Business Statistics 213 - Week 1

Foundations of Statistical Thinking, Arithmetic, and Distance Measures

• Long-Term Application of Logic

  • Statistical reasoning and formal logical frameworks extend beyond academic requirements into career analysis and daily decision-making.

  • The primary methodologies covered in statistical analysis are divided into descriptive statistics and inferential statistics.

• Principles of Statistical Model Building

  • Statistical model building is conducted to extract underlying structural understanding from complex real-world data.

  • Fundamental Axiom of Modeling: "All models are wrong. Some models are useful."

  • Models do not aim to recreate exact real-world mechanics; rather, they extract meaningful patterns, dynamics, and relationships despite structural simplifications.

• Essential Mathematical Operations

  • Fractions and Exactness: Fractions represent exact mathematical quantities. Decimals derived from division often result in infinite repeating values (e.g., $\frac{9}{8}$ is exact, whereas decimal representation can require infinite or rounded representations).

  • Algebraic Manipulation of Fractions: Fractions allow direct algebraic cancellation and simplification:         $\frac{10}{2} \times \frac{2}{5} = \frac{10 \times 2}{2 \times 5} = \frac{10}{5} = 2$

  • Conceptual Interpretation of Fractions: A fraction $\frac{N}{D}$ answers the fundamental question: "How many times does the denominator $D$ go into the numerator $N$?"

• Number Line Dynamics and Standardized Distance

  • A basic number line displays relative directional orientation: values to the right are in the positive direction (larger), and values to the left are in the negative direction (smaller).

  • The origin point ($0$) is completely arbitrary out of an infinite continuum of values; any reference point can serve as a baseline, such as $\bar{x}$ (x-bar).

  • Relative Distance:

    • If a point $x_1$ lies to the right of $\bar{x}$, the distance $x_1 - \bar{x}$ is positive ($x_1 > \bar{x}$).

    • If a point $z_2$ lies to the left of $\bar{x}$, the distance $z_2 - \bar{x}$ is negative ($z_2 < \bar{x}$).

    • In statistical analysis, positive and negative denote relative position (above or below a baseline reference value), not absolute algebraic sign.

  • Understanding relative distance along a baseline continuum constitutes roughly $50\%$ of statistical course content and $75\%$ of statistics overall.

• Standardized Distance Formula ($\sigma$)

  • To measure how many units of a standard distance metric (standard deviation, $\sigma$) a value lies from a baseline ($\bar{x}$), a ratio is constructed:         $\text{Standardized Distance} = \frac{x - \bar{x}}{\sigma}$

  • This ratio applies the core fraction definition: it calculates how many units of denominator $\sigma$ fit into the numerator distance $(x - \bar{x})$. This serves as the primary metric of distance in statistical analysis.

• Pedagogical Purpose of Hand Calculations

  • While modern computation is executed using spreadsheet software like Excel, hand-calculating basic problems instills structural comprehension of what the mathematical output means.

  • Understanding the underlying meaning of mathematical structures ensures retention across cumulative concepts throughout multi-week academic curricula.

  • Scalar Distance Difference: Calculating $c - (a \times b)$ determines scalar distance. For instance, if $c = 4$, $a = 1$, and $b = 2$, then $4 - (1 \times 2) = 2$, representing the scalar distance between the two terms.

Spurious Correlations, Data Interpretation, and Economic Concepts

• Problem Formulation and Question Design

  • Critical analytical reasoning requires questioning underlying definitions before attempting to interpret empirical claims (e.g., asking "What precisely is meant by 'makes the world go round'?")

• Case Study: North American Cheese Consumption vs. BlackRock Stock Price

  • Market Scale: The North American cheese market (Canada and the United States) accounts for approximately $\$50\,billion$ per year.

  • Corporate Profile: BlackRock is the largest private equity and asset management firm globally, managing trillions of dollars in assets.

  • Observed Data: A 20-year longitudinal dataset demonstrates that American cheese consumption and BlackRock's stock price moved in near-perfect lockstep.

  • Constructed Spurious Narrative: Increased cheese consumption raises baseline consumer happiness $\rightarrow$ optimistic citizens increase purchases of goods and services $\rightarrow$ Gross Domestic Product ($GDP$) increases $\rightarrow$ national income increases $\rightarrow$ corporate valuation increases $\rightarrow$ BlackRock asset values rise.

• Statistical Metrics for Evaluating Data Relationships

  • Correlation Coefficient ($r$):

    • A bounded quantitative metric between $-1.0$ and $+1.0$.

    • Values approaching $+1.0$ indicate strong direct linear tracking; values near $-1.0$ indicate strong inverse tracking; values near $0.0$ indicate no linear relationship.

    • The calculated correlation coefficient between American cheese consumption and BlackRock stock price over the 20-year period is $r = 0.93$.

  • Probability Value ($p$-value):

    • Calculates the mathematical probability that an observed statistical relationship between datasets occurred purely due to random chance.

    • Values closer to $0\%$ ($0.0$) indicate that the relationship is highly unlikely to be random noise.

    • Standard academic publishing thresholds require $p \le 0.05$ ($5\%$) or $p \le 0.10$ ($10\%$) to reject random chance.

    • The calculated $p$-value for the cheese/BlackRock correlation is $8.08 \times 10^{-10}$ ($0.000000000808$), which is functionally $0$.

  • Deconstructing False Causation and Graphical Manipulation:

    • Graphical Scale Manipulation: Truncating the y-axis creates an exaggerated visual impression of lockstep tracking. Adjusting the y-axis scale to start strictly at absolute zero ($0$) visually eliminates the apparent lockstep relationship.

    • Confounding Third Variables: A logical explanation relies on underlying economic mechanisms rather than direct causation between cheese and asset prices:

      • Normal Good: In microeconomics, a normal good is defined as any product whose demand increases when real consumer income increases.

      • Opportunity Cost: The fundamental microeconomic concept representing the foregone alternative when making a choice.

      • True Confounding Mechanism: Total national income/$GDP$ expansion causes asset prices to rise (driving BlackRock stock prices up) while simultaneously increasing consumer purchasing power, leading citizens to buy more normal goods (driving cheese consumption up).

The History of Statistical Deception and Ethical Applications

• Historical Perspectives on Statistics and Deception:

  • Mark Twain: Stated in his autobiography: "There are three kinds of lies: lies, damned lies, and statistics." This creates an ordinal ranking where standard lies are least severe, damned lies are worse, and statistical manipulation is the most severe.

  • Joan Robinson: Mid-20th-century Cambridge economist who noted: "The reason we teach economics is so people don't get fooled by economists."

  • Charles Wheelan / Swedish Mathematician Paradox: "The disease is to lie with statistics, but it's hard to tell the truth without them."

• Darrell Huff and How to Lie with Statistics (1954):

  • Author Background: Written by Darrell Huff, a journalist with a Master's degree in Economics, rather than an academic statistician.

  • Commercial Reach: Remains the best-selling statistics book in history, selling millions of copies. Decades post-publication, it maintained rankings such as #5 in Amazon's general statistics category and #11 in academic statistics and probability, remaining on syllabi at elite universities.

  • Core Metaphor: Huff described the text as a "manual for swindlers," comparing himself to a "retired burglar who published reminiscences amounting to a graduate course in how to pick a lock and muffle a footfall." He declared: "The crooks already know these tricks; honest men must learn them in self-defense."

  • Statisculate: A term coined by Huff defined as misinforming people through the intentional misuse of statistical materials.

  • Key Chapter Titles: "The Sample with Built-in Bias", "The Well-Chosen Average", "The Little Figures That Are Not There", "How to Statisculate".

• Epidemiological Case Study: Austin Bradford Hill and Richard Doll:

  • Context: Late 1940s post-WWII study conducted by medical doctors and epidemiologists Austin Bradford Hill and Richard Doll.

  • Sample Size: Tracked $40,000$ medical doctors in the United Kingdom out of a total national population of $\sim 59,000\text{--}60,000$ doctors (covering approximately $\frac{4}{5}$ or $80\%$ of the demographic).

  • Baseline Demographic: $85\%$ of UK medical doctors smoked cigarettes at the time of the study.

  • Findings: Cigarette smoking increased lung cancer mortality rates by $16\,\text{times}$ compared to non-smokers.

  • Anecdote: When a physician approached Hill at a gathering stating, "You're the chap who wants us to stop smoking," Hill replied: "Not at all. I'm interested if you go on smoking to see how you will die. I'm interested if you stop because I want to see how you will die. You can choose for yourself… I shall score up your death anyway." (Hill began his professional career as an economist).

  • Impact: UK medical doctors became the first documented social demographic in history to reduce smoking at a statistically measurable rate.

• Darryl Huff's Tobacco Industry Involvement:

  • Following his 1954 publication, Huff was retained by the tobacco industry as a consultant.

  • In 1965, Huff testified before the US Senate Select Committee to refute the 1964 US Surgeon General's report linking smoking to lung cancer.

  • In 1968, Huff completed a funded manuscript titled How to Lie with Smoking Statistics. The text remained unpublished and was uncovered due to the 1998 Tobacco Master Settlement Agreement, which mandated public disclosure of internal industry files.

• Tim Harford's Framework (10 Easy Rules to Make Sense of Statistics):

  • Tim Harford (economist, journalist, and Order of the British Empire recipient) argued that absolute statistical cynicism is a tragedy.

  • Unchecked cynicism leads individuals to discard data-driven analysis entirely, leaving them unable to comprehend complex real-world phenomena.

Formal Logic, Syllogisms, and Argument Structure

• The Anatomy of Thinking:

  • John Locke (17th/18th-century English philosopher) termed logic the "anatomy of thinking."

  • Just as biological anatomy dissects physical organs to understand internal mechanics, logic dissects cognitive reasoning to isolate correct from incorrect structural forms.

  • Logicians operate at the intersection of philosophy and mathematics.

• Core Components of Logical Reasoning:

  • Reasoning: A formal method of argumentation.

  • Argument: A structural construction where a Premise ($\text{Evidence}$) implies a Conclusion ($\text{Claim}$).

  • Symbolic Representation:
            $\text{Premise} \rightarrow \text{Conclusion}$

  • An argument may contain multiple premises but resolves to a single primary conclusion.

• Typology of Logic:

  • Formal Logic: Employs abstract algebraic symbols to evaluate the structural form of an argument independent of semantic meaning.

  • Informal Logic: Analyzes the concrete content and truth value of supporting premises.

• Aristotelian Logic and the Classical Syllogism:

  • Aristotle (4th century BCE Greek philosopher, c. 380–350 BCE) developed formal logic systems.

  • John Stuart Mill (19th-century economist) summarized Aristotelian syllogisms into algebraic symbolic notation.

  • Classical Syllogism Structure:         $\text{Premise 1: All men } (M) \text{ are mortal } (O) \quad (M \rightarrow O)$         $\text{Premise 2: Socrates } (S) \text{ is a man } (M) \quad (S \rightarrow M)$         $\text{Conclusion: Therefore, Socrates } (S) \text{ is mortal } (O) \quad (\therefore S \rightarrow O)$

  • Algebraic Substitution: Because validity depends purely on structure, terms can be substituted arbitrarily:         $\text{Premise 1: All plants } (P) \text{ have cells } (C) \quad (P \rightarrow C)$         $\text{Premise 2: A tree } (T) \text{ is a plant } (P) \quad (T \rightarrow P)$         $\text{Conclusion: Therefore, a tree } (T) \text{ has cells } (C) \quad (\therefore T \rightarrow C)$

Deductive vs. Inductive Reasoning and Analytical Criteria

• Deductive Reasoning Framework:

  • Derives specific conclusions from general premises.

  • Certainty: If premises are true and structural form is valid, the conclusion is guaranteed to be $100\%$ true.

  • Evaluation Criteria: Evaluated on Validity (structural mechanics) and Soundness (content truth).

  • Valid Argument: A deductive structure where, assuming premises are true, it is logically impossible for the conclusion to be false.

  • Invalid Argument: A deductive structure where premises do not logically necessitate the conclusion ("does not follow").

  • Sound Argument: An argument that is structurally valid AND possesses premises that are $100\%$ empirically true.

  • Unsound Argument: An argument that is structurally invalid OR possesses at least one false premise.

  • Example of Unsound Valid Deductive Argument:         $\text{Premise 1: My dog eats seeds.}$
            $\text{Premise 2: All seed eaters can fly.}$
            $\therefore \text{My dog can fly.}$
            (Valid deductive structure, but unsound due to empirically false premises).

• Inductive Reasoning Framework:

  • Derives probabilistic conclusions from specific factual evidence.

  • Certainty: Conclusions are never $100\%$ absolute; they are evaluated as probably true.

  • Evaluation Criteria: Evaluated on Strength (structural probability) and Cogency (factual truth).

  • Strong Argument: An inductive structure where true premises make the conclusion highly probable.

  • Weak Argument: An inductive structure where premises fail to make the conclusion probable.

  • Cogent Argument: An inductive argument that is strong AND possesses true premises.

  • Not Cogent Argument: An inductive argument that is weak OR contains false supporting premises.

  • Example of Weak Inductive Argument:         $\text{Premise: The Red Sox won the last game I attended.}$
            $\therefore \text{The Red Sox are likely to win the next game I attend.}$         (Weak argument due to omitted statistical variables).

• Domain Applications of Inductive Logic:

  • Statistics: Uses inductive reasoning via calculated $p$-values to evaluate whether empirical data supports hypotheses or represents random noise.

  • History: Distinguishes between "the past" (everything that has ever occurred) and "history" (the highly probable subset of past events verified through empirical evidence).

  • Historical Inductive Argument Example:         $\text{Premise 1: Julius Caesar wrote 1st-century BCE accounts of his exploits in Gaul.}$
            $\text{Premise 2: Sallust was a 1st-century BCE Roman historian who documented Julius Caesar.}$
            $\therefore \text{Julius Caesar existed.}$         (Strong inductive argument supported by independent historical attestations).

  • Social Sciences: Relies on inductive probability models because deterministic $100\%$ proof is unobtainable in complex social systems.

• Non-Premise Statements:

  • Statements incapable of serving as premises in logical arguments include: Questions, Suggestions, Proposals, Commands, and Explanations/Advertising claims.

  • Logical falsification occurs by gathering empirical evidence to test the accuracy of predictions generated by argument conclusions.

Rules of Logical Inference and Fallacies

• Symbolic Rules of Formal Logic:

  • Law of Excluded Middle:
            $A \lor \neg A$
            (A statement $A$ is either true or false; no middle logical state exists).

  • Law of Non-Contradiction:
            $\neg(A \land \neg A)$
            (A and not-$A$ cannot both be true simultaneously).

  • Disjunctive Addition (Or-Introduction):
            $\text{Premise 1: } A \rightarrow (A \lor B)$
            $\text{Premise 2: } A$
            $\therefore A \lor B$

  • Conjunction Simplification:
            $\text{Premise: } A \land B$
            $\therefore A \quad (\text{and } \therefore B)$

  • Conjunctive Negation:
            $\text{Premise: } \neg A$
            $\therefore \neg(A \land B)$
            (If $A$ is false, the joint occurrence of $A$ and $B$ is false regardless of $B$'s state).

  • Denying a Disjunct (Disjunctive Syllogism):
            $\text{Premise 1: } A \lor B$
            $\text{Premise 2: } \neg A$
            $\therefore B$

• Classical Rules of Inference:

  • Modus Ponens (Affirming the Antecedent):
            $\text{Premise 1: } A \rightarrow B$
            $\text{Premise 2: } A$
            $\therefore B \quad \text{(Valid)}$

  • Modus Tollens (Denying the Consequent):
            $\text{Premise 1: } A \rightarrow B$
            $\text{Premise 2: } \neg B$
            $\therefore \neg A \quad \text{(Valid)}$

• Formal Logical Fallacies:

  • Denying the Antecedent:
            $\text{Premise 1: } A \rightarrow B$
            $\text{Premise 2: } \neg A$
            $\therefore \neg B \quad \text{(Invalid Fallacy)}$

    • Example: "If it rains ($A$), I bring an umbrella ($B$). It is not raining ($\neg A$). Therefore, I do not bring an umbrella ($\neg B$)." Invalid because an umbrella could be brought for non-rain reasons (sun protection, habit).

  • Affirming the Consequent:
            $\text{Premise 1: } A \rightarrow B$
            $\text{Premise 2: } B$
            $\therefore A \quad \text{(Invalid Fallacy)}$

    • Conditional Requirement: For Affirming the Consequent to be structurally valid, an explicit bi-conditional premise ($A \leftrightarrow B$) must be established, proving $A$ is the sole cause of $B$ (e.g., "If and only if the car is turned on, the radio works").

  • Affirming a Disjunct (Inclusive Disjunction Fallacy):
            $\text{Premise 1: } A \lor B$
            $\text{Premise 2: } A$
            $\therefore \neg B \quad \text{(Invalid Fallacy)}$
            (Invalid under inclusive disjunction, as both $A$ and $B$ can be true simultaneously).

• Informal Fallacies and Non Sequiturs:

  • Formal Fallacies: Structural errors in symbolic logic equivalent to mathematical errors.

  • Informal Fallacies: Deceptive errors embedded in semantic language, content, or context.

  • Non Sequitur ("It does not follow"): Occurs when a conclusion fails to connect logically to premises due to missing premises or the insertion of irrelevant information.