Coefficient of Determination Formula

Coefficient of Determination Formula (Table of Contents)

• Formula
• Examples

What is the Coefficient of Determination Formula?

In statistics, coefficient of determination, also termed as R² is a tool which determines and assesses the ability of a statistical model to explain and predict future outcomes. In other words, if we have dependent variable y and independent variable x in a model, then R² helps in determining the variation in y by variation x. It is one of the key output of regression analysis and is used when we want to predict future or testing some models with related information. The value of R² lies between 0 and 1 and higher the value of R², better will be the prediction and strength of the model. R² is very similar to the correlation coefficient since the correlation coefficient measures the direct association of two variables. R² is basically a square of a correlation coefficient.

Formula For Coefficient of Determination:

There are multiple Formulas to calculate the coefficient of determination:

1. Using Correlation Coefficient :

Where:

• X – Data points in Data set X
• Y – Data points in Data set Y
• X_m – Mean of Data set X
• Y_m – Mean of Data set Y

So

2. Using Regression outputs

Coefficient of Determination (R²) = Explained Variation / Total Variation

Coefficient of Determination (R²) = MSS / TSS

Where:

• TSS – Total Sum of Squares = Σ (Yi – Ym)²
• MSS – Model Sum of Squares = Σ (Y^ – Ym)²
• RSS – Residual Sum of Squares =Σ (Yi – Y^)²

Y^ is the predicted value of the model, Yi is the ith value and Ym is the mean value

Examples of Coefficient of Determination Formula (With Excel Template)

Let’s take an example to understand the calculation of the Coefficient of Determination in a better manner.

Coefficient of Determination Formula – Example #1

Let’s say we have two data sets X & Y and each contains 20 random data points. Calculate the Coefficient of Determination for the data set X & Y.

Mean is calculated as:

• Mean of Data Set X = 48.7
• Mean of Data Set Y = 42.1

Now, we need to calculate the difference between the data points and the mean value.

Similarly, calculate for all the data set of X.

Similarly, calculate it for data set Y also.

Calculate the square of the difference for both the data sets X and Y.

Multiply the difference in X with Y.

Correlation Coefficient is calculated using the formula given below

Correlation Coefficient = Σ [(X – X_m) * (Y – Y_m)] / √ [Σ (X – X_m)² * Σ (Y – Y_m)²]

Coefficient of Determination is calculated using the formula given below

Coefficient of Determination = (Correlation Coefficient)²

Coefficient of Determination = 13.69%

Coefficient of Determination Formula – Example #2

Let say you are a very risk-averse investor and you looking to invest money in the stock market. You are not sure which stocks to invest in and also your risk appetite is low. So you want to invest in a stock which is safe and can mimic the performance of the index. Your friend, who is an active investor, has shortlisted 3 stocks for you, based on their fundamental and technical information and you want to choose 2 stocks among those three.

You have also collected information about their historical returns for the last 15 years.

Correlation Coefficient is calculated using the excel formula

Coefficient of Determination is calculated using the formula given below

Coefficient of Determination = (Correlation Coefficient)²

Based on the information, you will choose stock ABC and XYZ to invest since they have the highest coefficient of determination.

Explanation

Coefficient of determination, as explained above is the square of the correlation between two data sets. If R² is 0, it means that there is no correlation and independent variable cannot predict the value of the dependent variable. Similarly, if its value is 1, it means that independent variable will always be successful in predicting the dependent variable. But there are some limitations also. Although it tells us the correlation between 2 data sets, it does not tell us whether that value is enough or not.

Also, large value R² does not always imply that the 2 variables have strong relationships and it can be a fluke. For example: Let’s say R² value between a number of cars sold in a year and the number of ice cream boxes sold in a year is 80%. But there is no relation between these two. So one should be very careful while using R² and understand the data first and then apply the method

Relevance and Uses of Coefficient of Determination Formula

There are many practical applications of R². For example, R2 is very commonly used by investors to compare the performance of their portfolio with the market and try to predict future directions also. Similarly, Hedge Funds use R² helps them to model the risk in their models. But ultimately the outcome is based on pure numbers and statistics which can be misleading sometimes. As mentioned above, one needs to check first if the output of the R² makes sense in real life or not.

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