Chapter 1: Data Collection – Fundamentals of Statistics

1.1 Introduction to the Practice of Statistics

1.1.1 Define Statistics and Statistical Thinking

Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions. It also involves providing a measure of confidence in any conclusions.

• Data: Facts or propositions used to draw a conclusion or make a decision. Data describe characteristics of an individual.

• Variability: Data values vary among individuals; understanding and describing this variability is a key goal of statistics.

1.1.2 Explain the Process of Statistics

The process of statistics involves several key steps:

1. Identify the research objective: Clearly state the question(s) to be answered and the population to be studied.

2. Collect the data: Gather information needed to answer the research question, often by sampling.

3. Describe the data: Organize and summarize the information using descriptive statistics (numerical summaries, tables, and graphs).

4. Perform inference: Apply statistical methods to extend results from the sample to the population and measure the reliability of the result.

Statistic: A numerical summary based on a sample. Parameter: A numerical summary of a population. Descriptive statistics: Methods for organizing and summarizing data. Inferential statistics: Methods for drawing conclusions about a population based on sample data.

Example: Parameter vs. Statistic

• If 48.2% of all students own a car, this is a parameter (population summary).

• If 46% of a sample of 100 students own a car, this is a statistic (sample summary).

Example: The Process of Statistics

• Step 1: Identify the research objective (e.g., determine the percentage of adult Americans who trust their neighbors).

• Step 2: Collect data (e.g., survey a sample of 1628 adults).

• Step 3: Describe the data (e.g., 52% trust their neighbors – a descriptive statistic).

• Step 4: Draw conclusions (e.g., infer the percentage for all adults, accounting for margin of error).

1.1.3 Distinguish Between Qualitative and Quantitative Variables

Variables are characteristics of individuals within a population that can vary.

• Qualitative (Categorical) variables: Allow for classification based on some attribute or characteristic (e.g., gender, zip code).

• Quantitative variables: Provide numerical measures of individuals. Arithmetic operations can be performed on these values (e.g., temperature, number of days studied).

Example: Qualitative vs. Quantitative

• Gender: Qualitative

• Temperature: Quantitative

• Number of days studied: Quantitative

• Zip code: Qualitative

1.1.4 Distinguish Between Discrete and Continuous Variables

Quantitative variables can be further classified as:

• Discrete variable: Has a finite or countable number of possible values (e.g., number of heads in coin flips).

• Continuous variable: Has an infinite number of possible values, measurable to any desired level of accuracy (e.g., distance a car can travel on a tank of gas).

Example: Discrete vs. Continuous

• Number of cars at a drive-thru: Discrete

• Distance a car can travel: Continuous

1.1.5 Determine the Level of Measurement of a Variable

Variables can be measured at different levels:

• Nominal level: Values name, label, or categorize; no order (e.g., gender).

• Ordinal level: Values can be ranked or ordered (e.g., letter grades).

• Interval level: Differences in values have meaning; zero does not mean absence of quantity (e.g., temperature in Celsius).

• Ratio level: Ratios of values have meaning; zero means absence of quantity (e.g., number of days studied).

Example: Levels of Measurement

• Gender: Nominal

• Temperature: Interval

• Number of days studied: Ratio

• Letter grade: Ordinal

1.2 Observational Studies Versus Designed Experiments

1.2.1 Distinguish Between an Observational Study and an Experiment

• Observational study: Measures the value of the response variable without influencing the variables. Researchers observe behavior without intervention.

• Designed experiment: Researchers assign individuals to groups, manipulate the explanatory variable, and record the response variable.

Example: Cellular Phones and Brain Tumors

• Observational study: Following a group of women over time to compare brain tumor incidence based on mobile phone use.

• Experiment: Exposing rats to different types of radio-frequency radiation and comparing outcomes to a control group.

Summary Table: Types of Variables and Levels of Measurement

Type	Definition	Example
Qualitative	Describes attribute or category	Gender, Zip code
Quantitative	Numerical measure	Temperature, Number of days studied
Discrete	Countable values	Number of cars
Continuous	Infinite, measurable values	Distance traveled

Level of Measurement	Order?	Meaningful Differences?	True Zero?	Example
Nominal	No	No	No	Gender
Ordinal	Yes	No	No	Letter grade
Interval	Yes	Yes	No	Temperature (Celsius)
Ratio	Yes	Yes	Yes	Number of days studied

Key Formulas

• Sample Proportion: $ext{Sample Proportion} = rac{ ext{Number of successes in sample}}{ ext{Sample size}}$

• Margin of Error (for proportion): $ext{Margin of Error} = z^* imes rac{ ext{Standard deviation}}{ ext{Sample size}^{1/2}}$ Additional info: The exact formula depends on the confidence level and sample size.

Summary

• Statistics is about collecting, analyzing, and interpreting data to make informed decisions.

• Variables can be qualitative or quantitative, and quantitative variables can be discrete or continuous.

• Levels of measurement (nominal, ordinal, interval, ratio) determine the type of analysis possible.

• Observational studies observe without intervention; experiments involve manipulation of variables.