# Correlation and Linear Regression

## Correlation

Correlation is a statistical measure that quantifies the relationship between two or more variables. It is used to understand how changes in one variable are related to changes in another variable. Note that correlation does not imply causation, meaning that even if two variables are correlated, it does not necessarily mean that changes in one variable cause changes in the other.

## Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable (also known as the response variable) and independent variable(s). In linear regression, a linear equation is assumed between the dependent and independent variable(s).

For instance, if we have a dependent variable, y, and an independent variable, x, then a linear regression assumes a mathematical model of  y = a + bx, where a and b are some constants to be determined.

## Least Square Regression

Least squares regression is a statistical method used for fitting a linear model to a given set of data points. The goal of least squares regression is to find the best-fitting line that minimizes the sum of the squared vertical distances (residuals) between the observed data points and the predicted values from the linear model.

## Product Moment Correlation Coefficient ( r )

The product moment correlation coefficient, r, quantifies the strength and direction of the linear relationship between the dependent variable and the independent variable in a linear regression model. The correlation coefficient ranges from -1 to 1.

• When r = 1,  there is a perfect positive correlation. The two variables increase together in a perfect linear relationship.
• When r = -1, there is perfect negative correlation. There is a perfect linear relationship, and when the independent variable increases, the dependent variable decreases.
• When r = 0, there is no correlation, indicating that there is no linear relationship between the dependent and independent variables.

The closer |r| is to 1, the better the linear relationship between the dependent and independent variable.

## Plot a Scatter Plot using the graphic calculator Ti84

The graphic calculator Ti84 can be used to plot a scatter plot. This scatter plot allows us to determine the relationship between the independent variable and the dependent variable. For instance, we can tell whether the relationship is a linear one or not from such plots.

These are the steps to plot a scatter plot using the graphic calculator:

• Press [Stats]
• Select Edit…
• You’ll be shown a screen with columns of  L1, L2, etc.
• Type the independent variable into one column e.g.  L1, and the dependent variable into another column e.g. L2
• Go to Stat Plot by pressing [2nd][y=]
• Set the plot to ON, XList as L1 (or the list where you input the independent variable), and YList as L2 (or the list where you input the dependent variable)
• Press [Zoom] and select the option ZoomStat to obtain the scatter plot.

## Using the graphic calculator Ti84 to do a least square regression

• Input the values of the dependent and independent variables into the calculator by:
• Press [Stats]
• Select Edit…
• You’ll be shown a screen with columns of  L1, L2, etc.
• Type the independent variable into one column e.g.  L1, and the dependent variable into another column e.g. L2
• Under [Stats] go to Calc
• Search for LinReg(a+bx) and press enter
• Input the data:
• XList: refers to the independent variable (e.g. L1)
• Ylist: refers to the dependent variable (e.g. L2)
• Press the down arrow till Calculate and press enter

⇒ Values of a, b, r and r² will be given

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