Reasoning with Data

Chapter 2

Learning Objectives

Define reasoning.

Execute deductive reasoning.

Explain an empirically testable conclusion.

Execute inductive reasoning.

Differentiate between deductive and inductive reasoning.

Explain how inductive reasoning can be used to evaluate an assumption.

Describe selection bias in inductive reasoning.

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What is Reasoning?

Reasoning is the process of forming conclusions, judgments, or inferences from facts or data

Reasoning and logic are often used interchangeably

Logic is a description of the rules and/or steps behind the reasoning process

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Two Arguments

Argument 1:

The companies profits are up more than 10% over the past year. An increase in profits of 10% is the result of excellent management. You were the manager over the past year. Therefore, I conclude that you engaged in excellent management last year.

Argument 2:

Ten of your 300 employees came to me with complaints about your management. They indicated that you treated them unfairly by not giving them a raise they deserved. Therefore, I conclude that all of your employees are disgruntled with your management.

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Understanding Reasoning

In presenting the two arguments, the goal is not to make a definitive decision about which you believe (if either)

The goal is to think about and distinguish different “lines” of reasoning

In distinguishing between the different types of reasoning, you will be able to establish why you believe or question the claims made in the two arguments

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Two Major Types of Reasoning

Reasoning

Deductive ReasoningInductive Reasoning

Both play an important role in interpreting and drawing conclusions from data analysis

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Deductive Reasoning

Goes from the general to the specific

Also known as top-down logic

Seeks to prove statements of the form “If A, then B”

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Deductive Reasoning

Such reasoning always implies three underlying components: assumptions (“If A”), methods of proof (“then”), and conclusions (“B”)

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Deductive Reasoning

The purest applications of deductive reasoning are in the field of mathematics

Two of the most used approaches are direct proofs and transposition

Direct proofs

Proof that begins with assumptions, explains methods of proof, and states the conclusion(s)

Transposition

Any time a group of assumptions implies a conclusion, then it is also true that any time the conclusion does not hold, at least one of the assumptions must not hold

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Direct Proof

Let’s prove the following statement by direct proof:

If X and Y are odd numbers, then their sum (X + Y) is an even number

An Example:

If X = 5(odd) and Y = 9(odd), then their sum X + Y = 14 is an even number

Failing to find a contradiction is not the same a proving a statement is generally true

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Direct Proof: A Mathematical Approach

If X and Y are odd numbers, then their sum (X + Y) is an even number

X and Y are odd numbers

If X is an odd number, then X can be written as X=2K+1, where K is an integer. (Example: X=13 X=(2 × 6)+1)

If Y is an odd number, then Y can be written as Y =2M+1, where M is an integer, (Example: Y=23 Y=(2 × 11)+1)

X+Y=(2K+1)+(2M+1)=2K+2M+2=2(K+M+1)

K+M+1 is an integer so X+Y is 2 times an integer

Any number that is 2 times an integer is divisible by 2

This means X+Y is even

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Direct Proof: Common Sense Approach

“If McDonald’s offers breakfast all day, their revenues will increase.”

McDonald’s stores offer breakfast all day.

The addition of breakfast during lunch/dinner hours implies more choices.

Customers already choosing McDonald’s during lunch/dinner hours can continue buying the same meals at McDonald’s.

Customers not choosing McDonald’s during lunch/dinner hours may start eating at McDonald’s.

Retaining current customers and adding new ones, McDonald’s revenues will increase overall.

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Transposition

While direct proofs are sufficient to prove a point logically, an alternative approach, transposition, may be more effective

Transposition

Is the equivalence between the statements “If A, then B” and “If not B, then A”

Any time a group of assumptions implies a conclusion, then it is also true that any time the conclusion does not hold, then at least one of the assumptions must not hold

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Transposition

“If A, then B” AND “If not B, then not A”

ASSUMPTIONS

(A)

CONCLUSIONS

(B)

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Transposition: A Mathematical Approach

Prove the statement: If X2 is even, then X is even

Suppose X is not an even number; it is instead an odd number

If X is an odd number, then X= (2K +1), where K is an integer

X2 = (2K+1)2 = 4K2 + 4K+1.

4K2 + 4K = 4(K2 +K) and so is divisible by 2

4K2 + 4K is an even number

X2 = 4(K2 +K)+1 is an even number plus 1, meaning it is an odd number

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Transposition

The statement was: If X2 is even, then X is even

Using transposition, the opposite of the conclusion is used to proof the opposite of the assumption: If X is odd, then X2 is even would be incorrect

Transposition can also be used without using mathematics to prove statements like “If A, then B”.

Transposition can be particularly effective if an assumption seems indisputably obvious.

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Transposition: An Example

Proof the statement: “If McDonald’s stores offer breakfast all day, revenue will increase”

McDonald’s stores revenues will not increase

This means total revenues from current and new customers will not increase

This means either there will be no new customers or revenues from current customers will decrease

This means there could not have been an expansion in the menu

McDonald’s stores do not offer breakfast all day

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Direct Proof and Transposition

Direct Proof

State assumptions

Explain methods of proof (mathematics, common sense, etc.)

State conclusions

Transposition

Assume the opposite of the conclusion

Explain methods of proof (mathematics, common sense, etc.)

State assumption(s) that is (are) violated (not A)

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Deductive Reasoning

Used commonly in the application of law

If there is disagreement with a conclusion there are two possible sources:

The method of proof, OR

The assumption

There are two ways of resolving disputes about assumptions

Show robustness- the persistent accuracy of a conclusion despite variation in the associated assumption(s) within the context of a deductive argument

Assess consistency with a collected dataset

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Empirically Testable Conclusions

An empirically testable conclusion is a conclusion whose validity can be meaningfully tested using observable data.

Example:

A banana company’s management staffs are divided into two groups about their product’s placement in a major grocery store chain.

Group 1 believes that change in current location will increase its sales.

Group 2 believes that current location is good enough.

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Empirically Testable Conclusions

Company has the sales data in the current location.

Company chooses to move its product to a new location and collects sales data.

Now the company can meaningfully test the validity of the management’s competing conclusions.

Making the actual decision about the validity of an empirically testable conclusion based on observable data is an application of inductive reasoning

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Inductive Reasoning

Inductive reasoning

Reasoning that goes from the specific to the general; bottom-up logic

Population

The entire set of potential observations about which we want to learn

Data sample

A subset of population that is collected and observed

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Inductive Reasoning

Business regularly collect data samples to draw conclusions about the population after applying inductive reasoning.

The conclusion from inductive reasoning requires degree of support (also called inductive probability).

Degree of support is also called the strength of the inductive argument.

Example: if we are 50% confident about the conclusion, then the degree of support is 50%.

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Degrees of Support

Two Types of Degrees of Support

Both play an important role in interpreting and drawing conclusions from data analysis

SUBJECTIVE DEGREE OF SUPPORT

(IT IS BASED ON OPINION AND LACKING STATISTICAL FOUNDATION)

OBJECTIVE DEGREE OF SUPPORT

(IT HAS A STATISTICAL FOUNDATION AND THUS MORE CREDIBLE THAN SUBJECTIVE DEGREE OF SUPPORT)

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Evaluating Assumptions

Through deductive reasoning, an empirically testable conclusion is made

Collect a data sample

Test the conclusion by comparing the observed outcomes in the data samples to their corresponding probabilities

Use inductive reasoning to decide whether the conclusion passes or fails

If it fails, transposition implies we must reject

If it passes, then we must not reject

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Inductive Reasoning for Evaluating Assumptions

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Selection Bias in Inductive Reasoning

Improper use of inductive reasoning may lead to inaccurate, or biased conclusions

Data-generating process is typically the source of the bias

Survey questions constructed in a leading way

Confirmation bias is the tendency to confirm a claim

Predictable patterns are discovered

Predictable-world bias is the tendency to find order when none exists, and occurs when people “read too much” into perceived patterns from random data

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Selection Bias

Selection bias

The act of drawing conclusions about a population using a selected data sample, without accounting for the means of selection

There are two common types:

Collector selection bias occurs when the collector selects the members of the data sample in a systematic way

Availability bias occurs when the collector of the data sample selects the members of the data sample according to what is most readily available

Member selection bias occurs when potential members of the data sample self-select into, or out of, the sample

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