Introduction to Statistics

Statistical and Critical Thinking

Statistics is the science of collecting, analyzing, interpreting, and presenting data. A major use of statistics is to collect and use sample data to make conclusions about populations. Critical thinking is essential in statistics to ensure that conclusions drawn from data are valid and reliable.

Types of Data

Population vs. Sample

• Population: The complete collection of all elements (scores, people, measurements, etc.) to be studied.

• Sample: A subcollection of members selected from a population.

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

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

Quantitative vs. Categorical Data

• Quantitative (Numerical) Data: Consists of numbers representing counts or measurements. Examples: The weights of supermodels, the ages of respondents.

• Categorical (Qualitative or Attribute) Data: Consists of names or labels (not numbers that represent counts or measurements). Examples: The gender (male/female) of professional athletes, shirt numbers on uniforms.

Discrete vs. Continuous Quantitative Data

• 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 tails.

• 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 into four levels of measurement, each with increasing complexity and mathematical meaning.

Summary of Levels of Measurement

• Nominal: Categories only

• Ordinal: Categories with some order

• Interval: Differences but no natural zero point

• Ratio: Differences and a natural zero point

Big Data and Data Science

• Big Data: Refers to data sets so large and complex that their analysis is beyond the capabilities of traditional software tools. Analysis may require software running in parallel on many computers.

• Data Science: Involves applications of statistics, computer science, and software engineering, along with other relevant fields such as sociology or finance.

Missing Data

Types of Missing Data

• Missing Completely at Random (MCAR): The likelihood of a 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

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.