Statistics

Definition and Purpose

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

• Descriptive statistics focus on organizing and summarizing data using numerical summaries, tables, and graphs.

• Inferential statistics use data from a sample to make generalizations about a population and measure the reliability of these results.

Data

Definition and Characteristics

Data are facts or propositions used to draw conclusions or make decisions. Data describe characteristics of an individual.

• Examples of data questions:

  • Are we all the same height?

  • Do we all have the same eye color?

  • Do we all watch the same shows?

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

What data (information) do you see on social media?

What data do you keep track of about your own life or for your family?

Population, Individual, and Sample

Definitions

• 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.

Note: For most studies, it is unreasonable to access all individuals of interest, so a sample is used.

Statistic and Parameter

Definitions

• Statistic: A numerical summary based on a sample.

• Parameter: A numerical summary of a population.

Example:

• Suppose the proportion of all students at a college who have a job is 0.849 (parameter).

• Suppose a sample of 250 students is obtained, and from this sample, the proportion who have a job is 0.864 (statistic).

The Process of Statistics

Steps in a Statistical Study

1. Identify the research objective: Determine the question(s) to be answered and clearly identify the population.

2. Collect the data needed to answer the question(s): Proper data collection is crucial. If data are not collected correctly, conclusions may be invalid. A sample is often used due to the difficulty and expense of studying an entire population.

3. Describe the data: Organize and summarize the information to provide an overview and determine appropriate statistical methods.

4. Perform inference: Use techniques to extend results from the sample to the population and report the reliability of the results.

Example: In a study of high school student sleeping patterns, researchers investigated the association between start time of school and sleep duration. They sampled 383 randomly selected adolescents, found an average age of 15.5 years, and observed that students who started school at 8:30 a.m. or later had significantly longer sleep duration. The process included identifying the research objective, collecting information, describing the data, and drawing conclusions.

Types of Variables

Qualitative (Categorical) vs. Quantitative Variables

• Qualitative (Categorical) Variables: Allow for classification of individuals based on some attribute or characteristic (e.g., gender, name of a university).

• Quantitative Variables: Provide numerical measures of individuals. The values can be added or subtracted and provide meaningful results (e.g., number of vending machines, number of days per week a student eats lunch).

Discrete vs. Continuous Variables

• Discrete Variable: A quantitative variable with a finite or countable number of possible values (e.g., number of children in a classroom: 0, 1, 2, 3, 4).

• Continuous Variable: A quantitative variable with an infinite number of possible values that are not countable. It can take on every possible value between any two values (e.g., today's high temperature).

Types of Data (Observations a Variable Assumes)

• Qualitative data: Observations corresponding to a qualitative variable (e.g., gender: male or female).

• Quantitative data: Observations corresponding to a quantitative variable (e.g., gas mileage: 17 mpg, 22 mpg).

• Discrete data: Observations corresponding to a discrete variable.

• Continuous data: Observations corresponding to a continuous variable.

Levels of Measurement

Classification of Variables by Measurement Level

1. Nominal level of measurement: Values are names, labels, or categories. No ordering or ranking is implied (e.g., nationality).

2. Ordinal level of measurement: Values can be arranged in a ranked or specific order, but differences between values are not meaningful (e.g., income status: low, middle, high).

3. Interval level of measurement: Values have meaningful differences, but ratios are not meaningful. Zero does not mean the absence of the quantity (e.g., temperature in Celsius or Fahrenheit).

4. Ratio level of measurement: Values have meaningful differences and ratios. Zero means the absence of the quantity, and arithmetic operations can be performed (e.g., number of adults in a study).

Key Formulas

• Sample Proportion: where is the number of individuals in the sample with the characteristic of interest, and is the sample size.

• Population Proportion: where is the number of individuals in the population with the characteristic of interest, and is the population size.

Summary Table: Types of Variables and Data

Additional info: The above notes expand on the brief points in the original material, providing definitions, examples, and context for each concept. The tables and formulas are included for clarity and exam preparation.