Visualizing Numerical Data and Summary Statistics

Scatterplots: Graphically display relationships between two numerical variables to assess potential associations.
Stacked Dot Plots: Display single numerical distributions (such as GPA) by stacking individual observations vertically along a numerical scale.

Data Density: Bars represent frequency, with higher bars indicating values where data are relatively more concentrated.
Bin Width: Choice of bin width impacts the visual distribution, either hiding underlying patterns or adding excessive visual noise.
Modality: Evaluated by visualizing a smooth curve over the histogram bars:
- Unimodal: Displays a single prominent peak.
- Bimodal / Multimodal: Displays two or more prominent peaks.
- Uniform: Displays no distinct peaks.
Skewness: Determined by the direction of the long tail:
- Right-Skewed: Tail extends rightward toward higher values.
- Left-Skewed: Tail extends leftward toward lower values.
- Symmetric: Left and right sides are evenly balanced.

Five-Number Summary Components:
- Median ( $Q_2$ ): Midpoint value dividing data into equal halves ( $50\,\%$ below). For an even number of observations, it is calculated as the average of the middle two values: $\text{Median} = \frac{\text{Value}_1 + \text{Value}_2}{2}$
- First Quartile ( $Q_1$ ): Median of the lower half of data ( $25\text{th}$ percentile).
- Third Quartile ( $Q_3$ ): Median of the upper half of data ( $75\text{th}$ percentile).
- Interquartile Range (IQR): Covers the central $50\,\%$ of data, defined as $Q_3 - Q_1$ .
- Whiskers and Outliers: Whiskers extend to extreme non-outlier values; data points beyond whisker boundaries are plotted individually as outliers.

Categorical Comparison: Visualizing a numerical variable across categorical groups using side-by-side box plots enables direct comparison of distributions (e.g., comparing club membership counts across First-year, Sophomore, Junior, and Senior classes).