Exploring one- and two-variable data (AP level) Knowledge-Based Study

Study notes

1. Characterizing Quantitative Distributions

   When analyzing single-variable quantitative distributions, data must be described using four essential attributes: Shape, Outliers, Center, and Spread.

   Distributional Shape

   • Symmetric: The left and right halves of the distribution are approximately mirror images. In unimodal symmetric distributions, the mean and median are roughly equal.
   • Skewed Right (Positively Skewed): The right tail is substantially longer than the left tail. Unusually large values pull the mean upward, resulting in $\text{Mean} > \text{Median}$.
   • Skewed Left (Negatively Skewed): The left tail extends further than the right tail. Extreme low values pull the mean downward, resulting in $\text{Mean} < \text{Median}$.
   • Modality: Distributions can be classified as unimodal (one distinct peak), bimodal (two clear peaks), or uniform (roughly even frequencies across values).

   Measures of Center and Spread: Resistance

   A summary statistic is called resistant if extreme values or outliers do not exert a substantial influence on its value.

   • Non-resistant statistics: The sample mean $\bar{x} = \frac{1}{n}\sum x_i$ and the standard deviation $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ are sensitive to extreme observations and skewness. They should be used primarily for symmetric distributions lacking extreme outliers.
   • Resistant statistics: The median (the middle observation or average of the two middle observations) and the interquartile range ($\text{IQR} = Q_3 - Q_1$) resist extreme observations and are appropriate for skewed data or distributions with outliers.

   Outlier Criteria and Visual Summaries

   Identifying unusual values is critical for evaluating whether data collection errors exist or whether extreme phenomena are present.

   The $1.5 \times \text{IQR}$ Outlier Rule

   An observation is classified as an outlier according to the standard $1.5 \times \text{IQR}$ rule if it falls beyond the designated inner fences:

   • Lower Fence: $Q_1 - 1.5 \times \text{IQR}$
   • Upper Fence: $Q_3 + 1.5 \times \text{IQR}$

   Observations strictly less than the lower fence or strictly greater than the upper fence are flagged as potential outliers.

   The Five-Number Summary and Modified Boxplots

   The five-number summary consists of:

   1. Minimum
   2. First Quartile ($Q_1$)
   3. Median ($Q_2$)
   4. Third Quartile ($Q_3$)
   5. Maximum

   In a modified boxplot, individual outliers are plotted as separate dots or asterisks. The whiskers extend only to the smallest and largest observations that fall inside the fences, rather than to the absolute minimum and maximum.

   Comparing Distributions

   When comparing two or more distributions, explicitly use comparative language (e.g., greater than, less than, approximately equal to) for both center and spread, while explicitly contrasting their shapes and any present outliers in context.

   Analyzing Bivariate Categorical Data

   Two-variable categorical data are summarized using two-way frequency tables (contingency tables), cross-classifying individuals according to two distinct categorical variables.

   Marginal Distributions

   The marginal distribution of a categorical variable summarizes the distribution of values of that variable alone among all individuals in the complete dataset. It is calculated by dividing each row total or column total by the grand total.

   Conditional Distributions

   A conditional distribution shows the distribution of values of one variable restricted to a specific category of another variable. The row total or column total of that specific subpopulation serves as the denominator:

   $$P(A \mid B) = \frac{\text{Frequency of } A \text{ and } B}{\text{Total frequency in category } B}$$

   Statistical Independence

   Two categorical variables are said to be statistically independent if the conditional distribution of one variable is identical across all categories of the other variable.

   • If the proportion of category $A$ is uniform across all categories of $B$ (that is, $P(A \mid B_1) = P(A \mid B_2) = \dots = P(A)$), there is no association between the variables.
   • If the conditional distributions differ substantially between categories, an association exists. Note that association does not imply a causal relationship.

   Linear Modeling and Diagnostic Analysis

   Bivariate quantitative relationships are displayed using scatterplots and evaluated using direction, form, strength, and unusual features.

   The Correlation Coefficient ($r$)

   The Pearson correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables:

   $$r = \frac{1}{n - 1} \sum \left(\frac{x_i - \bar{x}}{s_x}\right)\left(\frac{y_i - \bar{y}}{s_y}\right)$$

   • $-1 \le r \le 1$. Values near $\pm 1$ indicate a strong linear pattern; values near $0$ indicate no linear pattern.
   • $r$ is unitless and non-resistant to outliers.
   • $r$ measures linear association only; a curved relationship can yield $r \approx 0$ despite a perfect deterministic link.

   Least-Squares Regression Line (LSRL)

   The regression model $\hat{y} = a + bx$ minimizes the sum of squared vertical residuals, where:

   $$b = r\frac{s_y}{s_x}, \quad a = \bar{y} - b\bar{x}$$

   • Slope ($b$): The predicted change in $y$ for each one-unit increase in $x$.
   • Coefficient of Determination ($r^2$): The proportion of the total variation in the response variable $y$ that is explained by the linear relationship with the explanatory variable $x$.

   Residuals and Residual Plots

   A residual is the vertical deviation of an observed point from the line: $e_i = y_i - \hat{y}_i$.

   • A linear model is appropriate if the residual plot displays a random scatter around the horizontal zero line with no discernible pattern.
   • A curved pattern or changing spread (fanning) in the residual plot indicates that a linear model is not appropriate.
   • Influential points substantially alter the slope, intercept, or correlation coefficient when included or omitted.