Measures of central tendency include the mean, median, and mode, which summarize the center of a data set:

• Mean: The average of the data points, \( \bar{x} = \frac{\sum x_i}{n} \).
• Median: The middle value when the data is ordered.
• Mode: The most frequently occurring value in the data set.

Measures of dispersion include the range, variance, and standard deviation, which describe the spread of the data:

• Range: Difference between the maximum and minimum values.
• Variance: Average of the squared differences from the mean, \( \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} \).
• Standard Deviation: Square root of the variance, \( \sigma = \sqrt{\sigma^2} \).

Probability measures the likelihood of an event occurring, expressed as:

\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Conditional probability is the probability of an event occurring given that another event has already occurred, denoted as \( P(A \mid B) \):

\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \]

The binomial distribution models the number of successes in a fixed number of independent trials, with a success probability \( p \) in each trial. The probability of \( k \) successes in \( n \) trials is:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

The normal distribution is a continuous probability distribution that is symmetric around the mean. The probability density function is:

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

The Poisson distribution models the number of events occurring within a fixed interval of time or space, with a given mean rate \( \lambda \). The probability of observing \( k \) events is:

\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]

A confidence interval for the mean \( \mu \) is an interval estimate, which provides a range of values within which the true mean is likely to fall. For a sample mean \( \bar{x} \) and standard deviation \( s \), the 95% confidence interval is:

\[ \bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}} \]

The null hypothesis \( H_0 \) is a statement of no effect or no difference, while the alternative hypothesis \( H_1 \) is a statement indicating the presence of an effect or difference. Hypothesis testing determines whether to reject \( H_0 \) in favor of \( H_1 \).

The p-value measures the strength of evidence against the null hypothesis. If the p-value is less than the significance level \( \alpha \), reject \( H_0 \). Common significance levels are 0.05 or 0.01.

The correlation coefficient \( r \) measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1:

• \( r = 1 \): Perfect positive correlation.
• \( r = -1 \): Perfect negative correlation.
• \( r = 0 \): No correlation.

Simple linear regression models the relationship between a dependent variable \( y \) and an independent variable \( x \) using the equation:

\[ y = \beta_0 + \beta_1 x + \epsilon \]

where \( \beta_0 \) is the y-intercept, \( \beta_1 \) is the slope, and \( \epsilon \) is the error term.

One-way ANOVA tests whether there are statistically significant differences between the means of three or more independent groups. The null hypothesis states that all group means are equal.

The Chi-square test for independence assesses whether two categorical variables are independent of each other. The test statistic is:

\[ \chi^2 = \sum \frac{(O - E)^2}{E} \]

where \( O \) is the observed frequency and \( E \) is the expected frequency.

The Wilcoxon Signed-Rank Test is a non-parametric test used to compare two related samples or repeated measurements on a single sample. It assesses whether their population mean ranks differ.

Moving averages smooth out short-term fluctuations and highlight longer-term trends in time series data. The simple moving average (SMA) for period \( t \) is:

\[ SMA_t = \frac{1}{n} \sum_{i=0}^{n-1} X_{t-i} \]

Bayes' Theorem relates the conditional probability of event \( A \) given \( B \) to the conditional probability of \( B \) given \( A \), and the individual probabilities of \( A \) and \( B \):

\[ P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)} \]

Simple random sampling is a method where each individual in the population has an equal chance of being selected. It is the most basic sampling technique.

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large.

Randomized Controlled Trials (RCT) are experiments where participants are randomly assigned to different treatment groups to test the effect of an intervention.

Control charts are used in quality control to monitor whether a process is in statistical control. Common types include X-bar charts (for the mean) and R-charts (for range).