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Descriptive Statistics: Variability Lesson 5 Theories & Statistical Models Theories Describe, explain, & predict realworld 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? Amount of error in model 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) ~ Xi Distributions: 3 useful features Summarizes important characteristics of data 1. What is shape of the distribution? 2. Where is middle of distribution? 3. How wide is distribution? 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 Variability How much do scores vary from the mean? ~ 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 ~ From the Dictionary Deviation: departure from a standard or norm. Variance: the state, quality, or fact of being variable, divergent, different, or anomalous. Error: a deviation from accuracy or correctness Variability: something that may or does vary; a variable feature or factor Variation: something that may or does vary; a variable feature or factor ~ Calculating the Standard Deviation Why only conceptually mean of deviation scores? Xi If Xi What is mean deviation? S(Xi – ) = 0 ~ 1 2 3 4 5 Xi - 4 Steps to Standard Deviation 1. Calculate deviation scores 2. Sums of squared deviations Or Sums of squares (SS) 3. Variance mean of squared deviations (MS) 4. Standard deviation square root of variance ~ Xi SS ( X i )2 2 (X i ) 2 N (X i ) N 2 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 s 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 ~ 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 ~