The null hypothesis in an ANOVA test is that the means of all groups are equal. If the p-value obtained from the ANOVA test is less than the significance level, the null hypothesis is rejected, implying that there is a significant difference between at least two of the group means.

In a scatter plot, an upward trend suggests a positive relationship between the variables. This means as one variable increases, the other variable also increases.

In factor analysis, eigenvalues represent the total variance explained by each factor. A larger eigenvalue indicates that more of the total variance is accounted for by that factor.

Pearson's Correlation Coefficient assumes that the variables are normally distributed. It's one of the key assumptions made when calculating the coefficient, and it refers to the shape of the distribution of the values.

Polynomial regression models relationships as curves, not straight lines. This allows polynomial regression to capture non-linear relationships, where the relationship changes direction at different levels of the independent variables. On the other hand, linear regression models relationships as straight lines, assuming a constant rate of change.

The Central Limit Theorem states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined (finite) expected value and finite variance, will be approximately normally distributed, regardless of the shape of the original distribution.

ANOVA stands for Analysis Of Variance. It's a statistical technique used to check if the means of two or more groups are significantly different from each other.

The 68-95-99.7 rule, also known as the empirical rule, states that for a normal distribution, 68% of the data fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. This rule provides a quick estimate of the probability of a certain event within the distribution.

In a skewed distribution, the mean tends to get pulled in the direction of the skew. Since the mean involves every value in the distribution, extreme values (values far from the others) have a big influence. This results in skewness where the mean is drawn towards the tail, and is a common occurrence in distributions that are not symmetric.

The cumulative distribution function (CDF) of a random variable is the probability that the variable takes a value less than or equal to a certain value. The CDF of a function increases monotonically, and its limit is one as it approaches positive infinity.

The probability of an impossible event is 0. In the probability scale, 0 denotes impossibility, while 1 denotes certainty. An event with a probability of 0 is said to be impossible because it cannot happen.

For Pearson's Correlation Coefficient to be valid and reliable, the following assumptions should be met: both variables should be continuous, they should be linearly related, and both variables should be approximately normally distributed. Independence of observations is also required.