Midterm 3 - Statistics and Experimental Design

A collection of vocabulary flashcards covering key definitions, formulas, assumptions, and principles of Pearson correlation and simple linear regression.

Pearson Correlation ($r$)

A parametric statistical test that quantifies the extent to which two continuous variables covary with one another, where $r = 1$ indicates a perfect positive correlation, $r = -1$ indicates a perfect negative correlation, and $r = 0$ indicates no correlation.

Sum of the Cross-Products ($SXY$)

A statistical quantity calculated by multiplying each pair of $X$ and $Y$ scores together and then adding all of these products together ($\sum X_i Y_i$).

Degrees of Freedom ($df$) for Pearson Correlation

The sample parameter used to determine critical $r$-values, calculated as the number of paired observations minus $1$ ($n - 1$).

Simple Linear Regression

A parametric statistical method used to test for a linear relationship between two variables, assessing whether the magnitude of the independent variable ($X$) reliably predicts the magnitude of the dependent variable ($Y$).

Fixed Effect Factor

A variable whose score magnitudes are fixed and manipulated directly by the experimenter, represented by the independent variable plotted on the $X$-axis.

Random Effect Factor

A variable whose scores are subject to random variation and must be normally distributed, represented by the dependent variable plotted on the $Y$-axis.

Least-Squares Line

The 'best-fit' regression line described by the equation $\hat{Y}_i = a + bX_i$, which mathematically minimizes the residual variation in the $Y$ values.

Slope ($b$) of a Regression Line

The rate of change in the dependent variable per unit change in the independent variable, calculated as the sum of cross-products divided by the sum of squares of $X$ ($b = \frac{\sum xy}{\sum x^2}$).

$Y$-Intercept ($a$)

The predicted value of $Y$ when $X = 0$, calculated by subtracting the product of the slope and the mean of $X$ from the mean of $Y$ ($a = \bar{Y} - b\bar{X}$).

Regression Sum of Squares

The amount of variability in the dependent variable ($Y$) that results from the linear relationship with $X$, calculated as the slope multiplied by the sum of cross-products ($\text{Regression } SS = b\sum xy$).

Residual Sum of Squares

The variance in $Y$ not explained by $X$ (error variance), calculated as the total sum of squares minus the regression sum of squares ($\text{Residual } SS = \text{Total } SS - \text{Regression } SS$).

Degrees of Freedom for Residual Error ($df_{\text{residual}}$)

The degrees of freedom associated with the unexplained error in simple linear regression, computed as the sample size minus the number of estimated parameters ($n - k$, which equals $n - 2$).

Coefficient of Determination ($r^2$)

A statistical measure ranging between $0$ and $1$ that quantifies how well the regression line fits the data, calculated as the linear regression sum of squares divided by the total sum of squares, representing the proportion of variation in $Y$ explained by the linear model.

Normality Assumption Difference (Correlation vs. Regression)

The distributional requirement stating that Pearson correlation requires both the $X$ and $Y$ variables to be normally distributed, whereas linear regression requires only the $Y$ variable to be normally distributed.

Residual

The deviation or difference between an observed data value and its expected or model-predicted value.