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1 Week 1 Statistics Background Measurement, testing Process of assigning numbers to objects so that specific properties of objects are represented by properties of numbers Psychological measurement concerned with attributes of persons (not person him/herself) Method of expressing individual differences (IDs) in a quantitative form Statistics allow for answering questions about meaning of numbers o Description of test scores o Make inferences about meaning of scores Scales of Measurement Numbers have different meanings depending on type of data collected Meanings based on scales of measurement Statistical procedure depends on level of measurement Ordinal level (vs. interval) in scale development 1. Nominal Most rudimentary Numbers classify, identify, code persons Sue belongs to Group 1; Megan and Kate are in Group 2 2. Ordinal Numbers demonstrate rank orders of individuals Sufficient for decision-making purposes No indication of absolute levels Sue - Megan - Kate 3. Interval Differences between numbers are equivalent Absolute values No fixed zero point 2 Difference in ability between Sue and Megan equals difference in ability between Megan and Kate 4. Ratio Most informative Absolute zero point Ratios between numbers are equivalent Megan is twice as agile as Sue Probability and Statistical Significance Science is conservative Alpha level (α) corresponds to p-value (or p-level) for statistical test o Stands for probability level o Finding from sample would occur by chance alone 5 times out of 100 (α = .05) o Usual levels used in social sciences are .05 or .01 (sometimes .10) Sampling Distributions Sample vs. sampling distributions Theoretical distributions used in making statistical decisions Shape based on sample size and number of variables in analysis Used to set α level for statistical tests and decide whether or not to reject H0 As N increases, the critical value decreases (i.e., significant difference easier to find) The Normal Distribution (μ = 0, σ = 1) Individual difference normally distributed within population Asymmetrical, bell-shaped curve Helps determine if value is extreme or not Central Tendency, Variability Average (mean, or M) score provides good summary but does not describe individual differences Variance, standard deviation (SD, or S) measures extent to which individuals differ 3 Z-score o Indicates – in SD units – how far score is from mean (standard normal distribution) o Provides direct and easily interpretable descriptions of scores Formulas Correlation Describes direction and magnitude of linear relationship between two scores Data into two vectors of numbers, X and Y Scatterplot Pearson product-moment correlation coefficient (r) o Ranges from -1 to +1 o The more the pairs of values vary together, the stronger the relationship and the farther from 0 r will be o Significance o N ≈ 20 usually sufficient to draw conclusions about sample data if moderate relationship exists Depends on strength of relationship Caution needs to be exercised when interpreting results Science is conservative o Formulas Influences o Restriction of range, i.e., r decreases (attenuation) o Combining groups, i.e., relationship between variables varies across groups Alternative measures of association o Both interval/ratio: Pearson coefficient o One nominal (truly dichotomized; i.e., item score): Point-biserial (same as Pearson’s) o Both nominal: Phi o Both ordinal: Spearman Linear Regression - Prediction One variable, X, is dependent (criterion) on the other(s), Y (predictor(s)) Regression line is mathematical function that relates predictor(s) to criterion o Minimizes the squared distances of each point from line o Formulas 4 Predicted score on Y Intercept, a, adjusts for different scales for X and Y Slope, b, i.e., expected change in Y per unit change in X Score on predictor, X Unstandardized vs. standardized (≈ correlation) coefficient values + significance R-value is multiple correlation between predictor(s) & criterion R-square & adjusted (i.e., shrunk) R-square values + significance Proportion of variance in Y accounted for by predictor(s) Standard error of estimate associated with predicting scores on criterion (see also residuals) F-value = (t-value)2 Simple vs. multiple regression Hierarchical vs. simultaneous multiple regression