100 cards
Statistics begins with the task of summarizing data, and the first step is choosing meaningful measures of central tendency. The mean, calculated as \(\bar{x} = \sum x_i / n\), captures the arithmetic average and is the most widely used summary, but it is sensitive to extreme values. The median, the middle value of sor...
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, which are subsets of that space. The complement of an event A, denoted A' or A^c, contains all outcomes not in A, and the complement rule \...
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, taking a finite or countable set of values, or continuous, taking any value in an interval. For discrete variables, the probability mass fu...
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 statistic, such as the sample mean, computed from all possible samples of a given size drawn from a population. Even though any individual samp...
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 there is a 95% probability that the specific interval contains the parameter. Rather, it means that if we repeated the sampling procedure ma...
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 \(H_0\) represents the status quo, typically stating that there is no effect or no difference (such as \(\mu = \mu_0\)). The alternative hypothesis \(H_1\)...
The Pearson correlation coefficient \(r\) measures the strength and direction of the linear relationship between two continuous variables, ranging from \(-1\) (perfect negative) through \(0\) (no linear association) to \(+1\) (perfect positive). It is computed as \(r = \sum (x_i - \bar{x})(y_i - \bar{y}) / \sqrt{\sum (...
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 representativeness. In simple random sampling, every member of the population has an equal chance of selection, often implemented using random n...