The main difference between R and R^{2} is the following:

- The quantity R is the Karl Pearson coefficient of correlation and it measures the degree of correlation between two variables X and Y.
- The quantity R
^{2}is the coefficient of determination. It measures the degree to which the independent variable X predicts the value of the dependent variable Y in linear regression. - The value of R
^{2}is always less than the absolute value of R.

**Interpretation of the Coefficient of Determination R**^{2}:

^{2}:

Suppose that we have an independent variable X and a dependent variable Y. We can use the technique of linear regression to quantify the relationship between one or more predictor variables and a response variable.

For example, we can have that:

**Y = Crop Yield.**

**X = Amount of Fertilizer.**

The yield of crops in an agricultural farm clearly depends on the amount of fertilizer used when growing the crops. Hence, Y is the dependent (response) variable and X is the independent (predictor) variable.

We can fit a linear regression model in order to predict the values of the variable Y. The regression model is given by the equation:

**Y = β _{1}X + β_{0} + e.**

- Here,
**β**and_{0}**β**are the regression coefficients._{1} - e is the error term.

We can use the linear model above to predict the value of Y. **The coefficient of determination R ^{2} measures the amount of variability in the dependent variable Y that is explained by the independent variable X.**

For instance, suppose that we are given the following set of data values:

Amount of Fertilizer | Crop Yield |

1 | 3 |

2 | 6 |

5 | 8 |

4 | 11 |

11 | 9 |

8 | 4 |

15 | 19 |

13 | 22 |

We can calculate the regression coefficients and the coefficient of determination.

Regression Equation: ŷ = 1.03357*X* + 2.62739.

Coefficient of determination: R^{2} = 0.6118.

**This means that only 61.118% of the variation in crop yield (Y) is explained by the variation in the amount of fertilizer (X). **This suggests to us that there are other factors affecting crop yield. For instance, one such factor affecting crop yield might be the amount of water.

**Thus we see that R ^{2} can help us decide if the model is missing some independent variables. **

**It can be used as a measure of the strength of the model.** If the value of R^{2} is high it means that the model has good predictive value. It can be used to predict the values of Y with a high degree of accuracy. If the value of R^{2} is low it means that the model has low explanatory power.

**Understanding the Meaning of R:**

**The coefficient of correlation R on the other hand only measures the degree of correlation between the two variables. It does not measure the extent to which one variable can be used to predict the other.**

For example, for the above set of data values for the crop yield and the amount of fertilizer, we have,

Correlation Coefficient: R = 0.7822.

This means that there is a high degree of correlation between the two variables. Note that we cannot say that X explains 78.822% of the variation in Y. As we have seen above the actual proportion of variation of Y due to X is 61.118%.