Statistical Methods PDF

STATISTICAL METHODS E. M. Baah (PhD) Department of Mathematics, Statistics and Actuarial Science Takoradi Technical University 2021  Statistical Methods  Objectives of Section Expose students to basic statistical terminologies, which cont...

STATISTICAL METHODS E. M. Baah (PhD) Department of Mathematics, Statistics and Actuarial Science Takoradi Technical University 2021  Statistical Methods  Objectives of Section Expose students to basic statistical terminologies, which contribute to the understanding of the field. Equip students with basic statistical concepts, procedures and skills to enable them understand the field of statistics.  Data  Data is the plural of the word Datum, which refers to a piece of fact, figure (number) or evidence, but  Data (in statistics) has assumed both the singular and plural connotation, so  Any piece of fact, figure (number relating to a phenomenon) or evidence can be referred to as Data.  Raw Data  So pieces of fact, figures or evidence relating to any group of units under discussion or that are of interest could be referred to as Data.  Data from an activity (observation, survey or experiment) which has not been organized or modified in any way is referred to as Raw Data.  Validity & Reliability  Data must be dependable, so the indicators of the levels of the attributes we want to measure or gauge should be:  Reliable  Valid  Validity & Reliability A Reliable measuring instrument will consistently assign the same value to a given level of an attribute time and again. A Valid measuring instrument is one that measures what it is suppose to measure.  Dependability of Data  Also related to dependability of data is the personnel used to collect the data.They must appreciate:  the group of units that are of interest or under discussion, and  The methods or procedures to be employed in acquiring data on the group under discussion.  What is Statistics?  What is Statistics? A mathematical science concerned with the collection, analysis, presentation and interpretation of (reliable) data, as well as making reasonable inferences and drawing valid conclusions from the data.  Organizing & Representing Data  Organization and Representation of data involve classification, ordering or arranging and presenting data in the form of Tables and Charts/Diagrams/Graphs.  Tables, charts, diagrams and graphs make it easy to interpret data.  Analysis of Data  Analysis of data involves organizing and representing data in useful and necessary forms and applying specialized statistical tools to the data to enable one interpret and draw conclusions from the data.  Statistics  Thus Statistics is a body of principles, methods and procedures for extracting useful information from (numerical) data, by molding data into forms that lend themselves to easy interpretation.  Statistics  The word Statistics is also used to refer to a collection of numerical values relating to a given activity or phenomenon.  Statistics  Thus we may refer to the set of values relating to number of goals scored, the number of yellow cards conceded, the number of corners, et cetera in a football match as the match statistics.  Number of goals: 3  Number of yellow cards: 5  Number of penalties: 2  Statistics  Finding the average monthly rainfall value for a given year from the monthly rainfall figures and  Using the monthly rainfall figures for the past ten (10) years to predict the likely monthly rainfall figures for the next year are examples of what statistics is concerned with.  Uncertainty  Any reason why one would be interested in the average monthly rainfall value for a given year or the likely monthly rainfall values for the next year?  Uncertainty  One would be interested in the average monthly rainfall value for a given year or the likely monthly rainfall values for the next year for planning purposes. However we cannot tell with certainty the amount of rain that is likely to fall in any given month because the amount of rain that falls from month to month is not fixed; it is variable.  Uncertainty  Atthe risk of belabouring the point, the element of variability makes us uncertain about what the monthly rainfall values for the next year would be.  Thus statistics is about Uncertainty.  Statistics helps us to gauge the degree of the attributes of items of interest under conditions of uncertainty.  Uncertainty  Statistics seeks to address the following under conditions of uncertainty :  what typifies a given attribute of a group of interest?  What is the extent of variability in this attribute of interest?  Uncertainty  The element of variability also means that most attributes of items of interest are subject to chance.  Statistics can, therefore, be said to facilitate the modeling of chance and uncertainty.  Statistics & Probability  Probability theory helps us to model chance, so there is a close relationship between Probability Theory and Statistical Theory.  The relationship between Statistics and probability will be elucidated further in the second semester.  Statistical Inference  The goal of statistical inference is to be able to describe the general situation from what has been specifically observed.  Putanother way, given what has been observed, what is the general situation likely to be?  Branches of Statistics  Statistics has two main branches:  Descriptive Statistics, and  Inferential Statistics  Descriptive Statistics It is concerned with organizing, summarizing, and presenting (numerical) data in a convenient form for interpretation.  Branches of Statistics  Inferential Statistics It is concerned with drawing conclusions (that is making inferences) about characteristics of a group of units of interest based on information available from a subset taken from the group under discussion.  Statistical concepts & Definitions  Population A collection of well-defined subjects that are of interest in a study.  Populations tend to be inaccessible, costly and time consuming to study. We therefore often turn to subsets of populations; what we refer to as Samples. Statistical Concepts & Definitions  Sample A representative subset of a population. Sample ⊂ Population  Any idea why the adjective representative is used?  Concepts & Definitions  The adjective representative is used because the sample must contain the characteristics of the population in the same proportion as they are found in the population, so that we can extrapolate any findings from the study of the sample to the population – which is part of what is referred to as statistical inference.  Concepts & Definitions  Apart from accessibility, cost and time constraint, what other factor can force an investigator to use a sample instead of the population?  Asa results of the factors enumerated above, we tend to describe the population in terms of average or mean values obtained by sampling.  Concepts & Definitions  Survey Studies that obtain data by interviewing people are called surveys.  Sample Surveys Studies in which representative samples of populations are studied are called sample surveys.  Concepts & Definitions  PilotSurvey A preliminary survey conducted to determine how a major survey would fare in order to effect any changes in the original planning, if necessary, and to test (referred to as pre-test) the instruments (questionnaire etc.) designed for the survey.  Concepts & Definitions  Census A survey in which each member of /element in a population is enumerated/examined.  Variable An attribute which can potentially assume a different value for each individual of a population.  Concepts & Definitions  Statistic A statistic is a quantity calculated from observations in a sample, that is, a function of observations in the sample.  Parameter A measure of an attribute of a population that defines the population or describes what the population is.  Types of Data  Quantitative Data Data for which numbers describe the extent or the degree of the attributes of the units under observation.  A set of numbers representing the weight of students in a class is an example a quantitative data.  Quantitative Data  The values in a quantitative data are interpretable relative to one another.  Of course a 30kg item is heavier than a 20kg item.  Qualitative Data  Qualitative Data Data for which numbers, labels or words indicate class membership or membership of one of several levels of an attribute of the units under study. Note that the levels of the attribute are of equal importance.  Qualitative Data  Ethnicity, as a variable, is qualitative, as the values it assumes shows which group one is affiliated with.  If M (or 1) represents male and F (or 0) represents female, then the resulting data is qualitative as the M (1) and the F (0) representation does not, as in reality, exalt one sex over the other.  Qualitative Data  Indeed interchanging the representation, viz: 0 for men 1 for women will not lead to loss of information or meaning. The 0 and 1 are for identification purposes only.  Types of Data  Rank Data Data for which numbers or labels not only indicate the various levels of the attribute of interest, but also the relative standings (importance) of the various levels of the attribute within the set of observations. For example, an A in a subject shows a better performance than a B in the same subject.  Types of Data Data Quantitative Qualitative Attribute Discrete Continuous  Data Scales  Nominal Scale On this scale data values (numbers or labels) are for identification purposes only.  The numbers on the shirts of the members of a football team are for identification purposes only. They are therefore nominal.  Nominal Scale  Giveanother example of data which is nominal.  Answer:  Data Scales  Ordinal Scale On this scale data values (numbers or labels) are not for only identification purposes but they also indicate the importance or the relative standing of the units under study.  Ordinal Scale  The members of a class can be put into two groups as the Pass Group and the Fail Group based on their performance in a quiz. Then the resulting data is ordinal since it does not only show which of two groups a member of the class belongs, but also which group are of superior performance.  Data Scales  Giveanother example of data which is ordinal.  Answer:  Data Scales  Interval Scale Values on the interval Scale are real numbers and express the degree or the extent of the attribute under discussion. Also on this scale, given any two unique values, one cannot interpret one value as expressing a multiple of the level of the attribute represented by the other value.  Interval Scale  IntervalScale Additionally the ‘zero’ on an interval scale is often arbitrary.  Temperature values are measured on the interval scale as one cannot say 30° represent a temperature twice as warm as the temperature represented by 15°.  Interval Scale  Also 0° does not mean that there is no heat.  Recall that 0°C is 32° F.  Give another example of quantity that is measured on the interval scale  Answer:  Data Scales  Ratio Scale Values on the Ratio Scale are real numbers and express the degree or the extent of the attribute under discussion. Also on this scale, one can interpret one value as expressing a multiple of the level of the attribute represented by another value.  Ratio Scale  Ratio Scale Additionally the ‘zero’ on an ratio scale is a real zero.  20km is half of 40km, so distance is measured on the ratio scale.  Give another example of a variable that is measured on the ratio scale.  Answer:  Data Scales  Review Question 1.What factor differentiates? a.A nominal scale from an ordinal scale? b. Ordinal from an interval scale? 2. Identify the scale type of the following: a. Numbers on the backs of baseball players’ uniform. b. Identification numbers of students enrolled in the Polytechnic.  Data Scales 2 (continuation): c. Prices of objects in a Mall. d. The time it takes for runners to complete the Milo marathon. e. The ranking of the completion times of runners competing in the Milo marathon.  References  Keller, G. & Warrack, B. (2003). Statistics for Management and Economics. 6thed. USA: Thompson – Brooks/Cole.  Johnson, R. R. & Siskin, B. (1988). Elementary Statistics for Business. 5th re. ed. USA: PWS-Kent Publishing Co.  Mason, R. D., Lind, D. A. & Marchal, W. G. (1999). Stats techniques in Bus. and Econs. 10thed. Bonston, USA: Irwin McGraw-Hill.  References  Bowerman, B. & O’Connel, R. T. (1997). Applied Statistics. 1st ed. Bonston, USA: Irwin.  Levin, R. I. & Rubin, D. S. (1998). Statistics for Management. 7thed. New Jersey, USA: Prentice Hall.  Harper, W. M. (1991). Statistics. 6th ed. London, UK: Longman Group Ltd. Thank You