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Statistics begins with the task of summarizing data, and the first step is choosing meaningful measures of central tendency. The mean, calculated as \(\bar{x} = \sum x_i / n\), cap...
Probability provides the mathematical framework for reasoning about uncertainty. Every probability problem starts with a sample space, the set of all possible outcomes, and events,...
A random variable assigns a numerical value to each outcome in a sample space, transforming random experiments into quantities we can analyze. Random variables are either discrete,...
Before drawing conclusions from data, we must understand how sample statistics behave across many possible samples. A sampling distribution is the probability distribution of a sta...
A confidence interval provides a range of plausible values for an unknown population parameter, constructed from sample data. A 95% confidence interval, for example, does not mean...
Hypothesis testing is a formal procedure for deciding whether sample data provides enough evidence to reject a default claim about a population. The null hypothesis \(H_0\) represe...
The Pearson correlation coefficient \(r\) measures the strength and direction of the linear relationship between two continuous variables, ranging from \(-1\) (perfect negative) th...
The validity of any statistical conclusion depends critically on how the data was generated, and different sampling methods offer different trade-offs between practicality and repr...