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Building Statistical Models Lesson 4 Theories & Models Theories Describe, explain, & predict real-world events/objects Models Replicas of real-world events/objects Can test predictions ~ Models & Fit Model not exact replica Smaller, simulated Sample Model of population Introduces error Fit How well does model represent population? Good fit more useful ~ Models in Psychology My research model Domestic chicks Effects of pre-/postnatal drug use Addiction & its consequences Who/What do most psychologists study? Rats, pigeons, intro. psych. students External validity Good fit with real-world populations? ~ The General Linear Model Relationship b/n predictor & outcome variables form straight line Correlation, regression, analysis of variance Other more complex models ~ The Mean as a Statistical Model Very simple model 1 number represents all the observations Often hypothetical value e.g., mean # friends = 2.6 Error introduced Actual # friends = mean + error Deviation (deviance) ~ X i Assessing the Fit of the Mean How well does it represent all observations? On average near or far from mean? Distance from mean Or width of distribution Mean Daily Temperature For which group is the mean a better fit for the data? 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Measures of Variability Deviation: for a single score Range Highest value – lowest value + 1 Standard deviation Conceptually: mean of all deviation scores average distance of scores from mean Variance Used to calculate standard deviation Also used in analysis of variance ~ Calculating the Standard Deviation Why only conceptually mean of deviation scores? If Xi What is mean deviation? S(Xi – ) = 0 ~ Xi 1 2 3 4 5 Xi - Variability: Notation & Formulas 3 steps to standard deviation Sums of squares (squared deviations) 2 SS = S(Xi – ) Variance = mean of squared deviations (MS) 2 2 ( X ) i N square root of variance = standard deviation ~ Standard Deviation (SD) (X i ) 2 N Conceptually mean deviation score for all data Gives width (dispersion) of distribution Describing a distribution Report mean & standard deviation , ~ Samples & Variability Usually study samples to learn about populations Sampling introduces error Change symbols & formula SS ( X X ) 2 s 2 2 ( X X ) N 1 s 2 ( X X ) N 1 Samples: Degrees of Freedom (df) df = N – 1 For a single sample (or group) s tends to underestimate Fewer Xi used to calculate Dividing by N-1 boosts value of s Also used for Confidence intervals for sample means Critical values in hypothesis testing ~ Degrees of Freedom: Extra Don’t lose any sleep over this df theory If n= 4 & sample mean = 10 th can be only 3 of Xi can be any value, 4 one value See Jane Superbrain 2.2 (pg 37) ~ Level Of Measurement & Variability Which can be used? nominal none ordinal range only interval/ratio all 3 OK range, standard deviation, & variance ~ Statistical Models Representation of the population We will focus on linear models Mean is a simple model One number represents all data Both and X Standard deviation measures fit of model Better fit more useful Smaller and s ~