Exhaustive Study Guide: Descriptive Statistics, Tabular & Graphical Displays, Data Visualization, and Dashboards

Learning Objectives

• LO1: Construct and interpret frequency, relative frequency, and percent frequency distributions for categorical data.
• LO2: Construct and interpret bar graphs and pie charts for categorical data.
• LO3: Construct and interpret frequency, relative frequency, and percent frequency distributions for quantitative data.
• LO4: Construct and interpret cumulative frequency, cumulative relative frequency, and cumulative percent distributions for quantitative data.
• LO5: Construct and interpret dot plots, histograms, and stem-and-leaf displays for quantitative data.
• LO6: Interpret the shape of a distribution of data and identify positive skewness, negative skewness, and symmetric distributions.
• LO7: Construct and interpret cross tabulations to summarize data for two variables.
• LO8: Construct and interpret a scatter diagram for two quantitative variables.
• LO9: Identify and explain Simpson's paradox from a cross tabulation.
• LO10: Construct and interpret side-by-side and stacked bar charts.
• LO11: Create and interpret choropleth maps and cartograms for applications to geospatial data.
• LO12: Apply concepts of data-ink ratio and decluttering to improve table and chart designs.

Statistics in Practice: Colgate-Palmolive Company

• Company Background: Founded in New York City in $1806$ as a small soap and candle shop. Today, Colgate-Palmolive employs over $33,000$ people across nearly $200$ countries and territories worldwide.
• Product Portfolio: Markets well-known consumer brands including Colgate, Palmolive, Softsoap, Irish Spring, Hill's Science Diet, Ajax, Fabuloso, Hill's Pet Nutrition, and Tom's of Maine.
• Statistical Quality Control Application: Colgate-Palmolive uses statistical quality assurance in the production of home laundry detergents.
  • Detergent cartons are filled by weight. However, variations in detergent powder density alter product volume.
  • High powder density leads to smaller volume for a specified weight, causing cartons to appear underfilled when opened by consumers.
  • To prevent underfilling perception, acceptable limits are placed on powder density.
• Sampling Procedure & Specifications:
  • Periodic statistical samples are drawn, and powder density is recorded.
  • Density levels exceeding $0.40$ are classified as unacceptably high.
  • In a sample of $n = 150$ detergent packages collected over a one-week period, the density frequency distribution and histogram yielded the following:
  • $0.29 - 0.30$: Frequency = $30$
  • $0.31 - 0.32$: Frequency = $75$
  • $0.33 - 0.34$: Frequency = $32$
  • $0.35 - 0.36$: Frequency = $9$
  • $0.37 - 0.38$: Frequency = $3$
  • $0.39 - 0.40$: Frequency = $1$ (representing less than $1\%$ of samples near the unacceptable $0.40$ limit)
  • Total Frequency: $150$
  • Quality Control Conclusion: Because all sampled densities are less than or equal to $0.40$, the production operation meets quality guidelines.

Fundamentals of Descriptive Statistics & Data Classification

• Categorical Data: Data that use labels or names to identify categories of like items.
• Quantitative Data: Numerical values that measure how much or how many.
• Data Visualization: The use of graphical displays to summarize and present information about a dataset.
• Scope of Analysis: Visualizations and tabular summaries can analyze a single variable (univariate analysis) or explore the relationship between two variables (bivariate analysis).

Summarizing Categorical Data: Tabular Methods

• Frequency Distribution: A tabular summary of data showing the number (frequency) of observations in each of several non-overlapping categories or classes.
• Soft Drink Purchase Sample Data ($n = 50$):
  • Data Values: Coca-Cola ($19$), Diet Coke ($8$), Dr. Pepper ($5$), Pepsi ($13$), Sprite ($5$).
  • Total Frequency: $50$
• Relative Frequency and Percent Frequency Distributions:
  • Relative Frequency Formula:     $\text{Relative Frequency of a Class} = \frac{\text{Frequency of the Class}}{n}$
  • Percent Frequency Formula:     $\text{Percent Frequency} = \text{Relative Frequency} \times 100$
  • Soft Drink Tabular Summary:
  • Coca-Cola: Frequency = $19$, Relative Frequency = $\frac{19}{50} = 0.38$, Percent Frequency = $38\%$
  • Diet Coke: Frequency = $8$, Relative Frequency = $\frac{8}{50} = 0.16$, Percent Frequency = $16\%$
  • Dr. Pepper: Frequency = $5$, Relative Frequency = $\frac{5}{50} = 0.10$, Percent Frequency = $10\%$
  • Pepsi: Frequency = $13$, Relative Frequency = $\frac{13}{50} = 0.26$, Percent Frequency = $26\%$
  • Sprite: Frequency = $5$, Relative Frequency = $\frac{5}{50} = 0.10$, Percent Frequency = $10\%$
  • Totals: Frequency = $50$, Relative Frequency = $1.00$, Percent Frequency = $100\%$
  • Substantive Insight: The top three soft drink brands (Coca-Cola, Pepsi, and Diet Coke) represent $38\% + 26\% + 16\% = 80\%$ of total purchases.
• Constructing Categorical Distributions in Excel:
  • Use Excel's Recommended PivotTables tool.
  • Data range: A2:A51 with brand labels in A1.
  • Procedure: Select cell in range $\rightarrow$ Insert tab $\rightarrow$ Tables group $\rightarrow$ Recommended PivotTables $\rightarrow$ OK.
  • In the PivotTable Fields pane, place brand purchased in Rows and Count of brand purchased in Values.
  • Formulas for Relative and Percent Frequency:
  • Cell C4 Relative Frequency formula: =B4/$B$9 (copy down through C8).
  • Cell D4 Percent Frequency formula: =C4*100 (copy down through D8).
  • Total sums in C9 and D9: =SUM(C4:C8) and =SUM(D4:D8).

Summarizing Categorical Data: Graphical Methods

• Bar Charts:
  • A graphical display depicting categorical data summarized in a frequency, relative frequency, or percent frequency distribution.
  • Labels are specified on one axis; frequency/relative frequency scale is placed on the other.
  • Bars must be drawn with fixed width and separated by spaces to emphasize that categories are discrete.
  • Column Chart vs. Bar Chart: Excel terms vertical bar charts as "Column Charts" and horizontal bar charts as "Bar Charts".
  • Sorted Bar Chart: Ranks categories from highest frequency on the left to lowest on the right, making brand relative preferences immediately clear.
• Pie Charts:
  • Divides a circle into sectors corresponding to the relative frequency or percent frequency of each category.
  • Sector Angle Formula:     $\text{Sector Angle (Degrees)} = \text{Relative Frequency} \times 360$
  • Soft Drink Sector Angles:
  • Coca-Cola: $0.38 \times 360 = 136.8^\circ$
  • Diet Coke: $0.16 \times 360 = 57.6^\circ$
  • Dr. Pepper: $0.10 \times 360 = 36.0^\circ$
  • Pepsi: $0.26 \times 360 = 93.6^\circ$
  • Sprite: $0.10 \times 360 = 36.0^\circ$
• Critique of 3D Charts and Visual Elements:
  • Three-dimensional pie charts add depth perspective that distorts visual proportional perception without offering new information.
  • Chart legends force the user's eyes to scan back and forth between key and chart.
  • Bar charts are generally superior to pie charts because humans compare linear heights more accurately than angles or area slices.

Summarizing Quantitative Data: Tabular Methods

• Steps to Construct Quantitative Frequency Distributions:
  1. Determine the number of non-overlapping classes: General guideline recommends between $5$ and $20$ classes depending on data size.
  2. Determine the width of each class:      $\text{Approximate Class Width} = \frac{\text{Largest Data Value} - \text{Smallest Data Value}}{\text{Number of Classes}}$
  3. Determine the class limits: Define lower and upper class limits so each observation falls into exactly one class.
• Sanderson & Clifford Audit Time Application ($n = 20$ sample audit completion times in days):
  • Raw Data: $12, 14, 19, 18, 15, 15, 18, 17, 20, 27, 22, 23, 22, 21, 33, 28, 14, 18, 16, 13$
  • Largest Value: $33$; Smallest Value: $12$
  • Class Width Calculation:     $\text{Approximate Class Width} = \frac{33 - 12}{5} = \frac{21}{5} = 4.2$     Round up to a convenient class width of $5$ days.
  • Audit Time Frequency Distribution Table:
  • Class $10 - 14$ days: Frequency = $4$, Relative Frequency = $0.20$, Percent Frequency = $20\%$
  • Class $15 - 19$ days: Frequency = $8$, Relative Frequency = $0.40$, Percent Frequency = $40\%$
  • Class $20 - 24$ days: Frequency = $5$, Relative Frequency = $0.25$, Percent Frequency = $25\%$
  • Class $25 - 29$ days: Frequency = $2$, Relative Frequency = $0.10$, Percent Frequency = $10\%$
  • Class $30 - 34$ days: Frequency = $1$, Relative Frequency = $0.05$, Percent Frequency = $5\%$
  • Totals: Frequency = $20$, Relative Frequency = $1.00$, Percent Frequency = $100\%$
• Class Midpoint:
  • The value halfway between the lower and upper class limits.
  • Midpoints for audit time classes: $12, 17, 22, 27, 32$
• Cumulative Frequency Distributions:
  • Shows the number of data items with values less than or equal to the upper class limit of each class.
  • Cumulative Relative Frequency: Proportion of items $\le$ upper class limit.
  • Cumulative Percent Frequency: Percentage of items $\le$ upper class limit.
  • Audit Time Cumulative Summary Table:
  • $\le 14$ days: Cumulative Frequency = $4$, Cumulative Relative Frequency = $0.20$, Cumulative Percent Frequency = $20\%$
  • $\le 19$ days: Cumulative Frequency = $12$, Cumulative Relative Frequency = $0.60$, Cumulative Percent Frequency = $60\%$
  • $\le 24$ days: Cumulative Frequency = $17$, Cumulative Relative Frequency = $0.85$, Cumulative Percent Frequency = $85\%$
  • $\le 29$ days: Cumulative Frequency = $19$, Cumulative Relative Frequency = $0.95$, Cumulative Percent Frequency = $95\%$
  • $\le 34$ days: Cumulative Frequency = $20$, Cumulative Relative Frequency = $1.00$, Cumulative Percent Frequency = $100\%$
• Important Rules & Exceptions:
  • Class Limits Precision: Depends on level of measurement rounding (e.g., nearest tenth $\rightarrow 10.0 - 14.9$; nearest hundredth $\rightarrow 10.00 - 14.99$).
  • Open-End Classes: Classes requiring only a lower limit or upper limit (e.g., "$35$ or more") used to handle extreme outlier values.
  • Cumulative Summary Rule: The final entry in a cumulative frequency distribution equals $n$ ($20$); in cumulative relative frequency equals $1.00$; in cumulative percent frequency equals $100$

Summarizing Quantitative Data: Graphical Methods

• Dot Plots:
  • Displays individual quantitative observations along a horizontal axis range.
  • Useful for displaying exact details and comparing distributions across variables.
  • Audit data plot: 3 dots above $18$; 2 dots above $14, 15, 22$; 1 dot above $12, 13, 16, 17, 19, 20, 21, 23, 27, 28, 33$
• Histograms:
  • A graphical display where rectangles are drawn over class intervals.
  • Rectangle height corresponds to frequency, relative frequency, or percent frequency.
  • Unlike bar charts, adjacent rectangles touch directly without spaces to reflect continuous quantitative ranges.
• Distribution Shapes and Skewness:
  • Moderately Skewed Left: Tail extends farther to the left (e.g., typical exam scores where most students perform well and few score low).
  • Moderately Skewed Right: Tail extends farther to the right (e.g., housing prices where few high-value homes skew the right tail).
  • Symmetric: Left tail mirrors right tail (e.g., SAT scores, human heights, human weights).
  • Highly Skewed Right: Extreme long right tail (e.g., retail store daily purchase totals, executive salaries).
• Stem-and-Leaf Display:
  • Displays rank order and shape of quantitative data simultaneously.
  • Stem: Leading digits placed to the left of a vertical line.
  • Leaf: Last single digit recorded to the right of the vertical line for each observation.
  • Haskins Technology Aptitude Test Data ($n = 50$ test scores between $68$ and $141$):
  • Stems: $6, 7, 8, 9, 10, 11, 12, 13, 14$
  • Row $6 \mid 8 \ 9$ represents $68, 69$
  • Row $7 \mid 2 \ 3 \ 3 \ 5 \ 6 \ 6$ represents $72, 73, 73, 75, 76, 76$
  • Stretched Stem-and-Leaf Display: Stretches data by using two rows per leading digit (first row for leaves $0 - 4$; second row for leaves $5 - 9$).
  • Leaf Unit: Indicates how to scale stem-leaf values to reconstruct original data magnitudes.
  • Example: Fast food hamburger sales (15-week data). Stem = $15$, Leaf = $6$, Leaf Unit = $10$
  • Reconstructed value: $(15 \times 10) + (6 \times 1) = 156 \times 10 = 1,560$ hamburgers.
  • If unspecified, Leaf Unit defaults to $1$

Summarizing Two Variables: Crosstabulations & Simpson's Paradox

• Crosstabulation: A tabular summary of data for two variables simultaneously (can combine categorical-categorical, categorical-quantitative, or quantitative-quantitative data).
• Los Angeles Restaurant Sample ($n = 300$ restaurants):
  • Variables: Quality Rating (Categorical: Good, Very Good, Excellent) and Meal Price (Quantitative: $\$10 - 19, \$20 - 29, \$30 - 39, \$40 - 49$).
  • Cross Tabulation Frequency Summary:
  • Good Quality: $\$10 - 19 \rightarrow 42$, $\$20 - 29 \rightarrow 40$, $\$30 - 39 \rightarrow 2$, $\$40 - 49 \rightarrow 0$ | Row Total = $84$
  • Very Good Quality: $\$10 - 19 \rightarrow 34$, $\$20 - 29 \rightarrow 64$, $\$30 - 39 \rightarrow 46$, $\$40 - 49 \rightarrow 6$ | Row Total = $150$
  • Excellent Quality: $\$10 - 19 \rightarrow 2$, $\$20 - 29 \rightarrow 14$, $\$30 - 39 \rightarrow 28$, $\$40 - 49 \rightarrow 22$ | Row Total = $66$
  • Column Totals: $\$10 - 19 \rightarrow 78$, $\$20 - 29 \rightarrow 118$, $\$30 - 39 \rightarrow 76$, $\$40 - 49 \rightarrow 28$ | Grand Total = $300$
  • Row Percentages Table (Distribution of price within each quality level):
  • Good: $\$10 - 19 \rightarrow 50.0\%$, $\$20 - 29 \rightarrow 47.6\%$, $\$30 - 39 \rightarrow 2.4\%$, $\$40 - 49 \rightarrow 0.0\%$
  • Very Good: $\$10 - 19 \rightarrow 22.7\%$, $\$20 - 29 \rightarrow 42.7\%$, $\$30 - 39 \rightarrow 30.6\%$, $\$40 - 49 \rightarrow 4.0\%$
  • Excellent: $\$10 - 19 \rightarrow 3.0\%$, $\$20 - 29 \rightarrow 21.2\%$, $\$30 - 39 \rightarrow 42.4\%$, $\$40 - 49 \rightarrow 33.4\%$
  • Column Percentages Table (Distribution of quality within each price class):
  • $\$10 - 19$: Good = $53.8\%$, Very Good = $43.6\%$, Excellent = $2.6\%$
  • $\$20 - 29$: Good = $33.9\%$, Very Good = $54.2\%$, Excellent = $11.9\%$
  • $\$30 - 39$: Good = $2.6\%$, Very Good = $60.5\%$, Excellent = $36.8\%$
  • $\$40 - 49$: Good = $0.0\%$, Very Good = $21.4\%$, Excellent = $78.6\%$
• Simpson's Paradox:
  • Occurs when conclusions drawn from two or more separate crosstabulations are reversed when the data are aggregated into a single summary table.
  • Judicial Trial Verdict Appeal Example (Judge Susan Luckett vs. Judge Dennis Kendall):
  • Aggregated Data Table (Both Courts Combined):
    • Judge Luckett: Upheld = $129$ ($86\%$), Reversed = $21$ ($14\%$) | Total = $150$
    • Judge Kendall: Upheld = $110$ ($88\%$), Reversed = $15$ ($12\%$) | Total = $125$
    • Aggregated Conclusion: Judge Kendall appears to perform better ($88\% > 86\%$ upheld).
  • Disaggregated Data Tables (By Court Type):
    • Common Pleas Court:
    • Luckett: Upheld = $29$ ($91\%$), Reversed = $3$ ($9\%$) | Total = $32$
    • Kendall: Upheld = $90$ ($90\%$), Reversed = $10$ ($10\%$) | Total = $100$
    • Result: Luckett superior ($91\% > 90\%$).
    • Municipal Court:
    • Luckett: Upheld = $100$ ($85\%$), Reversed = $18$ ($15\%$) | Total = $118$
    • Kendall: Upheld = $20$ ($80\%$), Reversed = $5$ ($20\%$) | Total = $25$
    • Result: Luckett superior ($85\% > 80\%$).
  • Paradox Explanation: Court type is a hidden/confounding variable. Reversal rates are higher in Municipal Court. Judge Luckett tried $118$ of $150$ cases (78.7\%$) in Municipal Court, whereas Judge Kendall tried 100$of$125$cases ($80.0\%$) in Common Pleas Court. Aggregating masks the underlying conditional relationships.

Summarizing Two Variables: Graphical Displays

• Scatter Diagram and Trend Line:
  • Used to show the relationship between two quantitative variables.
  • San Francisco Electronics Store Commercials ($x$) vs. Sales ($y$ in \$100s) Sample ($n = 10$ weeks):
  • Data pairs: $(2, 50), (5, 57), (1, 41), (3, 54), (4, 54), (1, 38), (5, 63), (3, 48), (4, 59), (2, 46)$
  • Scatter diagram pattern shows an upward-sloping trend line, indicating a positive relationship between TV commercials and weekly sales.
  • General Relationship Types:
  • Positive Relationship: Points scatter upward from left to right.
  • Negative Relationship: Points scatter downward from left to right ($y$ decreases as $x$ increases).
  • No Apparent Relationship: Points are scattered horizontally with no linear slope.
• Side-by-Side Bar Chart:
  • Displays multiple bars side-by-side on the same visual area to compare categories of two variables.
• Stacked Bar Chart:
  • Each bar represents $100\%$ total for a category, broken into stacked rectangular colored segments reflecting relative/percent frequency components.

Visualizing Geospatial Data

• Choropleth Maps:
  • Geographic visualizations using color shades or symbols to depict quantitative or categorical data across mapped regions.
  • Example: Weather map showing US daily high temperatures using warm red spectrum colors for $80^\circ \text{F} - 90^\circ \text{F}$ ranges in the South/Southwest and cooler purple spectrum colors for lower temperature zones.
• Cartograms:
  • Map-like diagrams that intentionally distort geographic land boundaries based on variable magnitudes.
  • Example: $2010$ US Population Cartogram ($\text{Total Population} = 308,745,538$) using grid squares where 1 square = $50,000$ people. States like California and New York appear massive, while vast land area states like Alaska, Nevada, Idaho, Montana, North Dakota, and South Dakota shrink significantly to convey low population density.

Data Visualization Best Practices & Guidelines

• Data-Ink Ratio:   $\text{Data-Ink Ratio} = \frac{\text{Data-Ink}}{\text{Total Ink Used for Display}}$
  • Data-Ink: Ink needed to convey intended dataset insights.
  • Non-Data-Ink: Unnecessary clutter, excess gridlines, decorative backgrounds, visual distractions.
• Decluttering Process (Gustin Chemical Sales Example):
  • Dataset: Planned vs. Actual Sales (in $\$1,000\text{s}$) for Northeast ($540$ plan, $447$ actual), Northwest ($420$ plan, $447$ actual), Southeast ($575$ plan, $556$ actual), Southwest ($360$ plan, $341$ actual).
  • Remove redundant gridlines and borders.
  • Add explicit axis labels and measurement units.
  • Avoid 3D perspective distortion.
• General Visual Design Rules:
  1. Provide a clear, concise title.
  2. Maximize the data-ink ratio.
  3. Remove unnecessary gridlines and duplicate labels.
  4. Do not use 3D visual effects when 2D is sufficient.
  5. Clearly label axes with variable names and units.
  6. Select distinct colors (avoid red/green pairings to accommodate colorblind users).
  7. Place chart legends close to plotted data points.
• Categorization of Display Types by Purpose:
  • Show Data Distributions: Bar Chart, Pie Chart, Dot Plot, Histogram, Stem-and-Leaf Display.
  • Make Comparisons: Side-by-Side Bar Chart, Stacked Bar Chart.
  • Show Relationships: Scatter Diagram, Trend Line, Choropleth Map, Cartogram.

Data Dashboards: Operational & Strategic Monitoring

• Data Dashboard Definition: A consolidated set of visual displays organizing key metrics to monitor organizational performance efficiently.
• Key Performance Indicators (KPIs / KPMs): Crucial performance metrics monitored continuously (e.g., inventory on hand, daily sales, on-time delivery percentages, call resolution times).
• Grogan Oil Information Technology Call Center Dashboard Case Study (Shift 1 starting 8:00 AM, Headquarters in Austin, branch offices in Houston and Dallas):
  • Stacked Bar Chart (Call Volume over Time): Shows hourly issue volume ($8:00 - 12:00$). Email calls peak at $8:00$ AM ($13$) and fall off ($1$ at $12:00$ PM); software calls peak midmorning ($10:00$ AM = $9$).
  • Time Breakdown Chart: Displays operational employee activity: Software = $47\%$, Email = $22\%$, Internet = $19\%$, Idle = $15\%$
  • Unresolved Cases Beyond 15 Minutes: Tracks prolonged unresolved calls: W59 (Internet, $20$ min), W24 (Software, $170$ min), W5 (Software, $280$ min), T57 (Software, $340+$ min, leftover from prior shift).
  • Call Volume by Office: Displays high email calls from Austin ($49$ email) and high software calls from Dallas ($166$ software) due to an ongoing enterprise software installation.
  • Resolution Time Histogram: Shows resolution distributions ($<1$ min = $10$, $1-2$ min = $13$, $2-3$ min = $10$, $3-4$ min = $9$, $4-5$ min = $5$, up to $32+$ min = $2$).
• Dashboard Management Levels:
  • Operational: Real-time decision-making (e.g., call center shift staffing adjustments).
  • Tactical: Mid-level operational oversight (e.g., logistics carrier mode selection).
  • Strategic: High-level corporate health evaluation (e.g., aggregate revenue, capacity utilization metrics).