Introduction to Statistics

• Statistics involves the processes of data collection, organization, analysis, interpretation, and presentation.
• As a mathematical science, statistics focuses on data collection, analysis, interpretation/explanation, and presentation.
• Statistics is a branch of mathematics that is concerned with data collection and interpretation.
• Statistics is applicable in diverse fields, including academic, industrial, and societal sectors.

Descriptive Statistics

• Descriptive statistics summarize and describe a dataset's characteristics.
• These statistics offer concise summaries of samples and their measurements.
• Common descriptive statistics include central tendency measures, dispersion measures, and data visualizations.
• Central tendency measures describe a dataset's "typical" value, using mean, median, and mode.
• The mean is the average of a dataset, found by summing values and dividing by the number of values.
• The median is the central value in an ordered dataset.
• The mode is the most frequently occurring value in a dataset.
• Dispersion measures describe data spread or variability, employing range, variance, and standard deviation.
• The range is the difference between a dataset’s largest and smallest values.
• Variance measures the average squared deviation from the mean, indicating data spread around it.
• Standard deviation, the square root of variance, measures the typical distance of values from the mean.
• Descriptive statistics use histograms, bar charts, pie charts, and scatter plots to visually represent data.
• Histograms display a single variable's distribution, showing value frequency within intervals.
• Bar charts compare category frequencies or values.
• Pie charts show each category's proportion relative to the whole.
• Scatter plots display relationships between two variables.

Inferential Statistics

• Inferential statistics makes inferences and generalizations about a population from a sample.
• These methods draw conclusions about a population using smaller sample observations.
• Common techniques are hypothesis testing, confidence intervals, and regression analysis.
• Inferential statistics uses probability to assess the accuracy of population inferences.
• Hypothesis testing assesses claims about population parameters using sample data.
• It involves a null hypothesis (no effect/difference) and an alternative hypothesis (contradicts null).
• Statistical tests determine if evidence is sufficient to reject the null hypothesis for the alternative.
• Common hypothesis tests: t-tests, chi-square tests, and ANOVA.
• Confidence intervals provide a range where a population parameter likely falls, with a certain confidence level.
• Confidence intervals are expressed as a point estimate (e.g., sample mean) plus/minus a margin of error.
• The margin of error depends on confidence level, sample size, and data variability.

Probability Theory

• Probability theory analyzes random phenomena.
• It provides a framework for quantifying and reasoning about uncertainty.
• Key concepts: probability, random variables, probability distributions, and expected value.
• Probability measures the likelihood of an event, ranging from 0 (impossible) to 1 (certain).
• A random variable's value is a numerical outcome of a random phenomenon.
• Random variables can be discrete (finite/countable values) or continuous (values within a range).
• A probability distribution describes the likelihood of each value of a random variable.
• Probability mass function (PMF) is the probability distribution for discrete random variables.
• Probability density function (PDF) is the probability distribution for continuous random variables.
• A random variable's expected value is the average over many trials.
• It is calculated by summing each possible value times its probability.

Regression Analysis

• Regression analysis models the relationship between a dependent variable and one or more independent variables.
• Used for predicting, forecasting, and understanding variable relationships.
• Types of regression analysis: linear, multiple, and nonlinear regression.
• Linear regression models the relationship as a linear function.
• Multiple regression extends linear regression to include multiple independent variables.
• Nonlinear regression models the relationship as a nonlinear function.
• The goal is to find the best-fitting line or curve describing variable relationships.
• Achieved by minimizing the sum of squared errors between observed and predicted values.
• The regression model predicts the dependent variable's value for given independent variable values.
• Regression analysis also indicates the strength and direction of variable relationships.

Hypothesis Testing

• Hypothesis testing evaluates claims about population parameters using sample data.
• It involves a null hypothesis (H0) and an alternative hypothesis (Ha).
• The null hypothesis states no effect or difference; the alternative contradicts it.
• The goal is to determine if there's enough evidence to reject the null hypothesis for the alternative.
• Hypothesis testing steps include stating hypotheses, and choosing a significance level (alpha).
• Alpha represents the probability of incorrectly rejecting the null hypothesis (Type I error).
• You must select a statistical test based on data type and hypotheses.
• Calculate the test statistic, measuring the difference between sample data and expectation under the null hypothesis.
• Determine the p-value, the probability of observing an extreme test statistic if the null hypothesis is true.
• Compare the p-value to alpha:
  • Reject the null hypothesis if the p-value is less than or equal to alpha.
  • Fail to reject the null hypothesis if the p-value is greater than alpha.
• Common hypothesis tests:
  • T-tests compare means of two groups or test one group's mean against a known value.
  • Chi-square tests assess associations between categorical variables.
  • ANOVA compares means of three or more groups.