he298c

2021-02-01

What is the correlation coefficient and what is its significance? Explain why correlation coefficient is bound between -1 and 1. How correlation coefficient depicts itself in scatterplots? Aside from probabilistic models, explain what is least squares line fitting.

Demi-Leigh Barrera

First, do a correlation analysis to determine how well two sets of data "go along." The symbol for it is r. The correlation coefficient's value ranges from -1 to +1. The two data sets are perfect and both point in the same direction, as indicated by the positive 1. The two data sets are perfect and point in opposing directions, as indicated by the negative 1. When there is no correlation between the two data sets, it will be 0.
R, the correlation coefficient
The Pearson's correlation coefficient, also known as the Karl Pearson's product-moment correlation coefficient or just the correlation coefficient, is a symbol that indicates how strongly two variables are correlated linearly. ${r}_{xy}$
The coefficient of correlation ${r}_{xy}$ between two variables x and y for the bivariate data set  is given below:
${r}_{xy}=\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left[n\left(\sum {x}^{2}\right)-\left(\sum {x}^{2}\right)\right]×\left[n\left(\sum {y}^{2}\right)-\left(\sum {y}^{2}\right)\right]}}$
Step 2

Scatterplot and correlation: A scatterplot is a sort of data visualization that demonstrates the association between two numerical variables. The coordinates of each individual member of the data set are shown as a point whose (x, y) values correspond to those of the two variables.
It may be claimed that there is a positive correlation between the variables when the y variable tends to rise when the x variable rises. In other words, there is a positive correlation between the variables when the scatterplot's dots form a lower left to higher right pattern.
It may be claimed that there is a negative correlation between the variables when the y variable tends to decline as the x variable rises. In other words, there is a positive correlation between the variables when the scatterplot's dots form an upper left to lower right pattern.
The two variables are considered to have a perfect correlation when every point on a scatterplot is on a straight line.
A zero correlation or near-zero correlation scatterplot is one where the dots do not exhibit a linear trend (either positive or negative).
Type of the relationship between the variables:
Whether the data points follow a linear pattern or some other complex curves depends on the association's nature. If it seems that a line would adequately summarize the general trend in the data, then do so. The relationship between the two variables is thus linear.
The direction of association is positive if a rise in one variable's values causes an increase in the values of another variable. The direction is negative if a rise in one variable results in a reduction in another variable's value.
Strength of the relationship
If all the points are somewhat near to the straight line, the relationship is considered to be strong. If all of the data points are far from the straight line, it is considered to be weak, and if some of the data points are somewhat near to the straight line, it is said to be moderate.
Less squares line fitting, third step:
The link between variables is estimated through regression analysis. The link between one dependent variable and one or more independent variables is thus estimated.
First-order regression model's standard form is $y-\cap ={\beta }_{0}+{\beta }_{1}x+ϵ$, Where variable x serves as the independent variable that is utilized to predict the dependent variable, variable y serves as the dependent variable to be modelled or forecasted, and variable serves as the error term.
The term "residual" refers to the variation between the observed and expected values of y. Consequently, the residual value is shown as $y–\left(y-\cap \right)$
The straight line satisfies the least squares property if the sum of the squares of the residuals is given as the smallest sum feasible. The least-squares property, which "best fits the points in a scatterplot," is satisfied by the regression line of the straight line.

Do you have a similar question?