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Introduction to Statistics Quantitative Methods in HPELS 440:210 Agenda n Roadmap n Basic concepts n Inferential statistics n Scales of measurement n Statistical notation Roadmap Descriptive Statistics Inferential Statistics •Central tendency •Parametric •Variability •Nonparametric Correlational Method Experimental Method Agenda n Roadmap n Basic concepts n Inferential statistics n Scales of measurement n Statistical notation Basic Concepts n Statistics: A set of mathematical procedures for organizing, summarizing and interpreting information n Statistics generally serve two purposes: ¨ Organize and summarize information n Descriptive statistics ¨ Answer questions (interpretation) n Inferential statistics Basic Concepts n Population: The set of all individuals or subjects of interest in a particular study n Sample: The set of individuals or subjects selected from a population intended to represent the population of interest n Parameter: A value that describes a population n Statistic or test statistic: A value that describes a sample Basic Concepts n Inferential statistics: Procedures that allow you to make generalizations about a population based on information about the sample ¨ Figure 1.1, p 6 Basic Concepts n Sampling error: The discrepancy that exists between a sample statistic and the population parameter ¨ Figure 1.2, p 8 Agenda n Roadmap n Basic concepts n Inferential statistics n Scales of measurement n Statistical notation Inferential Statistics n Statistical Inference: Statistical process that uses probability and information about a sample to make inferences about a population n Two Main Methods Method ¨ Correlational ¨ Experimental Method Correlational Method n Process: ¨ Observe two variables naturally ¨ Quantify strength and direction of relationship n Advantage: Simple and elegant n Disadvantage: Does not assume “cause and effect” ¨ Shoe size and IQ in elementary students? Experimental Method n Process: ¨ Manipulate one variable ¨ Observe the effect on the second variable n Advantage: A well controlled experiment can make a strong case for a “cause and effect” relationship n Disadvantage: Difficult to control for all “confounding” variables Experimental Method n Which variable is manipulated? ¨ Independent variable ¨ Treatment (not always a pill) n Which variable is observed? ¨ Dependent variable ¨ Measure or test n What is the effect of the IV on the DV? Agenda n Roadmap n Basic concepts n Inferential statistics n Scales of measurement n Statistical notation Scales of Measurement n The scales of measurement describe the nature/properties of data n The scale of measurement affects the selection of the test statistic n The are four scales of measurement: 1. Nominal 2. Ordinal 3. Interval 4. Ratio Scales of Measurement: Nominal n Characteristics of Nominal Data: n Assigns names to variables based on a particular attribute n Divides data into discrete categories n No quantitative meaning Scales of Measurement: Nominal n Example: Gender as a variable n Names assigned to variables based on particular attribute -Male or female n Divides data into discrete categories -Male or female (not both) n No quantitative meaning -Males cannot be quantified as “more or less” than girls Scales of Measurement: Ordinal n Characteristics of Ordinal Data: n Has quantifiable meaning n Intervals between values not assumed to be equal Scales of Measurement: Ordinal n Example: Likert Scales n UNI Teacher Evaluations: n “Does the instructor show interest . . .” ¨ Never ¨ Seldom ¨ Frequently ¨ Always Scales of Measurement: Ordinal n Example: Likert Scales n Has quantifiable meaning -”Never” is less than “seldom” -Values can be rank ordered n Intervals between values not assumed to be equal ? ? Never Seldom Frequently Always Scales of Measurement: Ordinal n Other examples: ¨ Small,medium, large sizes ¨ Low, medium, high performance Scales of Measurement: Interval n Characteristics of Interval Data: n Has quantifiable meaning n Intervals between values are assumed to be equal n Zero point does not assume the absence of a value n Values do not originate from zero n Values cannot be expressed as multiples or fractions Scales of Measurement: Interval n Example: Temperature (Fahrenheit or Celcius) n Has quantifiable meaning -10 C° is less than 20 C° n Intervals between values are assumed to be equal -The difference between 5 and 10 C° = difference between 15 and 20 C° n Zero point does not assume the absence of a value -0 C° does not mean absence of temperature n Values do not originate from zero -0 C° is arbitrary based on freezing point n Values cannot be expressed as multiples or fractions -10 C° is not twice as cold as 5 C° Scales of Measurement: Ratio n Characteristics: n Has quantifiable meaning n Intervals between values are assumed to be equal n Zero point assumes the absence of a value n Values originate from zero n Values can be expressed as multiples or fractions Scales of Measurement: Ratio n Example: Length n Has quantifiable meaning n Intervals between values are assumed to be equal n Zero point assumes the absence of a value n Values originate from zero n Values can be expressed as multiples or fractions Scales of Measurement n How do the scales of measurement affect the selection of the test statistic? n Bottom Line: ¨ Nominal and ordinal data à Nonparametric ¨ Interval and ratio data à Parametric Scales of Measurement n Parametric statistics: ¨ Definition:Statistical techniques designed for use when the data have certain specific characteristics in regards to: n Scale of measurement: Interval or ratio n Distribution: Normal ¨ More powerful n Nonparametric statistics: Statistical techniques designed to be used ¨ Definition: when the data are: n Scale of measurement: Nominal or ordinal or n Distribution: Nonnormal Agenda n Roadmap n Basic concepts n Inferential statistics n Scales of measurement n Statistical notation Statistical Notation n Textbook à progressive introduction of statistical notation n Summation = S Summation Example n SX = 3+1+7=11 X X2 n SX2 = 9+1+49=59 3 9 n S(X)2=11*11=121 1 1 7 7 Textbook Problem Assignment n Problems: 2, 8, 12a, 12c, 16, 20.