value which best represents the entire group

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Mean(Me)

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the arithmetic average of all the scores in a distribution. It is the most stable, sensitive, consistent and reliable measuring instrument

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Mode(Mo)

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the most frequent score

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Median(Md)

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the central value that divides the ordered data collection into two equal parts

-is the value of the middle term after arranging the data in ascending or descending order

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Measure of Variability

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a measure which aids the statisticians in making comparisons

- a measures which describes or show how far above or below is a score from the mean

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Variance

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the same as the standard deviation except that the square root is not taken

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Range

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the distance between the highest and lowest score in a array of data

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Inferential Statistics

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methods used ti describe a population by studying a random sample of that population

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Population

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any specified group taken as the subjects of the study or research

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Sample

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only a portion or subset of a population

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Types of Inferential Statistics

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Parametric Statistics

Nonparametric Statistics

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Parametric Statistics

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COme from a population with normal distribution

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Nonparametric Statistics

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does not require any assumption regarding the population where the sample is taken

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USes of Statistics

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-To be able to read professional literature with understanding

-To know how to treat research data adequately and present results corectly

-To develop an ability to evaluate data and to suspend correct judgment

-Training in statistics is a protection against errors in judgement

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Applications of Statistics

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-Education

-Government

-Business and Economics

-Medicine and Science

-Psychology

-Sociology and Population Dynamics

-Sports

-Engineers/Economists/Managers

-Astronomy

-Zoology

-Biology

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Statistical Data can be categorized as

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-Primary

-Secondary

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Primary

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refers to the informatin which are gathered directly from an original source or to the infoemation based in direct or first hand experiences

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Secondary

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taken from published or unpublished papers which were previously gathered by other individuals or agencies

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Methods used in the Collection of Data

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-Direct/Interview Method

-Indirect / Questionaire Method

-Registration Method

-Observation Method

-Experiment Method

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Direct or Interview MEthod

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a method of person to person exchanged between the interviewer or interviewee

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Indirect or Questionnaire Method

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int his method, written responses are given to prepared questions. Questions may be mailed or hand carried

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Registration Method

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in this method, the gathering of information is enforced by certain laws

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Observation Method

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in this method, the investigator observes the behavior of persons of organizations and their outcomes. It is usually used when the subjects can't talk or write

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Experiment Method

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this is a method used when the objective is to determine the cause and effect relationship of certain phenomena under controlled conditions

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VAriable

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the characteristic of the population investigated

-considered as causes and effects in a research project or experiment

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Variables' Functional relationship

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dependent

-independemt

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Independent Variable

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the controlling factors

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Dependent Variable

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depending on the controlling factors

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Types of Independent Variables

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-Experimental

-Manipulated

-Controlled

-Moderator

-Intervening

-Extraneous

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Experimental

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variable being analyzed in an experiment or research

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Manipulated

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variable which is reduced, increased, removed or added in the study to ascertain its effect on or relationship to the dependent variable

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Controlled

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variable being held constant in order to prevent it from affecting the dependent variable

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moderator

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variables which are not the main focus of the study but are analyzed in terms of their impact on the dependent variable

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intervening

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not mentioned in the problem or hypothesis but affects the results

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extraneous

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always figures out in any study

-cannot be controlled

-error or chance variable

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personal variable

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-age, sex, income education, family size, adresses, health, hobbies etc.

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environmental variable

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location, geographical nature, organizational climate

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Variables according to continuity

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-continuous

-discrete/discontinuous

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continuous

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variables that can theoretically assume any value between two given values.

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discrete/discontinuous

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vaariables with definite values and cannot take the form of decimals

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qualitative variable

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takes the form of attributes

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quantitative variables

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they come in emasurements

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Variables can be categorized according to scale of measurement

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-nominal scale

-ordinal scale

-interval scale

-ratio scale

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nominal scale

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numbers sued in scale are frequently used in everyday life

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ordinal scale

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scale of measurement is possible when the researcher can detect differing degree of an attribute or property in objects

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interval scale

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possible when the measures can distinguish not only different amounts of the property of objects but can also discern equal differences between objects

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ratio scale

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assigned in measurement reflect ratios amount od the property measured

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Presentation of Data

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Textual Form

Tabular Form

Graphical Form

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Textual Form

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in this form, the data us presented in paragraph form

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Tabular Form

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in this form, the data is presented in rows and columns

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Types of Tables

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-General or Reference Table

-Summary or Text Table

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General or Reference Table

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used as a warehouse of information in order to present data in such a way that individual items may easily be found and is usually found in appendix

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Summary or Text Table

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-designed to guide the reader in anlyzing the data

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Graphical Form

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in this form, numerical data are presented with the use of graphs`

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Kinds of Graphs

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-Bar graphs

-Line graphs

-Circle Graphs

-Pictographs

-Map Graph

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Bar graph

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represents the magnitude of the quantities being compared

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Line graph

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show the relationship between two or more sets of quantities

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Circle Graphs

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represent quantities that make up a whole

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Pictographs

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uses picture

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Map graph

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-best way to present geographical data

-also known as cartogram

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Frequency Distribution

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most fundamental technique used by statistician for putting into useful order a disarray of collected data

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Parts of frequency distribution

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-Class Interval/ Class Limit

-Class boundaries

-Class Size

-Class Frequency

-Class MArk

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Class Intervals/ Class limit

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category defines by a lower limit and uan upper limit

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Class boundaries

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refers to the value midway between the upper limit of a certain interval and the lower limit of the next

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Class Size

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-also known as class width

-difference between two successive lower or upper limits

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Class frequency

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-refers to the number of observations belonging to a class interval or the number of values that fall in a given interval

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Class MArk

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average of upper and lower clalss limits

-midpoint of the class interval

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i=Range/k

where

k = 1+3.3logn(rounded off to the next higher whole number)

n-> sample size

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upper and lower limit

number + interval - 1``

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class mark = upper limit + lower limit / 2

% rel. frequency/ total freq. * 100``

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Graphical Devicees Used to represent frequency distribution

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-Histogram

-Frequency Polygon

-Cumulative Frequency Distribution or Ogive(cfd)

-Relative Frequency Distribution

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Histogram

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freq. vs. class marks

-consist if rectangles

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Frequency Polygon

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frequency plotted against classmarks

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cfd/ogive

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tabular arrangement of data by class intervals whose frequency are cumulated

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the "less than cfd"

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whose sum of frequencies for each class interval is less than the upper class boundary

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the "greater than" cfd

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whose sum of freq. for each class interval us greater than the lower class boundary

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Relative frequency Distribution

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tabular arrangement of data showign the proportion in percent of each frequency to the total frequency

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Grouped

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number or cases/samples is more than 30

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Ungrouped Data

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if sample(n) is less than 30

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Me = summation of x/n

summ of x = sum of all values in the distribution

n = number of values in the distributiom

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Mean Formula (Ungrouped)

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Median(Ungrouped)

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arrange data in ascending order..

if odd Md = Middle most

if even Md = average of 2 middlemost valueus

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Mode(Ungrouped)

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the repeated data

11, 29, 29 mode = 29

it may not exist like: 1, 2, 3

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Range(Ungrouped)

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highest val - lowest val

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Variance(Ungrouped)

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V=summation(x-Me)^2/ N

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Standard Deviation(Ungrouped)

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SD = sqr(V)

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Mean(Grouped)

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Me = AM + i(summ(fd)/N)

AM = assumed mean

f = freq

d= deviation

i = class size

N = num of class`

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Median(Grouped)

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Md = LL[(N/2 - Fup)i] / f

Fup = cummulative freq corresponding to freq below the MD

f=freq of median

LL = lower true limit of the median class

N = num of classes

i= class size

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Mode(Grouped)

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Mo = Me - 3(Me - Md)

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Range(grouped)

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highest - lowest

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Standard Deviation(Grouped)

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SD = i[sqrt((summfd^2/N) - (summ of fd/N)^2]

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Variance(Grouped

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V= SD^2

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Quartile(Q)

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divides thes et of values into 4 equal parts

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Quartile formula

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Q = LL+((kapila na Q)N/4)-Fup/f)i

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Decile(D)

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divides the set of values into 10 equal parts

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Decile Formula

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D = LL + [(ikapila na decile(N)/10 - Fup / f)i

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Percentile(P)

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divides the set of values into 100 equal pasrts

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Percentile(P0 formula

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P = LL + (ikapila(N)/ 100 - Fup / f)i

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Percentile Rank(PR)

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Pr = 100 * [ts + (x-LL/i)f] / N

ts = the subtotal or sum of the frequencies up to, but not including, the frequencies of the class interval where the raw score in question is found

LL = the real lower limit of the class interval where the raw score in question is found

N = total number of cases

i= class size

x = raw score