Introduction to Statistics

Statistical and Critical Thinking

Statistics is the science of collecting, analyzing, presenting, and interpreting data. Critical thinking in statistics involves understanding the context of data, recognizing the limitations of statistical methods, and making informed decisions based on evidence.

• Statistics: The study of methods for collecting, analyzing, interpreting, and presenting empirical data.

• Parameter: A numerical measurement describing some characteristic of a population.

• Statistic: A numerical measurement describing some characteristic of a sample.

Example: Parameter vs Statistic

• Parameter: The mean age of all FAU students is 20.5 years (population).

• Statistic: The mean age of FAU students in a class is 18.75 years (sample).

Types of Data

Quantitative Data

Quantitative data consists of numbers representing counts or measurements. These data types are essential for most statistical analyses.

• Quantitative (Numerical) Data: Data expressed in numbers, such as weights, heights, or ages.

• Examples: The weights of supermodels, the ages of survey respondents.

Categorical Data

Categorical data consists of names or labels and does not represent counts or measurements. These data are used to classify subjects into groups.

• Categorical (Qualitative) Data: Data expressed as names, labels, or attributes.

• Examples: Gender (male/female), shirt numbers on athletes' uniforms.

Working with Quantitative Data: Discrete vs Continuous

Quantitative data can be further classified as discrete or continuous, which affects the choice of statistical methods.

• Discrete Data: Quantitative data with a finite or countable number of values. Example: Number of coin tosses before getting tails.

• Continuous Data: Quantitative data with infinitely many possible values, not countable. Example: Lengths of distances from 0 cm to 12 cm.

Levels of Measurement

Overview

Data can be classified into four levels of measurement: nominal, ordinal, interval, and ratio. The level of measurement determines the types of statistical analyses that are appropriate.

• Nominal: Categories only.

• Ordinal: Categories with some order.

• Interval: Differences are meaningful, but no natural zero point.

• Ratio: Differences and ratios are meaningful, with a natural zero point.

Nominal Level

Nominal data consists of names, labels, or categories only. The data cannot be arranged in any meaningful order.

• Example: Survey responses of yes, no, and undecided.

Ordinal Level

Ordinal data can be arranged in order, but differences between data values are either not meaningful or cannot be determined.

• Example: Course grades A, B, C, D, or F.

Interval Level

Interval data can be ordered, and differences between values are meaningful. However, there is no natural zero starting point.

• Example: Years 1000, 2000, 1776, and 1492.

Ratio Level

Ratio data can be ordered, differences and ratios are meaningful, and there is a natural zero point.

• Example: Class times of 50 minutes and 100 minutes.

Summary Table: Levels of Measurement

Ratio Level vs Interval Level

To distinguish between ratio and interval levels, consider whether the concept of 'twice as much' makes sense and whether there is a true zero point.

• Ratio Test: Does 'twice as much' make sense?

• True Zero: Is there a zero quantity?

Examples

• Time (minutes) to complete a test: Ratio level (0 minutes is a true zero; 'twice as much' makes sense).

• Body temperature (Celsius): Interval level (0°C is not a true zero; 'twice as much' does not make sense).

Questions: Levels of Measurement

Examples of categorizing data by level of measurement:

• Ratio: Age of students

• Nominal: Eye color (brown, blue, green, etc.)

• Nominal: Survey data with labels (1: Business student, 2: Art student, 3: Engineering student)

• Ordinal: Survey responses using Likert-type questions (e.g., Strongly Disagree to Strongly Agree)

Big Data and Data Science

Big Data

Big data refers to data sets that are so large and complex that traditional software tools cannot efficiently analyze them. Specialized software and parallel computing are often required.

Data Science

Data science is an interdisciplinary field that combines statistics, computer science, software engineering, and domain expertise to analyze and interpret complex data sets.

Missing Data

Types of Missing Data

• Missing Completely at Random (MCAR): The likelihood of a data value being missing is independent of its value or any other values in the data set.

• Missing Not at Random (MNAR): The missing value is related to the reason it is missing.

Correcting for Missing Data

1. Delete Cases: Remove all subjects with any missing values from the analysis.

2. Impute Missing Values: Substitute missing data values with estimated or predicted values.

Key Formulas and Concepts

Population Mean (Parameter)

The mean of a population is a parameter, denoted by :

Sample Mean (Statistic)

The mean of a sample is a statistic, denoted by :

Classification of Data Types

Additional info: The notes have been expanded with definitions, examples, and tables for clarity and completeness.