Identifying Outliers, Modified Boxplots, and Foundations of Probability

Five-Number Summary and Outlier Identification Procedure

• Components of a Boxplot:

  • The construction of a standard boxplot relies on the five-number summary: the Minimum, the first quartile ($Q_1$), the second quartile or median ($Q_2$), the third quartile ($Q_3$), and the Maximum.
  • In a standard boxplot, the central box spans from $Q_1$ to $Q_3$, a line divides the box at $Q_2$, and whiskers extend outwards to the Minimum and Maximum values.

• Interquartile Range ($IQR$):

  • Definition: The range of the middle $50\%$ of the data.
  • Formula:         $IQR = Q_3 - Q_1$

• Step-by-Step Procedure for Identifying Outliers:

  • Step 1: Determine the quartiles of the data set, specifically $Q_1$ and $Q_3$.
  • Step 2: Calculate the interquartile range using the formula $IQR = Q_3 - Q_1$.
  • Step 3: Evaluate the quantity $1.5 \times IQR$.
  • Step 4: Establish the upper and lower outlier fences and identify outlier data values:
    • An observation is an outlier if it is located above $Q_3$ by an amount strictly greater than $1.5 \times IQR$:             $\text{Upper Fence} = Q_3 + (1.5 \times IQR)$$\text{Outlier} > Q_3 + (1.5 \times IQR)$
    • An observation is an outlier if it is located below $Q_1$ by an amount strictly greater than $1.5 \times IQR$:             $\text{Lower Fence} = Q_1 - (1.5 \times IQR)$$\text{Outlier} < Q_1 - (1.5 \times IQR)$

Modified Boxplot Construction and Data Skewness

• Example Outlier Identification (October 13 Worksheet, Problem 12, Part 3):

  • The five-number summary for the data set:
    • $\text{Minimum} = 5$
    • $Q_1 = 13.5$
    • $Q_2 = 21$
    • $Q_3 = 30$
    • $\text{Maximum} = 80$
  • Execution of Step 2 (Calculate $IQR$):         $IQR = Q_3 - Q_1 = 30 - 13.5 = 16.5$
  • Execution of Step 3 (Evaluate $1.5 \times IQR$):         $1.5 \times IQR = 1.5 \times 16.5 = 24.75$
  • Execution of Step 4 (Determine outlier thresholds):
    • Upper threshold limit:             $Q_3 + (1.5 \times IQR) = 30 + 24.75 = 54.75$
    • Lower threshold limit:             $Q_1 - (1.5 \times IQR) = 13.5 - 24.75 = -11.25$
  • Evaluation against the data set:
    • Non-outlier interval: $[-11.25, 54.75]$.
    • Any data value falling beyond this interval ($< -11.25$ or $> 54.75$) is an outlier.
    • The highest values in the data set are $35$, $40$, $45$, and $80$.
    • Because $80 > 54.75$, the value $80$ is formally identified as an outlier.

• Modifying the Boxplot:

  • In a regular boxplot, the right whisker extends directly to $80$, treating it as the maximum.
  • In a modified boxplot:
    • Outliers are isolated and plotted as separate individual points outside the whisker structure (e.g., $80$ is plotted separately as an outlier).
    • The upper whisker no longer extends to the outlier; instead, it terminates at the next highest data value in the data set that falls within the boundary ($\le 54.75$), which is $45$.
    • The lower whisker, $Q_1$, $Q_2$, and $Q_3$ remain unchanged.

• Impact on Skewness Perception:

  • Boxplots visually convey the skewness of a distribution via whisker length and box symmetry.
  • A regular boxplot with a whisker reaching to $80$ depicts the data set as highly skewed to the right.
  • Plotting the modified boxplot reveals that the core distribution is only slightly skewed to the right, demonstrating that the apparent extreme skewness was an artifact driven by a single outlier ($80$).

Probability Foundations: Procedures, Events, and Sample Spaces

• Core Definitions:

  • Procedure: An experiment, operation, or process that leads to results or outcomes (e.g., flipping a coin, rolling a die, or surveying individuals).
  • Event: Any collection of results or outcomes of a procedure.
  • Simple Event: An outcome or an event that cannot be broken down into simpler components.
  • Sample Space: The set of all possible simple events. It encompasses all outcomes that cannot be broken down any further.

• Example: Two-Child Family Structure (Sex Assigned at Birth):

  • Notation: $b$ represents a boy, and $g$ represents a girl.
  • Procedure: A family has two children, categorized by birth order (eldest child, youngest child).
  • Sample space ($S$):         $S = \{bb, bg, gb, gg\}$
  • Differentiating Simple and Compound Events:
    • Event $A$: Both children are boys.
      • Outcomes: $\{bb\}$.
      • Classification: This outcome cannot be broken into simpler sub-components within the experiment; therefore, it is a simple event.
    • Event $B$: One child is a boy and the other child is a girl.
      • Outcomes: $\{bg, gb\}$.
      • Classification: This event can be decomposed into two distinct simple events ($bg$: eldest is a boy and youngest is a girl; $gb$: eldest is a girl and youngest is a boy); therefore, it is not a simple event.

• Standard Probability Notation:

  • $P$ denotes probability.
  • Uppercase letters ($A, B, C$) denote specific events.
  • $P(A)$ denotes the probability of event $A$ occurring.

Three Approaches to Calculating Probability

• 1. Classical Approach (Equally Likely Outcomes):

  • Application: Used when each simple event has an equal chance of occurring (e.g., fair dice, fair coins).
  • Definition: If a procedure has $n$ different simple events that are equally likely, and event $A$ can occur in $s$ of those ways:         $P(A) = \frac{s}{n} = \frac{\text{number of ways } A \text{ can occur}}{\text{number of different simple events}}$
  • Example 1: Three Flips of a Fair Coin:
    • Sample space consisting of $n = 8$ simple events:             $S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$
    • Calculating the probability of getting exactly two tails:
      • Favorable outcomes: $\{HTT, THT, TTH\}$ ($s = 3$).
      • $P(\text{exactly two tails}) = \frac{3}{8}$
    • Calculating the probability of getting at least one tail:
      • "At least one tail" corresponds to $1$, $2$, or $3$ tails (all outcomes except $HHH$).
      • Favorable outcomes: $s = 7$.
      • $P(\text{at least one tail}) = \frac{7}{8}$
  • Example 2: Rolling Two Fair Dice:
    • Sample space size: Each die has $6$ equally likely faces ($1$ through $6$), yielding $6 \times 6 = 36$ total outcomes.
    • Calculating the probability that the sum of the two dice equals $9$:
      • Event $A$ consists of the pairs: $\{(3, 6), (4, 5), (5, 4), (6, 3)\}$.
      • Outcomes such as $(1, 8)$ or $(2, 7)$ are impossible because a die face cannot exceed $6$.
      • The pairs $(4, 5)$ and $(5, 4)$ represent distinct simple events because they correspond to different dice (first die vs. second die).
      • Number of favorable outcomes: $s = 4$.
      • Probability:                 $P(\text{sum is } 9) = \frac{4}{36} = \frac{1}{9}$

• 2. Relative Frequency Approach (Empirical Probability):

  • Application: Relies on conducting trials, gathering sample data, and observing real-world occurrences.
  • Definition:         $P(A) \approx \frac{\text{number of times } A \text{ occurred}}{\text{number of times the procedure is repeated}}$
  • The symbol $\approx$ denotes that the computed value is an estimate or approximation derived from finite sampling rather than an absolute theoretical truth.
  • Example 1: Sleep Problem Survey:
    • Procedure: Randomly select and survey $500$ adults regarding sleep problems.
    • Observations: $120$ adults state they have no sleep problems.
    • Event $A$: An adult reports having no sleep problems.
    • Empirical probability estimate:             $P(A) \approx \frac{120}{500} = 0.24$
    • Sampling rationale: It is physically impossible to interview every adult globally; thus, sampling provides a workable estimate.
  • Example 2: Tire Quality Inspection (Worksheet Problem 4):
    • Procedure: An inspector tests $400$ tires and discovers that $18$ are defective.
    • Empirical probability of selecting a defective tire:             $P(\text{defective tire}) \approx \frac{18}{400} = 0.045$
  • Example 3: Student Bus Commuter Survey (Worksheet Problem 3b):
    • A survey of $300$ students finds $45$ take the bus.
    • The reported probability of approximately $0.15$ is classified under the relative frequency approach.

• 3. Subjective Probability Approach:

  • Application: Used when historical frequency or equally likely outcomes cannot be evaluated mathematically.
  • Definition: The probability of an event is estimated using relevant knowledge of prevailing circumstances and professional judgment.
  • Example 1: Meteorological Forecast:
    • A meteorologist predicts a $30\%$ chance of rain tomorrow.
    • $P(\text{rain}) = 30\% = 0.3$
    • This value represents an educated estimation based on atmospheric models and expert knowledge.
  • Example 2: Athletic Competition (Worksheet Problem 3c):
    • A coach asserts that her team has an $80\%$ chance of winning Friday's game.
    • This assessment is classified as subjective probability.

The Law of Large Numbers

• Principle Statement:

  • As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual (true theoretical) probability of that event.

• Relative Frequency Probability vs. Actual Probability:

  • Relative Frequency Probability: An empirical proportion obtained from conducting a procedure a finite number of times ($n$).
  • Actual Probability: The true, fixed probability obtained only if the procedure were applied to the entire population or repeated indefinitely.

• Illustration of Convergence:

  • Surveying adults regarding sleep issues across expanding sample sizes demonstrates this convergence toward a hypothetical true value (e.g., $0.20$):
    • At $n = 500$ trials: $P(A) \approx 0.24$
    • At $n = 100{,}000$ trials: $P(A) \approx 0.235$
    • At $n = 1{,}000{,}000$ trials: $P(A) \approx 0.234$
    • At $n = 2{,}000{,}000$ trials: $P(A) \approx 0.231$
  • Increasing the sample size narrows the gap between the empirical estimate and the actual population probability.

Probability Axioms and the Rare Event Rule

• Fundamental Range of Probability:

  • The probability of any event $A$ is always bounded between $0$ and $1$ inclusive:         $0 \le P(A) \le 1$
  • A probability of $P(A) = 0$ indicates an impossible event.
  • A probability of $P(A) = 1$ indicates a certain event.

• The Rare Event Rule for Inferential Statistics:

  • Formal Statement: If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct.
  • Application Scenario (Testing Coin Fairness):
    • Given Assumption: A coin is balanced and fair ($P(\text{heads}) = P(\text{tails}) = 0.5$).
    • Observed Event: The coin is flipped $100$ times, resulting in $90$ tails.
    • Analysis: Under the assumption of fairness, obtaining $90$ tails in $100$ tosses is an extremely improbable (rare) outcome.
    • Conclusion: The assumption of fairness is rejected; the coin is probably biased toward landing on tails.
  • Probabilistic Qualification:
    • The word probably is mandatory in statistical conclusions because extreme outcomes could theoretically occur as extreme coincidences, even if overwhelmingly unlikely.
    • If the observation had yielded $55$ tails and $45$ heads, this outcome is plausible under fair conditions, providing no evidence to dispute the assumption.

Complementary Events and Actual Odds

• Complements of Events:

  • Definition: The complement of event $A$, denoted by $A^c$ or $\bar{A}$, consists of all outcomes in the sample space in which event $A$ does not occur.
  • Axiom of Certainty: Because event $A$ must either occur or not occur, the sum of their probabilities equals $1$:         $P(A) + P(A^c) = 1$
  • Complement Formulas:$P(A^c) = 1 - P(A)$$P(A) = 1 - P(A^c)$

• Complement Examples Using a Fair Six-Sided Die ($S = \{1, 2, 3, 4, 5, 6\}$):

  • Example 1 (Rolling a Specific Value):
    • Event $A$: Rolling a $2$.
    • Probability: $P(A) = \frac{1}{6}$.
    • Complement $A^c$: Not rolling a $2$ ($\{1, 3, 4, 5, 6\}$).
    • Probability: $P(A^c) = \frac{5}{6}$.
    • Verification:             $P(A) + P(A^c) = \frac{1}{6} + \frac{5}{6} = \frac{6}{6} = 1$
  • Example 2 (Rolling an Even Number):
    • Event $B$: Rolling an even number ($\{2, 4, 6\}$).
    • Probability: $P(B) = \frac{3}{6} = \frac{1}{2}$.
    • Complement $B^c$: Rolling an odd number ($\{1, 3, 5\}$).
    • Probability using the complement formula:             $P(B^c) = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2}$

• Actual Odds Against vs. Actual Odds in Favor:

  • Actual Odds Against Event $A$:
    • Ratio prioritizing the non-occurrence of $A$ over the occurrence of $A$:             $\text{Actual odds against } A = \frac{P(A^c)}{P(A)} \quad \text{or expressed as} \quad P(A^c) : P(A)$
    • For rolling a $2$ on a fair die:             $\text{Odds against rolling a } 2 = \frac{\frac{5}{6}}{\frac{1}{6}} = \frac{5}{1} \implies 5:1$
  • Actual Odds in Favor of Event $A$:
    • Ratio prioritizing the occurrence of $A$ over the non-occurrence of $A$:             $\text{Actual odds in favor of } A = \frac{P(A)}{P(A^c)} \quad \text{or expressed as} \quad P(A) : P(A^c)$
    • For rolling a $2$ on a fair die:             $\text{Odds in favor of rolling a } 2 = \frac{\frac{1}{6}}{\frac{5}{6}} = \frac{1}{5} \implies 1:5$

Introduction to Compound Events

• Definition: A compound event is an event combining two or more simple events.

• Standard Operations:

  • $A \text{ or } B$: Indicates that event $A$ occurs, event $B$ occurs, or both events occur.
  • $A \text{ and } B$: Indicates that both event $A$ and event $B$ occur simultaneously.

Questions & Discussion

• Relationship Between Median and Quartiles:

  • Question: Is $Q_2$ the same as the median?
  • Response: Yes, the second quartile ($Q_2$) is mathematically identical to the median.

• Outlier Boundary Computation Directionality:

  • Question: When finding outliers, is it all about $Q_3$ and the $IQR$, and then on the other side, $Q_1$ and the $IQR$? Is it added to $Q_3$ and subtracted from $Q_1$?
  • Response: Yes. For the upper boundary, add $1.5 \times IQR$ to $Q_3$. For the lower boundary, subtract $1.5 \times IQR$ from $Q_1$.

• Two-Dice Sample Space Permutations for Sum of Nine:

  • Question: Why can't pairs like $1 + 8$ or $2 + 7$ be included when evaluating sums of nine for two dice?
  • Response: A standard die face only ranges from $1$ through $6$. Values of $7$ and $8$ do not exist on standard dice.

• Administrative Details:

  • Discussion Post Assignment: Students must submit their answers to Problem 3 from the worksheet (Problem 3a: Classical approach; Problem 3b: Relative frequency approach; Problem 3c: Subjective approach).
  • Project Milestone: The first deadline for the project is scheduled for next Thursday. The prompt and instructions covering this initial task will be posted before this Thursday.