# Chapter 9 : Correlation and Regression

### Topics covered in this snack-sized chapter:

#### Correlation arrow_upward

• A correlation is a single number that describes the degree of relationship between two variables.
• It is often measured with Pearson’s Correlation Coefficient and it is represented with the letter r.

• #### Pearson’s Correlation Coefficient Formula arrow_upward • Where, n equals the number of values.
• In this formula:

• Multiply by and sum Sum all the scores Sum all the scores Square all and sum Square all and sum Square the sum of  Square the sum of #### Types of Correlation arrow_upward

• The different types of correlation are:
• Positive Correlation
• Negative Correlation
• Linear Correlation
• Non Linear Correlation
• Simple Correlation
• Multiple Correlation
• Partial Correlation

#### Positive Correlation arrow_upward

• When the values of two variables x and y move in the same direction, the correlation is said to be positive.
• That is in positive correlation, when there is an increase in x, there will be an increase in y also. Similarly when there is a decrease in x, there will be a decrease in y also.

• #### Negative Correlation arrow_upward

• When the values of two variables x and y move in opposite direction, we say correlation is negative.
• That is in negative correlation, when there is an increase in x, there will be a decrease in y. Similarly when there is a decrease in x, there will be an increase in y.

• #### Linear Correlation arrow_upward

• When the change in one variable results in the constant change in the other variable, we say the correlation is linear.
• When there is a linear correlation, the points plotted will be in a straight line.

• #### Non-linear Correlation arrow_upward

• When the amount of change in one variable is not in a constant ratio to the change in the other variable, we say that the correlation is non-linear.

• #### Simple Correlation arrow_upward

• If there are only two variable under study, the correlation is said to be simple.

• #### Multiple Correlations arrow_upward

• When one variable is related to a number of other variables, the correlation is not simple. It is multiple if there is one variable on one side and a set of variables on the other side.
• ##### Example:
• Relationship between yield with both rainfall and fertilizer together is multiple correlations.

• #### Partial Correlation arrow_upward

• The correlation is partial if we study the relationship between two variables keeping all other variables constant.
• ##### Example:
• The relationship between yield and rainfall at a constant temperature is partial correlation.

• #### Regression arrow_upward

• A statistical measure that attempts to determine the strength of the relationship between one dependent variable (usually denoted by Y) and a series of other changing variables (known as independent variables).

• #### Types of Regression arrow_upward

###### Linear Regression:

• It analyzes the relationship between two variables: x and y.
• y = a + bx ###### Exponential Regression:

• It takes the input signal and fits an exponential function to it • where, t is the variable along the x-axis.
• • A quadratic function is a function f (x) of the form for ﬁxed constants a, b, and c.
• #### Simple Linear Regression arrow_upward

• Simple Linear Regression of one dependent variable (Y) and one independent variable (X).
• The model is:
• •     Where,

y = Values of the dependent variable

x = Values of the independent variable

a, b = “Regression coefficients” (what we want to find) = Residual or error

#### Coefficient of Determination arrow_upward

• The Coefficient of Determination is the proportion of the total variation in the dependent variable that is explained or accounted for by the variation in the independent variable • It is the square of the coefficient of correlation .
• It ranges from 0 to 1.
• It does not give any information about the direction of the relationship between the variables.
•  #### Regression: How good is the Fit? arrow_upward

• A line of best fit is a straight line that best represents the data on a scatter plot.
• We measure the fit with the coefficient of determination, r2
• r2 is the proportion of variation in Y explained by the regression.
• Values range from 0 to 1
• 0 indicates no relationship, 1 indicates perfect relationship.
• Find individual residuals or errors
• • Then, the sum of all the residual is
•  Observed value of the dependent variable for the ith observation. Estimated value of the dependent variable for the ith observation.

#### Slope for the Estimated Regression Equation arrow_upward

• The slope of the regression line is calculated by this formula:
• Where,

x = Value of independent variables

y = Value of dependent variables

#### Y- Intercept for the Estimated Regression Equation arrow_upward Where, = Mean value for independent variable = Mean value for dependent variable

#### Good Regression Fit arrow_upward

• Most of the points lie on the line:
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