Basic Statistcs Formula Sheet

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1 Basic Statistcs Formula Sheet Steven W. ydick May 5, 0 This document is only intended to review basic concepts/formulas from an introduction to statistics course. Only mean-based procedures are reviewed, and emphasis is placed on a simplistic understanding is placed on when to use any method. After reviewing and understanding this document, one should then learn about more complex procedures and methods in statistics. However, keep in mind the assumptions behind certain procedures, and know that statistical procedures are sometimes flexible to data that do not necessarily match the assumptions.

2 Descriptive Statistics Elementary Descriptives (Univariate & Bivariate) ame Population Symbol Sample Symbol Sample Calculation Main Problems Alternatives Mean µ x x = x Sensitive to outliers Median, Mode Variance σ x s x s x = (x x) Sensitive to outliers MAD, IQR Standard Dev σ x = s x Biased MAD Covariance σ xy y y = (x x)(y ȳ) Outliers, uninterpretable units Correlation Correlation ρ xy r xy r xy = sxy s y r xy = (zxz y) Range restriction, outliers, nonlinearity z-score z x z x z x = x x ; z = 0; s z = Doesn t make distribution normal Simple Linear Regression (Usually Quantitative IV; Quantitative DV) Part Population Symbol Sample Symbol Sample Calculation Meaning Regular Equation y i = α + βx i + ɛ i y i = a + bx i + e i ŷ i = a + bx i Predict y from x Slope β b b = sxy s x = (x x)(y ȳ) (x x) Predicted change in y for unit change in x Intercept α a a = ȳ b x Predicted y for x = 0 Standardized Equation z yi = ρ xyz xi + ɛ i z yi = r xyz xi + e i ẑ yi = r xyz xi Predict z y from z x Slope ρ xy r xy ( ) r xy = sxy s s y = b x sy Predicted change in z y for unit change in z x Intercept one one 0 Predicted z y for z x = 0 is 0 Effect Size P R r ŷy = r xy Variance in y accounted for by regression line

3 Inferential Statistics t-tests (Categorical IV ( or Groups); Quantitative DV) Test Statistic Parameter Standard Deviation Standard Error df t-obt One Sample x µ = Paired Samples D µd s D = Independent Samples x x µ µ s p = (x x) (D D) D (n )s +(n )s n +n s p t obt = x µ 0 sx s D D t obt = D µ D 0 s D D D n + n n + n t obt = ( x x ) (µ µ ) 0 s p n + n Correlation r ρ = 0 A A t obt = Regression (FYI) a & b α & β ˆσ e = r r (y ŷ) s a & s b t obt = a α 0 s a & t obt = b β 0 s b t-tests Hypotheses/Rejection Question One Sample Paired Sample Independent Sample When to Reject Greater Than? H 0 : µ # H 0 : µ D # H 0 : µ µ # Extreme positive numbers H : µ > # H : µ D > # H : µ µ > # t obt > t crit (one-tailed) Less Than? H 0 : µ # H 0 : µ D # H 0 : µ µ # Extreme negative numbers H : µ < # H : µ D < # H : µ µ < # t obt < t crit (one-tailed) ot Equal To? H 0 : µ = # H 0 : µ D = # H 0 : µ µ = # Extreme numbers (negative and positive) H : µ # H : µ D # H : µ µ # t obt > t crit (two-tailed) t-tests Miscellaneous Test Confidence Interval: γ% = ( α)% Unstandardized Effect Size Standardized Effect Size One Sample x ± t ; crit(-tailed) sx x µ 0 ˆd = x µ 0 Paired Samples D ± td ; crit(-tailed) s D D ˆd = D D s D Independent Samples ( x x ) ± t n +n ; crit(-tailed) s p n + n x x ˆd = x x s p 3

4 One-Way AOVA (Categorical IV (Usually 3 or More Groups); Quantitative DV) Source Sums of Sq. df Mean Sq. F -stat Effect Size Between Within Total g j= nj( xj xg) g SSB/dfB MSB/MSW η = SSB SST g j= (nj )s j g SSW/dfW i,j (xij xg). We perform AOVA because of family-wise error -- the probability of rejecting at least one true H 0 during multiple tests.. G is grand mean or average of all scores ignoring group membership. 3. x j is the mean of group j; n j is number of people in group j; g is the number of groups; is the total number of people. One-Way AOVA Hypotheses/Rejection Question Hypotheses When to Reject Is at least one mean different? H 0 : µ = µ = = µ k Extreme positive numbers H : At least one µ is different from at least one other µ F obt > F crit Remember Post-Hoc Tests: LSD, Bonferroni, Tukey (what are the rank orderings of the means?) Chi Square (χ ) (Categorical IV; Categorical DV) Test Hypotheses Observed Expected df χ Stat When to Reject Independence H 0 : Vars are Independent From Table p jp k (Cols - )(Rows - ) H : Vars are Dependent Goodness of Fit H 0 : Model Fits From Table p i Cells - H : Model Doesn t Fit. Remember: the sum is over the number of cells/columns/rows (not the number of people). For Test of Independence: p j and p k are the marginal proportions of variable j and variable k respectively 3. For Goodness of Fit: p i is the expected proportion in cell i if the data fit the model 4. is the total number of people R C (f O ij f E ij ) i= j= f E ij C i= (f O i f E i ) f E i Extreme Positive umbers χ obt > χ crit Extreme Positive umbers χ obt > χ crit 4

5 Assumptions of Statistical Models Correlation. Estimating: Relationship is linear. Estimating: o outliers 3. Estimating: o range restriction 4. Testing: Bivariate normality One Sample t-test. x is normally distributed in the population. Independence of observations Paired Samples t-test. Difference scores are normally distributed in the population. Independence of pairs of observations One-Way AOVA. Each group is normally distributed in the population. Homogeneity of variance 3. Independence of observations within and between groups Regression. Relationship is linear. Bivariate normality 3. Homoskedasticity (constant error variance) 4. Independence of pairs of observations Independent Samples t-test. Each group is normally distributed in the population. Homogeneity of variance (both groups have the same variance in the population) 3. Independence of observations within and between groups (random sampling & random assignment) Chi Square (χ ). o small expected frequencies Total number of observations at least 0 Expected number in any cell at least 5. Independence of observations Each individual is only in OE cell of the table Central Limit Theorem Given a population distribution with a mean µ and a variance σ, the sampling distribution of the mean using sample size (or, to put it another way, the distribution of sample means) will have a mean of µ x = µ and a variance equal to σ x = σ, which implies that σ x = σ. Furthermore, the distribution will approach the normal distribution as, the sample size, increases. Possible Decisions/Outcomes H 0 True H 0 False Rejecting H 0 Type I Error (α) Correct Decision ( β; Power) ot Rejecting H 0 Correct Decision ( α) Type II Error (β) Power Increases If:, α, σ, Mean Difference, or One-Tailed Test 5

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