introduction to statistical theory
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This document provides an introduction to statistical theory. It discusses why statistics are studied and defines key statistical concepts such as populations, samples, parameters, statistics, descriptive statistics, inferential statistics, and the different types of data and variables. It also covers experimental design, methods for collecting data such as surveys and sampling, and different sampling methods like random, stratified, cluster, and systematic sampling.
introduction to statistical theory
- 1. INTRODUCTION TO STATISTICAL THEORY BY UNSA SHAKIR
- 3. Why study statistics? 1. Data are everywhere 2. Statistical techniques are used to make many decisions that affect our lives 3. No matter what your career, you will make professional decisions that involve data. An understanding of statistical methods will help you make these decisions efectively
- 4. Statistics The science of collectiong, organizing, presenting, analyzing, and interpreting data to assist in making more effective decisions Statistical analysis – used to manipulate summarize, and investigate data, so that useful decision-making information results.
- 5. Data consists of information coming from observations, counts, measurements, or responses. A population is the collection of all outcomes, responses, measurement, or counts that are of interest. A sample is a portion, or part, of the population of interest
- 6. Populations & Samples •Example: •In a recent survey, 250 college students at Union College were asked if they smoked cigarettes regularly. 35 of the students said yes. Identify the population and the sample. Responses of all students at Union College (population) Responses of students in survey (sample)
- 7. Example:
- 8. Parameters & Statistics A parameter is a numerical description of a population characteristic. A statistic is a numerical description of a sample characteristic. Parameter Population Statistic Sample
- 9. Parameters & Statistics •Example: Decide whether the numerical value describes a population parameter or a sample statistic. a.) A recent survey of a sample of 450 college students reported that the average weekly income for students is $325. Because the average of $325 is based on a sample, this is a sample statistic. b.) The average weekly income for all students is $405. Because the average of $405 is based on a population, this is a population parameter.
- 10. Branches of Statistics The study of statistics has two major branches: descriptive statistics and inferential statistics. Statistics Descriptive statistics Inferential statistics Methods of organizing, summarizing, and presenting data in an informative way The methods used to determine something about a population on the basis of a sample
- 11. Inferential Statistics • Estimation • Hypothesis testing Inference is the process of drawing conclusions or making decisions about a population based on sample results
- 12. Descriptive Statistics • Collect data • e.g., Survey • Present data • e.g., Tables and graphs • Summarize data • e.g., Sample mean = iX n
- 13. Descriptive and Inferential Statistics •Example: •In a recent study, volunteers who had less than 6 hours of sleep were four times more likely to answer incorrectly on a science test than were participants who had at least 8 hours of sleep. Decide which part is the descriptive statistic and what conclusion might be drawn using inferential statistics. • The statement “four times more likely to answer incorrectly” is a descriptive statistic. • An inference drawn from the sample is that all individuals sleeping less than 6 hours are more likely to answer science question incorrectly than individuals who sleep at least 8 hours.
- 15. Statistical data The collection of data that are relevant to the problem being studied is commonly the most difficult, expensive, and time-consuming part of the entire research project. Statistical data are usually obtained by counting or measuring items. Primary data are collected specifically for the analysis desired Secondary data have already been compiled and are available for statistical analysis A variable is an item of interest that can take on many different numerical values. A constant has a fixed numerical value.
- 16. Types of Data Data sets can consist of two types of data: qualitative data and quantitative data. Data Qualitative Data Quantitative Data Consists of attributes, labels, or nonnumerical entries. Consists of numerical measurements or counts.
- 17. Qualitative data Qualitative data are generally described by words or letters. They are not as widely used as quantitative data because many numerical techniques do not apply to the qualitative data. For example, it does not make sense to find an average hair color or blood type. Qualitative data can be separated into two subgroups: dichotomic (if it takes the form of a word with two options (gender - male or female) polynomic (if it takes the form of a word with more than two options (education - primary school, secondary school and university).
- 18. Quantitative data Quantitative data are always numbers and are the result of counting or measuring attributes of a population. Quantitative data can be separated into two subgroups: • discrete (if it is the result of counting (the number of students of a given ethnic group in a class, the number of books on a shelf, ...) • continuous (if it is the result of measuring (distance traveled, weight of luggage, …)
- 19. Qualitative and Quantitative Data •Example: •The grade point averages of five students are listed in the table. Which data are qualitative data and which are quantitative data? Student GPA Sally 3.22 Bob 3.98 Cindy 2.75 Mark 2.24 Kathy 3.84 Quantitative dataQualitative data
- 20. Types of variables Variables QuantitativeQualitative Dichotomic Polynomic Discrete Continuous Gender, marital status Brands of clothes, hair color Children in family Amount of income tax paid, weight of a student
- 21. Levels of Measurement The level of measurement determines which statistical calculations are meaningful. The four levels of measurement are: nominal, ordinal, interval, and ratio. Levels of Measurement Nominal Ordinal Interval Ratio Lowest to highest
- 22. Nominal Level of Measurement Data at the nominal level of measurement are qualitative only. Levels of Measurement Nominal Calculated using names, labels, or qualities. No mathematical computations can be made at this level. Colors in the Pakistani flag Names of students in your class Textbooks you are using this semester
- 23. Ordinal Level of Measurement Data at the ordinal level of measurement are qualitative or quantitative. Levels of Measurement Arranged in order, but differences between data entries are not meaningful. Class standings: freshman, sophomore, junior, senior Numbers on the back of each player’s shirt Ordinal Top 50 songs played on the radio
- 24. Interval Level of Measurement Data at the interval level of measurement are quantitative. A zero entry simply represents a position on a scale; the entry is not an inherent zero. Levels of Measurement Arranged in order, the differences between data entries can be calculated. Temperatures Years on a timeline Interval
- 25. Ratio Level of Measurement Data at the ratio level of measurement are similar to the interval level, but a zero entry is meaningful. Levels of Measurement A ratio of two data values can be formed so one data value can be expressed as a ratio. Ages Grade point averages Ratio Weights
- 26. Summary of Levels of Measurement NoNoNoYesNominal NoNoYesYesOrdinal NoYesYesYesInterval YesYesYesYesRatio Determine if one data value is a multiple of another Subtract data values Arrange data in order Put data in categories Level of measurement
- 28. Designing a Statistical Study GUIDELINES 1. Identify the variable(s) of interest (the focus) and the population of the study. 2. Develop a detailed plan for collecting data. If you use a sample, make sure the sample is representative of the population. 3. Collect the data. 4. Describe the data. 5. Interpret the data and make decisions about the population using inferential statistics. 6. Identify any possible errors.
- 29. Methods of Data Collection In an observational study, a researcher observes and measures characteristics of interest of part of a population. In an experiment, a treatment is applied to part of a population, and responses are observed. A simulation is the use of a mathematical or physical model to reproduce the conditions of a situation or process. A survey is an investigation of one or more characteristics of a population. A census is a measurement of an entire population. A sampling is a measurement of part of a population.
- 30. Sampling A sample should have the same characteristics as the population it is representing. Sampling can be: • with replacement: a member of the population may be chosen more than once (picking the candy from the bowl) • without replacement: a member of the population may be chosen only once (lottery ticket)
- 31. Sampling methods Sampling methods can be: • random (each member of the population has an equal chance of being selected) • nonrandom The process of sampling may causes sampling errors.
- 32. Random sampling methods • simple random sample (each sample of the same size has an equal chance of being selected) • stratified sample (divide the population into groups called strata and then take a sample from each stratum) • cluster sample (divide the population into strata and then randomly select some of the strata. All the members from these strata are in the cluster sample.) • systematic sample (randomly select a starting point and take every n-th piece of data from a listing of the population)
- 33. Stratified Samples A stratified sample has members from each segment of a population. This ensures that each segment from the population is represented. Freshmen Sophomores Juniors Seniors
- 34. Cluster Samples A cluster sample has all members from randomly selected segments of a population. This is used when the population falls into naturally occurring subgroups. The city of Pakistan divided into city blocks. All members in each selected group are used.
- 35. Systematic Samples A systematic sample is a sample in which each member of the population is assigned a number. A starting number is randomly selected and sample members are selected at regular intervals. Every fourth member is chosen.
- 36. Convenience Samples A convenience sample consists only of available members of the population. Example: You are doing a study to determine the number of years of education each teacher at your college has. Identify the sampling technique used if you select the samples listed. 1.) You randomly select two different departments and survey each teacher in those departments. 2.) You select only the teachers you currently have this semester. 3.) You divide the teachers up according to their department and then choose and survey some teachers in each department. Continued.
- 37. Identifying the Sampling Technique Example continued: You are doing a study to determine the number of years of education each teacher at your college has. Identify the sampling technique used if you select the samples listed. 1.) This is a cluster sample because each department is a naturally occurring subdivision. 2.) This is a convenience sample because you are using the teachers that are readily available to you. 3.) This is a stratified sample because the teachers are divided by department and some from each department are randomly selected.
- 38. Reference Books 1. Ronald Walpole, Myers, Myers, Ye, “Probability & Statistics for Engineers & Scientists”, 8th edition, 2008, Prentice Hall Publisher. 2. Lay L. Devore, Probability and Statistics for Engineering and the Sciences, 2003, Duxbury Publishers. 3. G. Cowan, Statistical Data Analysis, 1998, Clarendon, Oxford.