Descriptive Statistics, Graphical Representations, and Contingency Table Analysis

Effect of Data Values on Standard Deviation

Kelly's Homework Assignment Data:
- Summary statistics calculated from Kelly's first $7$ homework assignments:
  - Sample Mean ( $\bar{x}$ ): $27.71$
  - Sample Standard Deviation ( $s$ ): $6.34$
- Standard Deviation as a Measure of Spread:
  - Standard deviation quantifies the dispersion or spread of individual data points relative to a central typical value (such as the mean or median).
  - It represents the typical distance of data values from the mean. For example, in an exam with a mean score of $90\%$ , a score of $60\%$ represents a large deviation or spread from the typical value.
Determining the Impact of Additional Data Values on Standard Deviation:
- Rather than manually recalculating the standard deviation formula or re-entering data into a calculator for every new observation, an interval-based method can determine whether a new score increases or decreases the standard deviation.
- Interval Construction Method ( $\mu \pm \sigma$ ):
  - Calculate the bounds for one standard deviation around the mean:
    - $\text{Lower Bound} = \text{Mean} - \text{Standard Deviation} = 27.71 - 6.34 = 21.37$
    - $\text{Upper Bound} = \text{Mean} + \text{Standard Deviation} = 27.71 + 6.34 = 34.05$
  - This establishes three distinct region intervals:
    1. $\text{Interval 1 (Below } 1\,SD\text{)}: x < 21.37$
    2. $\text{Interval 2 (Within } 1\,SD\text{)}: 21.37 \le x \le 34.05$
    3. $\text{Interval 3 (Above } 1\,SD\text{)}: x > 34.05$
Rules for Increasing vs. Decreasing Standard Deviation:
- Decreasing Standard Deviation: Any new data value falling within one standard deviation of the mean ( $[21.37, 34.05]$ ) pulls the overall spread closer to the center, thereby decreasing the standard deviation.
- Increasing Standard Deviation: Any new data value falling outside or beyond one standard deviation of the mean ( $< 21.37$ or $> 34.05$ ) adds relative spread to the dataset, thereby increasing the standard deviation.
Evaluation of Specific Homework Scores:
- Score of $41$ : Lies in Interval 3 ( $41 > 34.05$ ) $\rightarrow$ Increases standard deviation.
- Score of $26$ : Lies in Interval 2 ( $21.37 \le 26 \le 34.05$ ) $\rightarrow$ Decreases standard deviation.
- Score of $46$ : Lies in Interval 3 ( $46 > 34.05$ ) $\rightarrow$ Increases standard deviation.
- Score of $22$ : Lies in Interval 2 ( $21.37 \le 22 \le 34.05$ ) $\rightarrow$ Decreases standard deviation.

Matching Boxplots to Histograms and Estimating Standard Deviation

Technique for Matching Boxplots and Histograms:
- Compare the extreme values (range) of the distribution between the two graphical representations:
  - Identify the minimum (smallest value) and maximum (largest value) on the horizontal axis of the histogram.
  - In a vertical boxplot, locate the lower boundary or whisker (minimum value) and upper boundary or whisker/outlier (maximum value).
  - Match the histogram to the boxplot displaying the closest corresponding minimum and maximum values.
- Golf Ball Brand Example:
  - Unlabeled histogram feature bounds: minimum value near $25$ , maximum value near $60$
  - Comparison across brand boxplots (Nike, Callaway, etc.):
    - Drawing a horizontal line from the histogram minimum ( $25$ ) intersects the Nike boxplot lower boundary.
    - Drawing a horizontal line from the histogram maximum ( $60$ ) aligns with the top height of the Nike boxplot distribution.
    - Result: The histogram uniquely corresponds to the Nike boxplot.
Shortcut Formula for Estimating Standard Deviation from a Histogram:
- When precise summary statistics are unavailable, standard deviation can be estimated directly from a histogram using the range rule of thumb: $\text{Estimated Standard Deviation} = \frac{\text{Range}}{6} = \frac{\text{Maximum} - \text{Minimum}}{6}$
- Theoretical Rationale:
  - In a symmetric distribution, substantially all data falls within $3$ standard deviations on either side of the mean ( $\mu \pm 3\sigma$ ).
  - Dividing the total span into $6$ equal standard deviation intervals covers approximately the entire spread of the data.
- Sample Calculation:
  - $\text{Maximum} = 60$
  - $\text{Minimum} = 25$
  - $\text{Range} = 60 - 25 = 35$
  - $\text{Estimated Standard Deviation} = \frac{35}{6} \approx 5.8$

Comparative Analysis of Boxplots and Skewness Relationships

Comparing Interquartile Range (IQR) Visually:
- $\text{IQR} = Q_3 - Q_1$ (Upper Quartile minus Lower Quartile).
- On a boxplot, IQR is physically represented by the vertical height (or horizontal length) of the central box.
- Comparison: Comparing Callaway vs. Nike boxplots shows that the box height for Callaway is visibly taller than the box height for Nike. Consequently, $\text{IQR}_{\text{Callaway}} > \text{IQR}_{\text{Nike}}$ .
Evaluating Maximum Values with Outliers:
- When identifying the absolute maximum value of a distribution from a boxplot, individual outlier points plotted beyond the whiskers represent the true maximum data values.
- If an outlier exists above the upper whisker, the outlier point dictates the maximum value rather than the top whisker itself.
Determining Skewness and the Relationship Between Mean and Median:
- While the median ( $Q_2$ ) is explicitly drawn as a line inside a boxplot, the mean is not directly displayed. The relationship between mean and median is determined by inspecting the skewness of the boxplot:
  - Right-Skewed Distribution:
    - The upper section of the box (between median $Q_2$ and upper quartile $Q_3$ ) is noticeably larger/wider than the lower section (between $Q_1$ and median $Q_2$ ), or the upper whisker/tail is longer.
    - Extreme larger values pull the mean upward.
    - Mathematical Relationship: $\text{Mean} > \text{Median}$ .
  - Left-Skewed Distribution:
    - The lower section of the box (between $Q_1$ and median $Q_2$ ) is larger than the upper section, or the lower whisker/tail is longer.
    - Extreme smaller values pull the mean downward.
    - Mathematical Relationship: $\text{Mean} < \text{Median}$ .
  - Symmetrical Distribution:
    - The median line divides the box into equal upper and lower halves.
    - Mathematical Relationship: $\text{Mean} = \text{Median}$ .
- Application to Nike Brand Data:
  - The upper portion of the Nike box is larger than the lower portion above the median line, indicating a right-skewed distribution.
  - Conclusion: For Nike, $\text{Mean} > \text{Median}$ .
Comparing Proportions of Data from Boxplots:
- By definition of quartiles:
  - $\text{Proportion of data above the median } (Q_2) = 50\% \text{ or } 0.50$
  - $\text{Proportion of data below the median } (Q_2) = 50\% \text{ or } 0.50$
- Example (Comparing Customer Age relative to 40 years old):
  - For Callaway customers, age $40$ aligns exactly with the median ( $Q_2$ ) line. Therefore, exactly $50\%$ of Callaway customers are older than $40$
  - For Nike customers, the line representing age $40$ lies above the median line ( $Q_2$ ). Therefore, the proportion of Nike customers older than $40$ is less than $50\%$ ( $< 0.50$ ).
  - Conclusion: The proportion of Callaway customers older than $40$ ( $50\%$ ) is greater than the proportion of Nike customers older than $40$ ( $< 50\%$ ).

Visualizing Categorical Data and Sample Statistics

Distinguishing Graphical Tools by Data Type:
- Numerical (Quantitative) Variables: Visualized using dot plots, histograms, and boxplots.
- Categorical (Qualitative) Variables: Visualized using bar graphs, pie charts, and mosaic plots (for two categorical variables).
Bar Graphs for Categorical Data:
- Displays individual categories along one axis and counts/frequencies along the opposing axis.
- The height of each bar represents the total frequency (number of observations) falling into that specific category.
- Ames, Iowa Housing Dataset Example:
  - Variables: Exterior material (Vinyl Siding, Wood Siding, Brick Face, Cement Board) and Overall Quality Rating (High, Good, Fair, Poor).
  - Most Common Category: Identified by the tallest bar (Vinyl Siding).
  - Least Common Category: Identified by the shortest bars (Brick Face and Cement Board).
  - Frequency Estimation: High overall quality rating category corresponds to approximately $8$ homes out of a total sample of $45$ homes.
Categorical Proportions and Notation:
- A proportion represents the fraction of observations falling into a specific category: $\text{Proportion} = \frac{\text{Count in Category}}{\text{Total Number of Observations (Cases)}}$
- Notation Distinction between Sample and Population:
  - $\hat{p}$ ("p-hat"): Represents the sample proportion calculated from observed sample data.
  - $p$ : Represents the true population proportion for the entire population parameter.
- Sample Proportion Calculations:
  - High Quality Rating Homes: $\hat{p} = \frac{8}{45} \approx 0.178$
  - Wood Siding Homes: $\hat{p} \approx 0.30$ ( $30\%$ ) based on bar height relative to total sample size.

Contingency Tables and Two-Way Categorical Relationships

Evaluating Relationships Between Two Categorical Variables:
- To determine whether two categorical variables are related or associated, evaluate whether Variable A is independent of Variable B.
- Contingency Table (Two-Way Table):
  - A grid format used to summarize bivariate categorical data.
  - Rows represent the categories of one variable (e.g., Early Class status).
  - Columns represent the categories of a second variable (e.g., Class Year).
  - Cells contain joint counts (frequencies) for each combination of categories.
Student Survey Dataset Example:
- Sample Size: $253$ randomly selected students surveyed regarding class schedules and academic year.
- Variables: Early Class status (No Early Class vs. Early Class) and Class Year (Freshman, Sophomore, Junior, Senior).
- Joint and Marginal Counts from Table:
  - Total students surveyed = $253$
  - Total students with an early class = $168$
  - Freshmen with no early class = $8$
  - Freshmen with an early class = $39$
  - Total Freshmen surveyed = $47$ ( $8 + 39 = 47$ )
  - Seniors with an early class = $32$
- Proportion Calculations:
  - Overall proportion of surveyed students with an early class: $\hat{p}_{\text{overall}} = \frac{168}{253} \approx 0.664$
  - Proportion of Freshmen with an early class: $\hat{p}_{\text{Freshmen}} = \frac{39}{47} \approx 0.830$

Mosaic Plots and Testing for Independence

Mosaic Plot Construction:
- A mosaic plot displays bivariate categorical data using stacked rectangular segments.
- Vertical Columns: Each column represents a category of the primary explanatory variable (e.g., Class Year: Freshman, Sophomore, Junior, Senior). The width of each column reflects the relative sample size of that category.
- Stacked Segments: Each column is divided vertically into stacked segments representing the proportion of subcategories for the second variable (e.g., Early Class vs. No Early Class).
Determining Association vs. Independence from a Mosaic Plot:
- Associated (Dependent) Variables:
  - If the vertical segment heights for a specific response category differ noticeably across the primary variable's columns, the two variables are associated (a relationship exists).
  - Class Schedule Example: Freshmen have a significantly higher proportion segment for early classes ( $\hat{p} \approx 0.83$ ) compared to Sophomores, Juniors, and Seniors. Because segment heights vary across years, Class Year and Early Class status are associated.
- Independent Variables:
  - If the vertical segment heights for each category remain identical or equal across all columns of the primary variable, the variables are independent (no relationship exists).
Political Affiliation and Iraq War Support Example:
- Variables: Political Affiliation (Democrat, Independent, Republican) and Iraq War Support (Support vs. Oppose).
- Mosaic Plot Observation: Comparing the blue "Support" segment heights across columns reveals distinct differences between Democrats, Independents, and Republicans.
- Conclusion: Iraq War Support and Political Affiliation are associated (dependent).