Introduction to Statistics Basic Concepts

Intro. to Statistics What is Statistics? What is Statistics? “…a set of procedures and rules…for reducing large masses of data to manageable proportions and for allowing us to draw conclusions from those data”“…a set of procedures and rules…for reducing large masses of data to manageable proportions and for allowing us to draw conclusions from those data”

Intro. to Statistics What can Stats do? What can Stats do? Make data more manageableMake data more manageable Group of numbers: Group of numbers: 6, 1, 8, 3, 5, 4, 9 Average is: 36/7 = 5 1/7 Average is: 36/7 = 5 1/7 Graphs: Graphs:

Intro. to Statistics What can Stats do? What can Stats do? Allow us to draw conclusions from the dataAllow us to draw conclusions from the data Group of numbers #1: 6, 1, 8, 3, 5, 4, 9 Group of numbers #1: 6, 1, 8, 3, 5, 4, 9 Average is 5 1/7 Average is 5 1/7 Group of numbers #2: 8, 3, 4, 2, 7, 1, 4 Group of numbers #2: 8, 3, 4, 2, 7, 1, 4 Average is 4 ¼ Average is 4 ¼ Allows us to do this objectively and quantitativelyAllows us to do this objectively and quantitatively

Intro. to Statistics “Quantitative” “Quantitative” Involves measurementInvolves measurement Data in numerical formData in numerical form Answers “How much” questionsAnswers “How much” questions Objective and results in unambiguous conclusionsObjective and results in unambiguous conclusions “Qualitative” “Qualitative” Describes the nature of something Answers “What” or “Of what kind” questions Often evaluative and ambiguous

Intro. to Statistics Qualitative Distinctions: Qualitative Distinctions: “Good” versus “Bad”“Good” versus “Bad” “Right” versus “Wrong”“Right” versus “Wrong” “A Lot” versus “A Little”“A Lot” versus “A Little” Quantitative Distinctions: Quantitative Distinctions: 5 1/7 versus 4 ¼5 1/7 versus 4 ¼ 25% versus 50%25% versus 50% 1 hour versus 24 hours1 hour versus 24 hours

Basic Terminology Summarizing versus Analyzing Summarizing versus Analyzing Descriptive Statistics Descriptive Statistics Inferential Statistics Inferential Statistics Inference from sample to populationInference from sample to population Inference from statistic to parameterInference from statistic to parameter Factors influencing the accuracy of a sample’s ability to represent a population:Factors influencing the accuracy of a sample’s ability to represent a population: Size Size Randomness Randomness

Basic Terminology Size –Size – Sample of 5 cards from a deck of 52 Sample of 5 cards from a deck of 52 2 of Clubs, 10 of Diamonds, Jack of Hearts, 5 of Clubs, and 7 of Hearts2 of Clubs, 10 of Diamonds, Jack of Hearts, 5 of Clubs, and 7 of Hearts What could we conclude about the full deck from this sample about what the full deck looks like without any prior knowledge of a deck of cards? What could we conclude about the full deck from this sample about what the full deck looks like without any prior knowledge of a deck of cards? Compare this to a sample of 51/52 cards – What could we conclude from this sample? Compare this to a sample of 51/52 cards – What could we conclude from this sample?

Basic Terminology Randomness –Randomness – This time lets use the same 5 card sample, but this time the deck is unshuffled (nonrandom) This time lets use the same 5 card sample, but this time the deck is unshuffled (nonrandom) 2 of Clubs, 10 of Clubs, Jack of Clubs, 5 of Clubs, and 7 of Clubs2 of Clubs, 10 of Clubs, Jack of Clubs, 5 of Clubs, and 7 of Clubs What would we conclude about the characteristics of our population (the deck) this time versus when the sample was more random (shuffled)? What would we conclude about the characteristics of our population (the deck) this time versus when the sample was more random (shuffled)?

Basic Terminology Smaller/less random samples both poorly represent population of entire deck of cards Smaller/less random samples both poorly represent population of entire deck of cards Also result in inaccurate inferences about population – poor external validityAlso result in inaccurate inferences about population – poor external validity

Basic Terminology Most often, the aim of our research is not to infer characteristics of a population from our sample, but to compare two samples Most often, the aim of our research is not to infer characteristics of a population from our sample, but to compare two samples I.e. To determine if a particular treatment works, we compare two groups or samples, one with the treatment and one withoutI.e. To determine if a particular treatment works, we compare two groups or samples, one with the treatment and one without

Basic Terminology We draw conclusions based on how similar the two groups areWe draw conclusions based on how similar the two groups are If the treated and untreated groups are very similar, we cannot declare the treatment much of a success If the treated and untreated groups are very similar, we cannot declare the treatment much of a success Another way of putting this in terms of samples and populations is determining if our two groups/samples actually come from the same population, or two different ones Another way of putting this in terms of samples and populations is determining if our two groups/samples actually come from the same population, or two different ones

Basic Terminology Group A (Treated) and B (Untreated) are sampled from different populations/treatment worked: Group A (Treated) and B (Untreated) are sampled from different populations/treatment worked: Group A Population of Well People Group B Population of Sick People

Basic Terminology Group A and B are sampled from the same population/treatment didn’t work: Group A and B are sampled from the same population/treatment didn’t work: Group A Group B Population of Sick People

Basic Terminology What if Group A (who received the Tx) were sicker then Group B (who did not receive Tx), prior to treatment? What would their scores look like after Tx? What if Group A (who received the Tx) were sicker then Group B (who did not receive Tx), prior to treatment? What would their scores look like after Tx? The inability to attribute changes in the variable of interest to the manipulation – poor internal validityThe inability to attribute changes in the variable of interest to the manipulation – poor internal validity I.e. we can’t say for sure if our experiment worked or not I.e. we can’t say for sure if our experiment worked or not

Basic Terminology Quantitative Data Quantitative Data Dimensional/Measurement Data versus Categorical/Frequency Count DataDimensional/Measurement Data versus Categorical/Frequency Count Data Dimensional Dimensional When quantities of something are measured on a continuumWhen quantities of something are measured on a continuum Answers “how much” questionsAnswers “how much” questions I.e. scores on a test, measures of weight, etc.I.e. scores on a test, measures of weight, etc.

Basic Terminology Categorical Categorical When numbers of discrete entities have to be countedWhen numbers of discrete entities have to be counted Gender is an example of a discrete entity – you can be either male or female, and nothing else – speaking of “degree of maleness” makes little sense Gender is an example of a discrete entity – you can be either male or female, and nothing else – speaking of “degree of maleness” makes little sense Answers “how many” questionsAnswers “how many” questions I.e. number of men and women, percentage of people with a given hair colorI.e. number of men and women, percentage of people with a given hair color

Basic Terminology A dimensional variable can be converted into a categorical one A dimensional variable can be converted into a categorical one Convert scores on a test (0-100) into “Low”, “Medium”, and “High” groups – 0-33 = Low; = Medium, and = HighConvert scores on a test (0-100) into “Low”, “Medium”, and “High” groups – 0-33 = Low; = Medium, and = High The groups are discrete categories (hence “categorical”), and you would now count how many people fall into each category The groups are discrete categories (hence “categorical”), and you would now count how many people fall into each category

Basic Concepts Scales of Measurement: Scales of Measurement: NominalNominal labeling/classifying objects labeling/classifying objects i.e. your last name, names on jerseys, social security number, etc. i.e. your last name, names on jerseys, social security number, etc. not technically a scale of measurement since nothing is measured not technically a scale of measurement since nothing is measured OrdinalOrdinal labels that imply rank labels that imply rank i.e. place in a race, military rank – 1 st > 2 nd > 3 rd and General > Lieutenant > Private i.e. place in a race, military rank – 1 st > 2 nd > 3 rd and General > Lieutenant > Private doesn’t say how much more one is than the other doesn’t say how much more one is than the other

Basic Concepts Interval Interval provides labels that imply exactly how much different one label is than anotherprovides labels that imply exactly how much different one label is than another i.e. temperature - 15° F is 5 ° F more than 10 ° Fi.e. temperature - 15° F is 5 ° F more than 10 ° F lacks true zero point - 0 ° F does not represent the complete absence of heat because we have negative values of °Flacks true zero point - 0 ° F does not represent the complete absence of heat because we have negative values of °F Ratio Ratio has all of the above, plus a true zero pointhas all of the above, plus a true zero point i.e. height, weight, ° Kelvin – 0 lbs represents a true lack of weighti.e. height, weight, ° Kelvin – 0 lbs represents a true lack of weight can talk about 16 ° being four times 4 °, which is a proportion /ratio, hence the name of the scale - x = 4ycan talk about 16 ° being four times 4 °, which is a proportion /ratio, hence the name of the scale - x = 4y often very difficult to identify in practice if a true zero point existsoften very difficult to identify in practice if a true zero point exists

Basic Concepts Scales of Measurement Scales of Measurement NominalNominal OrdinalOrdinal IntervalInterval RatioRatio Qualitative Quantitative

Basic Concepts Variables Variables Discrete versus Continuous VariablesDiscrete versus Continuous Variables same as Categorical versus Dimensional variables same as Categorical versus Dimensional variables Not to be confused with “discreet” variables, that people simply do not think should be talked aboutNot to be confused with “discreet” variables, that people simply do not think should be talked about

Basic Concepts ConstantVariable QualitativeQuantitative Categorical/ Discrete Dimensional/ Continuous NominalOrdinalIntervalRatio

Basic Concepts Variables versus Constants Variables versus Constants A constant has only one possible value that it can assumeA constant has only one possible value that it can assume π = … π = … A variable can assume many possible valuesA variable can assume many possible values X = ? X = ? Independent Variables (IV’s) versus Dependent Variables (DV’s) Independent Variables (IV’s) versus Dependent Variables (DV’s) IV manipulated, DV measuredIV manipulated, DV measured Whether a variable is a DV or IV depends upon the design of the experimentWhether a variable is a DV or IV depends upon the design of the experiment

Basic Concepts Variables Variables In true experiments, the effects of one variable (the IV) are manipulated to see the effects on another variable (the DV)In true experiments, the effects of one variable (the IV) are manipulated to see the effects on another variable (the DV) All other factors other than the IV are kept constant so that we can attribute the change to the IV and not to something elseAll other factors other than the IV are kept constant so that we can attribute the change to the IV and not to something else Example: Influence of direct heat on the temperature of waterExample: Influence of direct heat on the temperature of water IV = presence or absence of heat IV = presence or absence of heat DV = temperature of water DV = temperature of water