The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

A 95% confidence interval means that if we were to take a large number of samples and calculate the confidence interval for each sample, we would expect the true population parameter to fall within the interval 95% of the time.

Distance measures, like Euclidean distance or Manhattan distance, play a crucial role in cluster analysis. They determine the similarities or differences between data points. They influence how the clusters will be formed, as the most similar or closest data points get clustered together.

Spearman's rank correlation coefficient is a nonparametric measure of rank correlation. It's preferred over Pearson's correlation when the variables are not normally distributed, the relationship is nonlinear, or the data contains outliers. It assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about the frequency distribution of the variables.

A residual is the difference between the observed value of the dependent variable (y) and the predicted value (ŷ), given by the regression model. It represents the error of the estimate.

The correlation coefficient is not affected by changes in the center (mean) or scale (standard deviation) of the variables. This is because correlation measures the strength of a relationship between variables relative to their variability. It's a dimensionless quantity, so changes in the scale of measurements of the variables do not change it.

The Chi-square test for goodness of fit assumes that the observations are independent, which means that the outcome of one observation does not affect the outcome of another.

The null hypothesis in a Chi-square test for independence states that there is no association between the variables - they are independent.

Bartlett's Test of Sphericity is used in factor analysis to test the hypothesis that the variables are uncorrelated in the population. In other words, the population correlation matrix is an identity matrix. A significant test indicates that a factor analysis may be useful with your data.

In multiple linear regression, each coefficient represents the average change in the dependent variable for one unit of change in the independent variable, while holding all other independent variables constant.