Introduction to Statistics

Statistical and Critical Thinking

Statistics is the science of collecting, analyzing, interpreting, and presenting data. Critical thinking in statistics involves evaluating the validity of data sources, methods, and conclusions. Understanding the types of data and their properties is essential for choosing appropriate statistical methods.

• Statistics refers to the study and use of data to make informed decisions.

• Critical thinking is necessary to assess the reliability and relevance of statistical findings.

Key Statistical Concepts

Parameters and Statistics

In statistics, it is important to distinguish between parameters and statistics, as they refer to measurements from different groups.

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

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

Example

• Parameter: The mean age of all FAU students is 19.5 years.

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

Types of Data

Quantitative Data

Quantitative (or numerical) data consist of numbers representing counts or measurements. These data are used for mathematical calculations and statistical analysis.

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

Categorical Data

Categorical (or qualitative or attribute) data consist of names or labels that do not represent counts or measurements. These data classify individuals into groups or categories.

• Examples: The gender (male/female) of professional athletes, shirt numbers on professional athletes (as substitutes for names).

Working with Quantitative Data

Quantitative data can be further classified as discrete or continuous based on the nature of the values.

• Discrete Data: Result when the data values are quantitative and the number of values is finite or countable. Example: The number of tosses of a coin before getting heads.

• Continuous Data: Result from infinitely many possible quantitative values, where the collection of values is not countable. Example: The lengths of distances from 0 cm to 12 cm.

Levels of Measurement

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

Nominal Level

The nominal level of measurement is characterized by data that consist 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

The ordinal level of measurement involves data that can be arranged in some order, but differences between data values either cannot be determined or are not meaningful.

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

Interval Level

The interval level of measurement involves data that can be arranged in order, and the differences between data values can be found and are meaningful. However, there is no natural zero starting point.

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

Ratio Level

The ratio level of measurement is characterized by data that can be arranged in order, differences can be found and are meaningful, and there is a natural zero starting point. Both differences and ratios are meaningful.

• 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 the use of the term 'twice' make sense?

• True Zero: Is there a zero quantity that means none of the variable is present?

Examples

• Ratio Level: Time (minutes) taken to complete a statistics exam. It makes sense to say 'one student took twice as much time as another,' and 0 minutes is a true zero.

• Interval Level: Body temperatures (Celsius) of students. It does not make sense to say 'twice as much,' and 0°C is not a true zero.

Questions: Levels of Measurement

Examples of categorizing data by their level of measurement:

• Ratio: Age of students

• Nominal: Data with eye colors (brown, blue, green)

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

• Ordinal: Survey using Likert-type questions

Big Data and Data Science

Big Data

Big data refers to data sets so large and complex that traditional software tools cannot analyze them efficiently. Analysis of big data often requires parallel processing on multiple computers.

Data Science

Data science involves the application of statistics, computer science, software engineering, and other relevant fields to analyze and interpret complex data sets.

Handling 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 that it is missing.

Correcting for Missing Data

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

• Impute Missing Values: Substitute missing values with estimated or calculated values.

Additional info: Imputation methods may include mean substitution, regression imputation, or more advanced techniques such as multiple imputation.