6.9: Detailed Figure Captions - Chapter 6

Statistical Foundation for Analytics -- Figure Specifications

Overview

• Chapter: 6 -- Statistical Foundation for Analytics
• Total Figures Proposed: 8
• Figure Types Used: Conceptual Illustration, Data Chart, Comparison Table, Flowchart, Infographic

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Figure 6.1.1: Title: Measures of Central Tendency in Skewed vs. Symmetric Distributions 

Detailed Caption: Three density curves are arranged side by side, each sharing the same horizontal baseline but telling a different story about where mean, median, and mode land relative to one another. In the left-skewed panel, the long tail pulls leftward and the three dashed reference lines fan out in the order Mode (rightmost, navy) → Median (middle, teal) → Mean (leftmost, gold), reflecting how the mean chases the tail while the mode anchors the peak. The symmetric center panel collapses all three measures onto a single navy dashed line — Mean = Median = Mode — because perfect bilateral symmetry leaves no tail to distort any of them. The right-skewed panel mirrors the left-skewed arrangement in reverse: Mode sits leftmost at the peak, Median steps right, and Mean stretches furthest into the long right tail, producing the familiar Mode < Median < Mean ordering. A shared color-coded legend at the bottom ties the dashed-line colors to their labels across all three panels. By placing the three scenarios on the same canvas and using position rather than annotation to show divergence, the figure converts an abstract inequality into a visual fact: the mean is pulled toward whichever tail exists, the mode stays at the peak regardless, and the median always sits between them — a spatial relationship students can recall without reciting a rule. 

Figure 6.1.2: Comparing Datasets with Same Mean but Different Variability 

Detailed Caption: Two side-by-side histograms share an identical mean of $500, marked by the same navy dashed vertical line in each panel, but their bar shapes tell opposite stories about spread. Store A's teal bars rise steeply to a tall central column and taper sharply within a $400–$600 window, producing a narrow mountain that signals daily sales rarely deviate far from the average. Store B's gold bars are shallower and spread across a $100–$900 span, forming a broad, low plateau where every outcome from near-zero to nine hundred dollars appears with non-trivial frequency. Because the mean line falls at exactly the same x-position in both charts, the visual contrast between the tight teal cluster and the sprawling gold plateau cannot be attributed to differences in center — it can only be attributed to differences in dispersion. The figure makes the chapter's core claim about descriptive statistics tangible: two businesses can share identical average sales while one manager faces a predictable, plannable revenue stream and the other faces a volatile one, and no single number captures that distinction unless spread is measured alongside center. 

Figure 6.2.1

Title: The Normal Distribution and the Empirical Rule (68-95-99.7) 

Detailed Caption: A smooth navy bell curve spans the full width of the figure, centered at μ on a labeled x-axis that runs from −3σ through +3σ in unit steps. Three nested shaded regions fill the area under the curve in concentric layers: the innermost gold band between −1σ and +1σ carries the label 68.2%, the middle teal layer extending to ±2σ is labeled 95.4%, and the pale outermost teal wash reaching to ±3σ is labeled 99.7%, each percentage floating inside its own region so the reader can associate the number with its visual extent rather than hunting for a legend. The Probability Density label on the vertical axis and the σ tick-marks on the horizontal axis together set the coordinate system without requiring any additional annotation. By rendering the three bands as nested fills rather than separate zones, the figure communicates that each wider band includes everything inside the narrower ones — a cumulative structure that the phrase "within two standard deviations" implies but that a table of numbers cannot show. The 68-95-99.7 rule is the single most-used mental shortcut in applied statistics, and grounding it in the actual shape of the curve gives students a spatial intuition they can invoke whenever they need to judge how unusual an observed value is. 

Figure 6.2.2

Title: Binomial Distribution Bar Chart (n=10, p=0.5) 

Detailed Caption: Eleven teal bars rise from a common baseline, one for each integer outcome from k = 0 to k = 10, with their heights precisely encoding the Binomial(10, 0.5) probability mass function: near-invisible slivers at the extremes (0.001 at k = 0 and k = 10), steadily growing bars through k = 1–4, and a matching mirror image on the right side through k = 6–9. The k = 5 bar stands alone in warm gold and reaches the chart's apex at 0.246, its color singling it out as the distribution's mode and mean simultaneously, a coincidence unique to the symmetric p = 0.5 case. A gold dashed vertical line rises through that bar and carries the annotation "Mean = np = 5," tying the visual peak to its algebraic definition. Small probability values printed above every bar transform the chart into a reference table without adding a separate table, so a student can read both the shape and the exact numbers in one glance. The perfect left-right mirror symmetry visible across the eleven bars is itself a theorem — Binomial(n, 0.5) is symmetric for any n — and making that symmetry conspicuous by eye prepares students to recognize asymmetry immediately when they encounter distributions with p ≠ 0.5. 

Figure 6.3.1

Title: Confidence Interval Construction and Interpretation

Detailed Caption: A vertical navy dashed line anchors the center of the figure, labeled "True Population Mean (μ)" at its top, and twenty horizontal line segments stack vertically in sample-number order, each representing one 95% confidence interval drawn from an independent random sample with a dot at its midpoint marking that sample's mean. Nineteen of the twenty intervals are rendered in teal and extend far enough left or right to cross the central reference line, visually confirming that they capture the true parameter. The eighth interval is drawn in gold and ends before reaching the dashed line, annotated with the label "Does not capture μ" — the single miss that the 95% confidence level predicts will occur roughly once in twenty trials. The label "19 out of 20 capture μ (95%)" appears on the right side of the figure, converting the color-coded count into the frequentist definition of confidence. By showing variability in sample-mean position (the dots shift left and right across rows) alongside variability in whether the interval captures the true value, the figure makes concrete the distinction between a sample mean and a population parameter — and it supplies a visual counterargument to the common misreading that "95% confident" means a single interval has a 95% probability of being correct. 

Figure 6.3.2

Title: Hypothesis Testing Decision Framework 

Detailed Caption: A top-to-bottom flowchart descends through four identical navy rounded rectangles that encode the procedural backbone of any hypothesis test: state H₀ and Hₐ, collect sample data, compute a test statistic (z, t, χ², etc.), and calculate the p-value from the null distribution. From the fourth step an arrow drops into a teal diamond decision node bearing the question "p-value ≤ α?" — the single binary fork on which the entire procedure turns. A YES branch exits the diamond leftward and terminates in a warm-gold rectangle reading "Reject H₀ / Statistically significant," while a NO branch exits rightward into a teal rectangle reading "Fail to Reject H₀ / Insufficient evidence." A small note box beneath the diamond states "α is typically 0.05," grounding the abstract threshold in the value students will encounter most often in practice. The chart's visual grammar — identical rectangles for rote steps, a colored diamond for the one judgment call, and color-coded terminals that distinguish outcomes — encodes a rule of thumb: the first four steps require only arithmetic, but the fifth requires a decision governed by a pre-chosen standard, and that standard must be set before the data are collected for the test to retain its logical integrity. 

Figure 6.3.3

Title: Type I and Type II Errors in Hypothesis Testing 

Detailed Caption: A two-by-two matrix organizes the universe of hypothesis-test outcomes under column headers "H₀ is Actually True" and "H₀ is Actually False," and row headers "Decision: Fail to Reject H₀" and "Decision: Reject H₀," all rendered in navy text against light-gray label cells. The top-left cell carries a teal-tinted background and the label "Correct Decision (True Negative) — Probability: 1 − α," pairing the correct outcome with its complement-of-alpha probability. The top-right cell sits on a gold-tinted background and is labeled "Type II Error (β) — False Negative — Fail to detect real effect," naming the cost of under-sensitivity. The bottom-left cell mirrors the gold tint and reads "Type I Error (α) — False Positive — Incorrectly reject true H₀," naming the cost of over-sensitivity. The bottom-right cell returns to teal and reads "Correct Decision (True Positive) — Power = 1 − β," completing the symmetry between the two correct outcomes and tying statistical power to its formal definition. A note strip beneath the grid defines α as the significance level and Power as the probability of correctly detecting an effect. By using color (teal for correct, gold for error) consistently across both diagonals, the matrix makes the structure of the four-way outcome space visually navigable — students can locate any cell by its background rather than reading every label, which is precisely what is needed when reasoning quickly under the pressure of a live analysis. 

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Figure 6.4.1

Title: Correlation vs. Causation -- Three Possible Explanations

Detailed Caption: Three equal-sized panels rest side by side inside a shared white canvas, each framing a different explanation for why two variables X and Y might appear correlated. In Scenario 1 a navy circle labeled X and a teal circle labeled Y are connected by a single rightward arrow, encoding the simplest causal story: X produces Y, and nothing else needs to be said. In Scenario 2 the same two circles reappear but the arrow reverses direction, pointing from Y back to X, a reminder that observed correlation is symmetric even though causation is not — the data alone cannot distinguish which panel is true. Scenario 3 introduces a gold circle labeled Z above and between the X and Y circles, with gold arrows descending from Z to each of them and a gray dashed double-headed line running between X and Y labeled "Observed correlation (not causal)," geometrically encoding the confounding structure: Z is the common cause, X and Y are common effects, and their correlation is a statistical shadow cast by Z rather than a direct link. A full-width navy banner beneath all three panels carries the caption "Correlation does not imply causation — always consider alternative explanations," anchoring the visual argument in the chapter's most important methodological warning. Placing all three scenarios on the same canvas forces a comparison that reading the text alone discourages: the correlation between X and Y is identical in every panel, but the interventional implications are completely different, which is why no regression coefficient, however precisely estimated, can by itself justify a causal claim. 

Figure 6.4.2

Title: Hypothesis Testing Decision Framework

Detailed Caption: The diagram is a vertical flowchart illustrating the step-by-step process of hypothesis testing, flowing from top to bottom and ending with a decision branch. A box at the bottom states "α is typically 0.05", indicating the conventional significance threshold used in most hypothesis tests (a 5% chance of incorrectly rejecting a true null hypothesis). The flowchart captures the universal logic of hypothesis testing used across analytics — from A/B testing in business to quality control to social science research. Regardless of the specific test statistic used, the core workflow is always: state hypotheses → collect data → compute statistic → find p-value → compare to α → decide.

Figure 6.4.3

Title: Type I and Type II Errors in Hypothesis Testing (2×2 Decision Matrix)                                                             
                                                                       
Detailed Caption: The diagram is a two-by-two matrix (contingency table) showing the four possible outcomes of a hypothesis test, organized by the actual truth of the null hypothesis versus the decision made. α = significance level (typically 0.05); Power = probability of correctly detecting an effect. The matrix clarifies the trade-off in hypothesis testing — lowering α (being stricter about false positives) tends to increase β (missing real effects), and vice versa. The two correct outcomes sit on the diagonal (top-left and bottom-right), while the two errors sit on the off-diagonal. Statistical power (1 − β) is the ability to detect a real effect when one truly exists.