The assumptions of factor analysis include normality (the variables used in the analysis should be normally distributed), linearity (the relationship between the factors and the variables should be linear), and homoscedasticity (the variances of the errors should be constant).

The Gaussian or normal distribution has several important properties and is widely used in statistics and natural sciences. It's significant because it is the distribution that maximizes entropy among all distributions with given mean and variance, making it the most "uninformative" and often serving as a good default choice in many scenarios. Also, according to the central limit theorem, the sum of many independent and identically distributed (i.i.d.) random variables tends toward a normal distribution.

Pearson's correlation coefficient measures linear relationships, while Spearman's Rank Correlation measures monotonic relationships. Monotonic relationships are ones where the variables tend to change together, but not necessarily at a constant rate. Pearson's Correlation is used when the data is normally distributed, whereas Spearman's Rank Correlation is used when the data does not assume normal distribution.

If the Chi-square statistic is significantly larger than the critical value, we reject the null hypothesis in favor of the alternative hypothesis. This suggests that there is a significant association between the variables.

Qualitative data has several limitations in data analysis. Firstly, it cannot be easily quantified for statistical analysis which limits its utility in certain research settings. Secondly, collecting and analyzing qualitative data often requires substantial resources and time, which can be a challenge for large-scale studies. Lastly, qualitative data may be influenced by researcher bias, particularly during data collection and interpretation.

The method of least squares is commonly used to find the best fitting line in simple linear regression. It minimizes the sum of the squares of the residuals (the vertical distances between the observed and predicted values).

A Type II error occurs when the null hypothesis is false, but it is not rejected. It is also known as a "false negative" result.

The Chi-square statistic in a Chi-square test for independence represents the sum of the squared differences between observed and expected frequencies, divided by the expected frequencies. This statistic measures the degree to which the observed frequencies deviate from the frequencies that would be expected under the null hypothesis of independence.

In a Chi-square test, the expected frequency for each cell in the contingency table is calculated by multiplying the row total and column total and then dividing by the total number of observations.

The Pearson Correlation Coefficient measures the linear relationship between two variables and can range from -1 to 1. A value of -1 means there is a perfect negative correlation, while a value of 1 means there is a perfect positive correlation.