The standard deviation is the square root of the variance. It measures the average distance between each data point and the mean. Like the variance, it expresses the dispersion of data around the mean, but unlike the variance, its units are the same as the original data, making it easier to interpret.

In hierarchical clustering, a dendrogram is used to visualize the hierarchy of clusters. It is a tree-like diagram that records the sequences of merges or splits.

The main difference between one-tailed and two-tailed tests is the directionality of the hypothesis. One-tailed tests look for an effect in a specific direction, while two-tailed tests look for an effect in either direction.

The Multiplication Rule is used when we want to find the probability of two events happening at the same time. Specifically, it states that the probability of two independent events both occurring is the product of their individual probabilities.

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. Positive skewness indicates a distribution with an asymmetric tail extending towards more positive values. Negative skewness indicates a distribution with an asymmetric tail extending towards more negative values.

Yes, a symmetrical distribution can have nonzero kurtosis. Kurtosis is a measure of the weight in the tails, or the extreme values, which can occur in both directions, thus not affecting the symmetry. For example, a normal distribution is symmetric and has a kurtosis greater than zero.

Spearman's rank correlation coefficient is a non-parametric measure of correlation, meaning it can be used when the data is not normally distributed. It is also less sensitive to outliers compared to Pearson's coefficient. Further, it can be used to measure monotonic relationships, whether they are linear or not.

The Mann-Whitney U test is a common non-parametric statistical method used to compare two independent groups when the dependent variable is either ordinal or continuous, but not normally distributed.

A non-significant result in the Chi-square test for goodness of fit indicates that the observed distribution does not significantly differ from the expected distribution. In other words, we do not have enough evidence to reject the null hypothesis.

The Chi-square test for goodness of fit is used to test whether the observed data fits a specific distribution. It compares the observed data with the values that would be expected under the theoretical distribution.