The power of a test is the ability of the test to detect an effect if it truly exists. It's the probability that the test correctly rejects a false null hypothesis. High power is desirable because it means the test is less likely to make a Type II error (false negative). When designing a study, it's important to choose a sample size and significance level that will provide enough power to detect an effect if one exists.

Standard deviation is a measure of how spread out the numbers in a dataset are around the mean. It measures the average distance between each data point and the mean. The higher the standard deviation, the more spread out the data is.

The centroid is the geometric center of a cluster. In other words, it's the mean value of all the points in a specific cluster.

The mean is most affected by extreme values or outliers. The mean calculates the average of all data points in a dataset. Therefore, when an extreme value (either very high or very low compared to the rest) is introduced, it can significantly impact the mean. The median and mode are less affected by outliers.

In the Sign Test, ties between paired samples are generally discarded and not included in the final calculation.

The Sign Test is a non-parametric test used to test the median of a distribution, or to compare the medians of two distributions.

In the context of statistics, a 'population' refers to the entire set of individuals or items that we are interested in studying. This could range from all the people living in a country, to all the particles in a chemical solution, to all the sales transactions in a store, and so forth. When we collect data from every individual or item in the population, it's called a census. The information gained from a population is reliable and accurate because it accounts for everyone or everything we want to study.

An event in the context of probability refers to a possible outcome or combination of outcomes from a random experiment. For instance, if we toss a coin, the possible outcomes are 'Heads' and 'Tails'. Each of these outcomes is an event.

The probability of the union of two mutually exclusive events A and B is given by P(A ∪ B) = P(A) + P(B). This is because mutually exclusive events cannot occur at the same time, so the probability of either event A occurring or event B occurring is the sum of their individual probabilities.

The alternative hypothesis proposes that there is a difference or an effect in the population and the sample observations are influenced by some non-random cause.