Introduction to Statistics: Key Concepts and Data Types

Introduction to Statistics and Data

• Statistics is the process of taking and analyzing data.

Populations, Samples, and Characteristics

• There are two primary types of groups studied in statistics:
  • Population: The entire group about which information is desired. For example, if a study aims to understand all people with United States citizenship, the population includes every single person holding that citizenship.
  • Sample: A subset of the population from which data is actually collected. Obtaining data from a full population is often unfeasible (e.g., surveying every single U.S. citizen).
• Key terminologies for measurements:
  • Parameter: Information or measurements derived from a whole population.
  • Statistic: Information or measurements drawn specifically from a sample.
• Individuals: These are the specific objects or entities from which information is collected. While often people, individuals can be anything under study.
• Variable: The specific characteristic or attribute of the individuals that is of interest. In mathematical terms, these are often represented as $x$ and $y$, especially in topics like linear correlation or graphing, though they are referred to broadly as variables.

Identification of Individuals and Variables: A Case Study

• Consider a study collecting data from graduates of Armstrong Atlantic State University who majored in mathematics or focused on statistics. The goal is to identify the industry in which these graduates are employed.
• Individuals: The graduates themselves, as they are the source of the information.
• Variable: The industry of employment (e.g., genetic risk assessment, law, insurance, marketing, and pharmacology).
• Note that a study may involve numerous variables for a single individual.

Logical Fallacies: Correlation and Causation

• Statistics can be misused, a concept sometimes referred to as 'lies, damn lies, and statistics.'
• A common error in news reporting is confusing correlation (two things happening together) with causation (one thing causing another).
• Historical Example (Polio): At one time, it was believed that polio was caused by ice cream since cases rose as ice cream consumption increased. In reality, both were linked to a lurking variable: it was summertime and the weather was hot. Polio was more commonly contracted in the heat.

Diversity of Individuals and Variables

• Individuals do not have to be living organisms. Examples include:
  • Computers (outputting different data).
  • Vehicles.
  • Financial instruments (stocks, bonds).
  • Inanimate objects (candles).
• Each individual often has multiple associated variables. For instance, in a medical context (e.g., nursing majors checking in patients), multiple variables are recorded for one person:
  • Sex.
  • Height.
  • Weight.
  • Blood pressure (Systolic and Diastolic).
  • Pulse.
  • Temperature.
  • Specific symptoms of sickness.
• Doctors use these variables and symptoms (e.g., sniffling, fever, cough) to determine the probability of specific illnesses, moving from common causes like the common cold to other possibilities. The book How Doctors Think explores these probabilistic determination processes.

Classification of Variables: Qualitative vs. Quantitative

• Qualitative Variables (Categorical): These place individuals into categories.
  • Examples: Employment industry (law, marketing), Sex.
  • Likert Scales: These use numbers (e.g., scale of 1 to 5 where 1 is poor and 5 is great) but are considered qualitative because the numbers represent categories of sentiment rather than true measurements.
• Quantitative Variables (Numerical): These involve numerical values or measurements.
  • Examples: Height, Weight, Pulse, Temperature, Blood Pressure.
• Note: Terminologies can vary by textbook or professor; qualitative data is frequently referred to as categorical data.

Types of Quantitative Variables: Discrete and Continuous

• Discrete Data: A dataset where there is a finite number of possible values or there are distinct gaps between any two possible values on a number line.
  • Example: Number of people in a restaurant (you can have 2 or 3 people, but not 2.5 people).
  • Example: Pulse rate (measured in whole beats per minute, such as 62 or 63).
• Continuous Data: A dataset where values can take on any value within a finite or infinite interval; there are no gaps, and values can be increasingly precise decimals.
  • Example: Height (can be $72\,inches$, $72.5\,inches$, or $72.578\,inches$ depending on the accuracy of the device).
  • Example: Weight (can be $200.12\,lbs$).
  • Example: Age (can be broken down into years, months, days, minutes, and milliseconds; functionally $22.56\,years$).
  • Example: Average height above sea level.
• Money and Decimals: Money is often treated as continuous because, while we use cents in stores, financial contexts like the stock market or gas stations utilize fractions of a cent (e.g., $\$0.9999$).

Summarizing Data Distributions

• Distribution of a Variable: A description that specifies which values a variable takes and how often it takes them.
• Frequency Table: A table that records the count (frequency) of how many times each category or value occurs.
• Relative Frequency Table: Displays the proportion or percentage of the total for each category.
  • Calculation: $\text{Relative Frequency} = \frac{\text{Frequency of Category}}{\text{Total Frequency}}$.
  • To convert to a percentage: multiply the relative frequency by $100$.
• Precision Rule: To avoid inaccuracies, always save rounding until the final step of a calculation. Rounding sub-steps can lead to totals not equaling $100\%$ or significant errors in large datasets.

Visualizing Qualitative Data

• Pie Charts: Show the distribution of a categorical variable as slices of a circle.
  • Drawback: They lose specific raw data. Without labels, you cannot determine individual counts (e.g., how many students are nursing majors).
• Bar Graphs: Use rectangles to represent frequencies or relative frequencies.
  • The x-axis contains categories (categories like Social Psychology, Abnormal Psychology).
  • The y-axis represents the frequency (count) or relative frequency (percentage).
  • Bar graphs make it easier to compare the proportions of different categories compared to pie charts.

Visualizing Quantitative Data

• Histograms: Similar in appearance to bar graphs but used for quantitative data.
  • Bins: The x-axis is divided into "bins" or intervals representing ranges of values (e.g., ages $75$ to $80$).
  • Frequency: The height of the rectangles on the y-axis represents the count within that bin.
• Stem Plots (Stem and Leaf Plots): A method of displaying quantitative data where each value is split into a "stem" (the first digit or digits) and a "leaf" (the last digit). It effectively acts as a sideways histogram composed of actual numbers.
• Interpreting Histograms:
  • Example: Average life expectancy by country. If the bin $75$ to $80$ has the highest bar, that is the most common range.
  • Frequency analysis: To find how many countries have a life expectancy of $50$ years or less, you must add the heights of all bars to the left of the $50$ mark.

Questions & Discussion

• Question: Is it feasible to get a full sample of data for every single person with United States citizenship?
• Answer: No, it is not feasible; a sample must be used instead.
• Question: Can individuals be inanimate objects, such as stocks or candles?
• Answer: Yes, individuals are simply the objects we collect information from, living or not.
• Question: Is the rate of a Likert scale (1 to 5) a quantitative variable?
• Answer: No, it is qualitative because the numbers are used to represent categories of feeling or opinion.
• Question: Why is pulse considered discrete but weight continuous?
• Answer: Pulse has gaps (you cannot have half a pulse beat), whereas weight can be measured to infinite decimal places given a sensitive enough scale.
• Question: Is the y-axis always frequency on a histogram?
• Answer: In this course, it generally will be, though in advanced statistics, it can represent relative frequency or density. On a histogram, the x-axis must represent a quantitative variable.