Data Collection and Statistical Fundamentals

Learning Objectives

• Define statistics and statistical thinking.

• Explain the process of statistics.

• Distinguish between qualitative and quantitative variables.

• Distinguish between discrete and continuous variables.

• Determine the level of measurement of a variable.

Fundamentals of Statistics, Data, and Populations

• Statistics is defined as the science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions, as well as providing a measure of confidence in any conclusions.

• Data are defined as a fact or proposition used to draw a conclusion or make a decision, describing characteristics of an individual.

• A primary goal of statistics is to describe and understand sources of variability.

• Key concepts and terminology:

  • Population: The entire group of individuals to be studied.

  • Individual: A person or object that is a member of the population being studied.

  • Sample: A subset of the population that is being studied.

Statistics vs. Parameters and Types of Statistics

• Statistical Measures:

  • Statistic: A numerical summary of a sample.

  • Parameter: A numerical summary of a population.

• Branches of Statistics:

  • Descriptive Statistics: Consist of organizing and summarizing data through numerical summaries, tables, and graphs.

  • Inferential Statistics: Uses methods that take a result from a sample, extend it to the population, and measure the reliability of the result.

• Illustrative Example (Car Ownership):

  • Scenario A: If $48.2\%$ of all students on a campus own a car, this value represents a parameter because it is a numerical summary of a population.

  • Scenario B: If a sample of $100$ students is obtained and $46\%$ own a car, this value represents a statistic because it is a numerical summary of a sample.

The Four-Step Process of Statistics

• Step 1: Identify the Research Objective

  • A researcher must determine the question(s) to be answered.

  • The question(s) must clearly identify the specific population to be studied.

• Step 2: Collect the Data Needed to Answer the Question(s)

  • Conducting research on an entire population is often difficult and expensive, so researchers typically study a sample.

  • Data collection is a vital step because if data are collected incorrectly, any conclusions drawn are meaningless. The importance of appropriate data collection cannot be overlooked.

• Step 3: Describe the Data

  • Descriptive statistics allow the researcher to obtain an overview of the data.

  • Helps determine the type of statistical methods the researcher should use.

• Step 4: Perform Inference

  • Apply appropriate techniques to extend results obtained from the sample to the population.

  • Report a level of reliability of the results.

Detailed Case Study: Pew Research Neighbor Trust Survey

• Research Objective: Pew Research conducted a poll asking, "Do you trust all or most of your neighbors?" The researchers wanted to determine the percentage of adult Americans who trust all or most of their neighbors. The target population was all adult Americans.

• Data Collection: Because surveying the more than $200\,000\,000$ adult Americans is unreasonable, researchers surveyed a sample of $1628$ adult Americans. Among those surveyed, $847$ stated they trust all or most of their neighbors.

• Data Description:

  • Proportion calculation:     $\frac{847}{1628} \approx 52\%$

  • Exactly $52\%$ of the $1628$ surveyed individuals stated they trust all or most of their neighbors. This represents a descriptive statistic as a summary of sample data.

• Perform Inference:

  • Pew Research sought to extend the results of the survey to all adult Americans.

  • Generalizing results from a sample to a population introduces uncertainty. To account for this uncertainty, Pew reported a margin of error of $2.5\%$.

  • This margin of error means Pew feels fairly certain (specifically $95\%$ certain) that the percentage of all adult Americans who trust all or most of their neighbors falls within a specific range:

  • Lower bound calculation:     $52\% - 2.5\% = 49.5\%$

  • Upper bound calculation:     $52\% + 2.5\% = 54.5\%$

  • Conclusion: The percentage of all adult Americans who trust all or most of their neighbors is estimated to be between $49.5\%$ and $54.5\%$.

Classification of Variables

• Variables defined: Variables are the characteristics of the individuals within the population.

• Principle of Variability:

  • Variables vary by definition.

  • Example: Consider tomato weights. If every tomato had the exact same weight, knowing the weight of one tomato would allow determination of the weights of all tomatoes. However, tomato weights vary. One goal of research is to learn the causes of variability to cultivate plants yielding optimal tomatoes.

• Primary Classifications of Variables:

  • Qualitative (Categorical) Variables: Allow for classification of individuals based on some attribute or characteristic.

  • Quantitative Variables: Provide numerical measures of individuals. Values can be added or subtracted and yield mathematically meaningful results.

• Variable Classification Examples:

  • Race: Qualitative

  • Temperature: Quantitative

  • Number of days during the past week that a college student studied: Quantitative

  • Zip code: Qualitative

Quantitative Variables: Discrete vs. Continuous

• Discrete Variable:

  • Definition: A quantitative variable that has either a finite number of possible values or a countable number of possible values.

  • Meaning of countable: Values result from counting (e.g., $0, 1, 2, 3, \dots$).

  • Key restriction: A discrete variable cannot take on every possible value between any two possible values.

• Continuous Variable:

  • Definition: A quantitative variable that has an infinite number of possible values that are not countable.

  • Key property: A continuous variable may take on every possible value between any two values.

• Examples of Discrete vs. Continuous Classification:

  • Number of heads obtained after flipping a coin five times: Discrete

  • Number of cars that arrive at a McDonald's drive-thru between 12:00 p.m. and 1:00 p.m.: Discrete

  • Distance a 2023 Tesla Model Y can travel in city driving conditions with a full charge: Continuous

Case Study: Seattle Parking Meter Sample Data Analysis

• Context: Data presented from a sample of on-street parking meters within the City of Seattle.

• Complete Sample Data Set:

  • Car 1: Payment Method = Credit Card, Amount Paid = \$3.75, Duration = 30 minutes, Side of Street = W, Parking Space Number = 458

  • Car 2: Payment Method = Credit Card, Amount Paid = \$2.00, Duration = 240 minutes, Side of Street = E, Parking Space Number = 37

  • Car 3: Payment Method = Credit Card, Amount Paid = \$2.00, Duration = 240 minutes, Side of Street = NE, Parking Space Number = 18

  • Car 4: Payment Method = Phone, Amount Paid = \$1.38, Duration = 225 minutes, Side of Street = SW, Parking Space Number = 382

  • Car 5: Payment Method = Phone, Amount Paid = \$0.50, Duration = 60 minutes, Side of Street = W, Parking Space Number = 770

  • Car 6: Payment Method = Credit Card, Amount Paid = \$0.25, Duration = 10 minutes, Side of Street = S, Parking Space Number = 75

  • Car 7: Payment Method = Credit Card, Amount Paid = \$1.00, Duration = 120 minutes, Side of Street = E, Parking Space Number = 136

  • Car 8: Payment Method = Credit Card, Amount Paid = \$0.50, Duration = 20 minutes, Side of Street = S, Parking Space Number = 59

  • Car 9: Payment Method = Phone, Amount Paid = \$0.50, Duration = 20 minutes, Side of Street = S, Parking Space Number = 62

  • Car 10: Payment Method = Phone, Amount Paid = \$0.75, Duration = 30 minutes, Side of Street = S, Parking Space Number = 15

  • Car 11: Payment Method = Phone, Amount Paid = \$1.71, Duration = 204 minutes, Side of Street = SW, Parking Space Number = 382

• Identification of Structural Elements:

  • Individuals: There are $11$ individuals in the table, corresponding to cars $1, 2, 3, \dots, 11$

  • Variables: There are $5$ variables measured for each car:

  • Payment Method (Qualitative / Categorical - categorizes the individual)

  • Side of Street (Qualitative / Categorical - categorizes the individual)

  • Parking Space Number (Qualitative / Categorical - categorizes the individual)

  • Amount Paid (Quantitative)

  • Duration in minutes (Quantitative)

  • Observations: There are $11$ observations, one for each individual. For example, the observation for the third car consists of: Credit Card, \$2.00, 240, NE, and 18.

  • Data Values: There are $55$ total data values, calculated by multiplying the number of individuals by the number of variables:     $11 \times 5 = 55$

Levels of Measurement

• Nominal Level of Measurement:

  • Definition: A variable is at the nominal level of measurement if the values of the variable name, label, or categorize.

  • Key property: The naming scheme does not allow for the values of the variable to be arranged in a ranked or specific order.

• Ordinal Level of Measurement:

  • Definition: A variable is at the ordinal level of measurement if it has the properties of the nominal level of measurement and its naming scheme allows for the values to be arranged in a ranked or specific order.

• Interval Level of Measurement:

  • Definition: A variable is at the interval level of measurement if it has the properties of the ordinal level of measurement and differences in the values of the variable have meaning.

  • Meaning of zero: A value of zero does not signify the absence of the quantity.

  • Arithmetic operations: Addition and subtraction can be performed meaningfully on values of the variable.

• Ratio Level of Measurement:

  • Definition: A variable is at the ratio level of measurement if it has the properties of the interval level of measurement and ratios of the values of the variable have meaning.

  • Meaning of zero: A value of zero signifies the complete absence of the quantity.

  • Arithmetic operations: Multiplication and division can be performed meaningfully on values of the variable.

• Examples of Levels of Measurement:

  • Race: Nominal

  • Letter grade earned in a statistics class: Ordinal

  • Temperature: Interval

  • Number of days during the past week that a college student studied: Ratio