Statistics Quiz

Normal Distribution and Standardization

• A normal distribution is a bell-shaped distribution of scores.
• In a normal distribution, about 68% of cases fall between the values -1 and 1 standard deviations.
• About 1% of cases fall above 3 standard deviations.
• A standard deviation value of 345 or 100 + 725 is equivalent to 1 if transformed into z-scores.
• A standard normal distribution has a mean of 0 and a standard deviation of 1.

Z-Scores

• A z-score is a score expressed in standard deviation units.
• Transforming raw scores into z-scores implies a linear transformation.
• Z-scores can be used to:
  • Know the percentile of a score on a variable.
  • Compare a score on a variable with a score on another variable.
  • Know which score on a variable is at a specific percentile.

Correlation

• A correlation coefficient (Pearson's r) indicates the strength and direction of the relationship between two variables.
• A correlation coefficient can take values from -1 to 1.
• A correlation coefficient of -0.61 indicates a negative relationship.
• A correlation coefficient of 0.00 indicates no relationship.
• A correlation coefficient of 0.06 indicates a small positive relationship.
• The strength of a relationship refers to how closely bunched around the imaginary line the dots are in the scattergram.

Pearson's r Formula

• The formula for Pearson's r is defined as the mean of the products of paired z-scores.
• The numerator in the Pearson's r formula refers to the sum of the products of the paired z-scores.
• If one of the variables is constant, the correlation is undefined.

Scattergram

• The two axes of a scattergram are called the ordinate (vertical axis) and abscissa (horizontal axis), or Y-axis and X-axis.
• If the relationships between two variables depart from linearity, Pearson's r will underestimate the correlation.

Correlation in Restricted Range

• When the correlation is calculated using a restricted range, the correlation will either be reduced or inflated.

Outliers

• The value of Pearson's r will be affected by outliers, either inflated or reduced.

Perfect Positive Correlation

• A perfect positive correlation (r = 1) can only occur when X and Y distributions have exactly the same shape.

Skewed Distribution

• A skewed distribution is a distribution where the most frequently occurring scores are clustered at one end of the scale.

Kurtosis

• Kurtosis refers to the peakedness of a distribution.

Assumptions for Pearson's r

• Interval data is required for Pearson's r.
• If assumptions are not satisfied, tests specifically designed for non-interval data can be used, or alpha level can be lowered.

Regression Analysis

• Regression analysis is used to predict the value of a variable (Y) based on the value of another variable (X).
• The regression line is a straight line drawn through a scatterplot of two variables, that comes as close to the data points as possible.
• The method used to find the regression line is called the method of least squares.
• The intercept in regression analysis is the point at which the regression line cuts across the Y-axis.
• The slope, or regression coefficient, represents the change in Y for a one-unit change in X.
• The true linear regression equation is y = a + b * x.

Simple Regression

• In simple regression, the effect of a single predictor (X) on an outcome (Y) is assessed.
• The value of the regression coefficient (b) depends on the steepness of the slope.
• The value of the standardized regression coefficient (β) depends on how closely clustered around the line the data points are.

Multiple Regression

• In multiple regression, the effect of multiple predictors (X1, X2, …) on an outcome (Y) is assessed.
• The number of predictors required is at least 2.
• The ANOVA table is used to assess the goodness-of-fit of the overall model.
• The coefficients table provides information about the unique effects of each predictor.
• The R-Square value represents the proportion of the total variance in Y explained by the predictors.
• The F-value represents the ratio between the portion of total variance accounted for by the regression line and the variance not accounted for by the regression line.

Hierarchical Multiple Regression

• In hierarchical multiple regression, the effects of multiple predictors on an outcome are assessed in a step-by-step manner.
• The R-Square Change value represents the additional variance explained by a specific model.
• The Sig. F Change value is used to determine if the R-Square Change value is statistically significant.
• The model with the highest number of predictors is referred to as Model 3.

Categorical Predictors

• A categorical predictor is a predictor measured on a categorical (or nominal) scale.
• Dummy coding is used to represent categorical predictors.
• The b value of a categorical predictor with two categories is equal to the difference between the mean scores obtained on the outcome by the two categorical groups.

Mediation Analysis

• Mediation analysis assesses the effect of a predictor (X) on an outcome (Y) through a mediator (M).
• The indirect effect of X on Y through M is represented by paths a and b.
• The direct effect of X on Y is represented by path c.
• The standardized coefficient for the indirect effect can be obtained using PROCESS.
• The mediation model can be used to test the hypothesis that a mediator (M) completely mediates the effect of a predictor (X) on an outcome (Y).

Moderation Analysis

• Moderation analysis assesses the effect of a predictor (X) on an outcome (Y) as a function of a moderator (M).
• The interaction term is the product of the predictor and the moderator.
• The coefficient for the effect of M on Y represents the effect of M on Y for a case with an average score on X.
• The coefficient for the effect of X on Y represents the effect of X on Y for a case with an average score on M.
• A simple slope analysis shows the effect of the predictor on the outcome at different levels of the moderator.

Regression Diagnostics

• Regression diagnostics are used to assess the accuracy of the regression model.
• Outliers are cases that differ substantially from the main trend.
• Influential cases are cases that affect the precision of the estimation of the regression coefficient.
• Multicollinearity occurs when two or more predictors are highly correlated.
• Normality of residuals, independence of errors, and homoscedasticity are assumptions of regression analysis.

Test your knowledge of normal distributions, standard deviations, and z-scores in statistics. Questions cover characteristics of normal distributions and transformations.