The standard error is the standard deviation of the sampling distribution of a statistic. It measures the dispersion of the sample means around the true population mean.

The two main branches of statistics are descriptive and inferential. Descriptive statistics involves methods of organizing, picturing, and summarizing information from data. It provides simple summaries about the sample and measures, such as mean, median, mode, etc. Inferential statistics, on the other hand, involves methods of using information from a sample to draw conclusions (inferences) about the population. It includes various techniques like hypothesis testing, regression analysis, etc.

The Mann-Whitney U test assumes that the samples are independent (not paired or related) and heterogeneous (can have different variances).

In probability theory, the "complement" of an event A is the event that A does not occur, often denoted as A'. If the probability of event A happening is P(A), then the probability of it not happening, or its complement, is P(A') = 1 - P(A).

In a two-way ANOVA, the main effect refers to the effect of one independent variable on the dependent variable, adjusting for the effects of the other independent variables. It provides the overall effect of one factor on the outcome, irrespective of the levels of other factors.

For a Chi-square test for independence to be valid, the observations must be independent, and the expected frequency of each category must be at least 5.

The coefficient in Spearman's Rank Correlation indicates the strength and direction of the association between two ranked variables. This coefficient can range from -1 (perfect negative correlation) to 1 (perfect positive correlation).

A scatter plot is a graphical representation that uses dots to represent the values obtained for two different variables - one plotted along the x-axis and the other plotted along the y-axis. It helps to identify the type of relationship (if any) between two numerical variables.

Yes, PCA requires the features to be on the same scale. If features are on different scales, PCA might end up giving higher weightage to features with higher variance, which could lead to incorrect principal components. So, it's typically a good practice to standardize the data before applying PCA.

In factor analysis, communalities are the proportion of variance in each variable that is accounted for, or shared among the factors. They represent the shared variance between variables.