Comprehensive Notes on Preliminary Definitions and Fundamentals of Statistics

Fundamental Concepts of Statistics

Data: Collected facts, observations, measurements, or responses that help describe situations, identify patterns, and make informed decisions.
Statistics: The science of collecting, organizing, analyzing, interpreting, and presenting data.
Scope of Statistics: The field encompasses collecting, storing, summarizing, presenting, analyzing, drawing inferences from, making predictions from, and basing decisions on data or information.

Preliminary Definitions and Terminologies

Subjects: The objects from whom, or about whom, data is collected.
- Subjects can be living or nonliving, tangible or intangible.
- Subjects may consist of events or abstract concepts.
- Identification Strategy: First identify the sample size $n$ , then ask: " $n$ of whom or what are we getting information about?"
Observation / Response / Data Point: A single collected data value (a single piece of information).
- Identification Strategy 1: First identify the subjects, then ask: "From each subject, what piece of information are we getting?"
- Identification Strategy 2: Identify the variable in question by identifying the missing word in the sentence: "This story involves the distribution of _____."
- Note: The words used to describe the subjects and the observations can sometimes be identical.
Variable: A characteristic belonging to the subjects. When data is collected about subjects, values of variables are collected.
- Different subjects often give rise to different values of the same variable.
- Example: When studying Technology Enhanced Classrooms (TECs) across buildings on the Florida State University (FSU) campus, the buildings are the subjects, and the number of TECs in each building is the variable being measured.
Distribution: The distribution of a variable specifies what values the variable can possibly take and how often it takes these values.
- Distributions can be set out in tables, charts, graphs, or written in statistical notation.
Frequency: Count or number (e.g., if the count of long-haired persons in a class is $10$ , the frequency is $10$ ).
Relative Frequency: Proportion or percentage, calculated using the formula: $\text{Relative Frequency} = \frac{\text{Count}}{\text{Total}}$
- Example: In a class of $40$ students with $10$ long-haired students, the relative frequency is: $\frac{10}{40} = 0.25 = 25\%$
Population: The entire group of subjects about which information is sought in a single study. Obtaining data from an entire population can be costly or time-consuming, necessitating the use of samples.
Sample: A subgroup of subjects selected from within a population used to make inferences about the larger population.
Sample Size ( $n$ ): The number of subjects involved in a study or included in the sample.
Parameter: A numerical measurement that describes a characteristic of an entire population.
Statistic: A numerical measurement that describes a characteristic of a sample. Sample statistics are frequently used as estimates of corresponding population parameters.
Sampling Variation: The phenomenon where different random samples of the same size drawn from the exact same population yield different values.
- As a result of sampling variation, sample statistics vary from sample to sample (e.g., producing different sample means or sample standard deviations).

Population vs. Sample Symbols and Framework

Population Parameter Notation (Population represented structurally as a overarching group):
- Mean: $\mu$ (mu / myoo)
- Variance: $\sigma^2$ (sigma squared)
- Standard Deviation: $\sigma$ (sigma)
- Proportions / Percentages: $p$
- Correlation: $\rho$ (rho)
Sample Statistic Notation (Sample represented structurally as a subgroup within the population):
- Sample Size: $n$
- Mean: $\bar{x}$ (x-bar)
- Variance: $s^2$
- Standard Deviation: $s$
- Proportions / Percentages: $\hat{p}$ (p-hat)
- Correlation: $r$
Numerical Worked Example 1:
- Population values: $\{14, 21, 51, 45, 38, 56, 49, 13\}$ (Population size $N = 8$ )
- Selected sample values: $\{51, 45, 56\}$ (Sample size $n = 3$ )
- Population average ( $\mu$ ): $\mu = \frac{14 + 49 + 51 + 45 + 56 + 21 + 38 + 13}{8} = \frac{287}{8} = 35.875$
- Sample average ( $\bar{x}$ ): $\bar{x} = \frac{51 + 45 + 56}{3} = \frac{152}{3} \approx 50.667$
Categorical Worked Example 2:
- Population category elements: $\{Y, B, G, Y, R, R, G, Y\}$ (Where Yellow = $Y$ , Blue = $B$ , Green = $G$ , Red = $R$ ; Total size $N = 8$ )
- Selected sample elements: $\{G, Y, R\}$ (Sample size $n = 3$ )
- Population proportion of Green ( $p$ ): $p = \frac{2}{8} = 0.25 = 25\%$
- Sample proportion of Green ( $\hat{p}$ ): $\hat{p} = \frac{1}{3} \approx 33.3\%$

Variable Types and Classification Rules

Categorical Variables:
- Non-numerical observations that are never used directly in mathematical calculations.
- Consist of category names or labels assigned to categories.
- Each individual subject belongs to exactly one category.
Quantitative Variables:
- Numerical observations that can be used directly in mathematical calculations.
- Almost always feature a unit of measurement (though the unit may not always be immediately obvious).
Contextual Classification Rules:
- Rule 1: If numbers are used exclusively as labels for categories, they are classified as categorical data.
- Rule 2: If quantitative data are utilized for the purpose of categorization, they are classified as categorical data within that specific context.
- Rule 3: If a variable can be interpreted or used quantitatively at all, it should be listed as quantitative.

Applied Studies and Practical Case Analysis

Case 1: Statistics Students Enrolled in French
- Context: Identifying the percentage of current term Statistics students taking French.
- Sample: Random sample of $n = 40$ students asked for a YES or NO answer. Result: $15$ answered YES.
- Sample Statistic ( $\hat{p}$ ): $\hat{p} = \frac{15}{40} = 0.375 = 37.5\%$
- Population: All Statistics students taking the course during the term.
- Population Parameter ( $p$ ): $p = 29.8\%$
- Subjects: Statistics students.
- Observations: The YES/NO responses (Categorical data, categorizing subjects into YES or NO).
Case 2: Household Age Study
- Context: Estimating the mean age of all persons residing in a household.
- Population: All persons living in the household.
- Population Parameter ( $\mu$ ): $\mu = 16.4\text{ years}$
- Sample: Sample of size $n = 1$ (a single 20-year-old household member).
- Sample Statistic ( $\bar{x}$ ): $\bar{x} = 20\text{ years}$
- Subjects: Persons living in the household.
- Observations: Ages of the household members (Quantitative data).
Case 3: Webflicks DVD Rental Variable Analysis
- Subjects: Movies owned by the Webflicks DVD mail-rental company.
- Variables, Data Types, and Units:
  - Genre: Categorical
  - Number of copies owned: Quantitative (Unit: copies)
  - Duration: Quantitative (Unit: minutes)
  - Primary language: Categorical
  - MPAA rating: Categorical
  - Current retail price: Quantitative (Unit: dollars)
Case 4: Variable Classification and Observation Identification
- (a) Amount of air inside balloons at a party:
  - Type: Quantitative
  - Observation Statement: For each balloon, the amount of air inside it is recorded.
- (b) Ice-cream flavors favored by FSU students:
  - Type: Categorical
  - Observation Statement: For each FSU student, her/his favorite ice-cream flavor is recorded.
- (c) Ranks held by officers at a military base:
  - Type: Categorical
  - Observation Statement: For each officer at the base, her/his military rank is recorded.
- (d) Number of textbooks in each room at a dormitory:
  - Type: Quantitative
  - Observation Statement: For each room in the dorm, the number of textbooks in it is recorded.
- (e) Current temperature inside classrooms on campus:
  - Type: Quantitative
  - Observation Statement: For each classroom on campus, the current temperature inside it is recorded.
- (f) Time it takes to fully charge a cell phone every day:
  - Type: Quantitative
  - Observation Statement: For each day, the time it takes to fully charge the cell phone is recorded.
- (g) Maximum legal highway speed in major European cities:
  - Type: Quantitative
  - Observation Statement: For each major European city, the maximum legal highway speed is recorded.
- (h) Most-watched TV channel in each household in a neighborhood:
  - Type: Categorical
  - Observation Statement: For each household in the neighborhood, the most-watched TV channel is recorded.
- (i) Country that earned the highest number of gold medals at each Olympic games:
  - Type: Categorical
  - Observation Statement: For each occurrence of the Olympic games, the country that earned the highest number of gold medals is recorded.
- (j) Systems of government in countries of the world:
  - Type: Categorical
  - Observation Statement: For each country of the world, its system of government is recorded.