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## Coefficient of determination

**Coefficient of determination**, in statistics, *R*^{2} (or *r*^{2}), a measure that assesses the ability of a model to predict or explain an outcome in the linear regression setting. More specifically, *R*^{2} indicates the proportion of the variance in the dependent variable (*Y*) that is predicted or explained by linear regression and the predictor variable (*X*, also known as the independent variable).

In general, a high *R*^{2} value indicates that the model is a good fit for the data, although interpretations of fit depend on the context of analysis. An *R*^{2} of 0.35, for example, indicates that 35 percent of the variation in the outcome has been explained just by predicting the outcome using the covariates included in the model. That percentage might be a very high portion of variation to predict in a field such as the social sciences; in other fields, such as the physical sciences, one would expect *R*^{2} to be much closer to 100 percent. The theoretical minimum *R*^{2} is 0. However, since linear regression is based on the best possible fit, *R*^{2} will always be greater than zero, even when the predictor and outcome variables bear no relationship to one another.

*R*^{2} increases when a new predictor variable is added to the model, even if the new predictor is not associated with the outcome. To account for that effect, the adjusted *R*^{2} (typically denoted with a bar over the *R* in *R*^{2}) incorporates the same information as the usual *R*^{2} but then also penalizes for the number of predictor variables included in the model. As a result, *R*^{2} increases as new predictors are added to a multiple linear regression model, but the adjusted *R*^{2} increases only if the increase in *R*^{2} is greater than one would expect from chance alone. In such a model, the adjusted *R*^{2} is the most realistic estimate of the proportion of the variation that is predicted by the covariates included in the model.

When only one predictor is included in the model, the coefficient of determination is mathematically related to the Pearson’s correlation coefficient, *r*. Squaring the correlation coefficient results in the value of the coefficient of determination. The coefficient of determination can also be found with the following formula: *R*^{2} = *MSS*/*TSS* = (*TSS* − *RSS*)/*TSS*, where *MSS* is the model sum of squares (also known as *ESS*, or explained sum of squares), which is the sum of the squares of the prediction from the linear regression minus the mean for that variable; *TSS* is the total sum of squares associated with the outcome variable, which is the sum of the squares of the measurements minus their mean; and *RSS* is the residual sum of squares, which is the sum of the squares of the measurements minus the prediction from the linear regression.

The coefficient of determination shows only association. As with linear regression, it is impossible to use *R*^{2} to determine whether one variable causes the other. In addition, the coefficient of determination shows only the magnitude of the association, not whether that association is statistically significant.

## Coefficient of Determination (R Squared): Definition, Calculation

## Coefficient of Determination (R Squared)

The coefficient of determination, R^{2}, is used to analyze how differences in one variable can be explained by a difference in a second variable. For example, *when *a person gets pregnant has a direct relation to when they give birth.

More specifically, R-squared gives you the percentage variation in y explained by x-variables. The range is 0 to 1 (i.e. 0% to 100% of the variation in y can be explained by the x-variables).

The coefficient of determination, R^{2}, is similar to the **correlation coefficient**, R. The correlation coefficient formula will tell you how strong of a linear relationship there is between two variables. R Squared is the square of the correlation coefficient, *r* (hence the term r squared).

## Finding R Squared / The Coefficient of Determination

**Step 1: ***Find the correlation coefficient, r (it may be given to you in the question).* Example, r = **0.543**.

**Step 2: ***Square the correlation coefficient.*

0.543^{2} = **.295**

**Step 3: ***Convert the correlation coefficient to a percentage*.

.295 = **29.5%**

That’s it!

## Meaning of the Coefficient of Determination

The coefficient of determination can be thought of as a percent. It gives you an idea of how many data points fall within the results of the line formed by the regression equation. The higher the coefficient, the higher percentage of points the line passes through when the data points and line are plotted. If the coefficient is 0.80, then 80% of the points should fall within the regression line. Values of 1 or 0 would indicate the regression line represents all or none of the data, respectively. A higher coefficient is an indicator of a better goodness of fit for the observations.

The CoD can be **negative**, although this usually means that your model is a poor fit for your data. It can also become negative if you didn’t set an intercept.

## Usefulness of R^{2}

The usefulness of R^{2} is its ability to find the likelihood of future events falling within the predicted outcomes. The idea is that if more samples are added, the coefficient would show the probability of a new point falling on the line.

Even if there is a strong connection between the two variables, determination does not prove causality. For example, a study on birthdays may show a large number of birthdays happen within a time frame of one or two months. This does not mean that the passage of time or the change of seasons causes pregnancy.

## Syntax

The coefficient of determination is usually written as R^{2}_p. The “p” indicates the number of columns of data, which is useful when comparing the R^{2} of different data sets.

## What is the Adjusted Coefficient of Determination?

The Adjusted Coefficient of Determination (Adjusted R-squared) is an adjustment for the Coefficient of Determination that takes into account **the number of variables in a data set.** It also penalizes you for points that don’t fit the model.

You might be aware that few values in a data set (a too-small sample size) can lead to misleading statistics, but you may not be aware that **too many **data points can also lead to problems. Every time you add a data point in regression analysis, R^{2} will increase. **R ^{2} never decreases.** Therefore, the more points you add, the better the regression will seem to “fit” your data. If your data doesn’t quite fit a line, it can be tempting to keep on adding data until you have a better fit.

Some of the points you add will be significant (fit the model) and others will not. R^{2} doesn’t care about the insignificant points. **The more you add, the higher the coefficient of determination**.

The adjusted R^{2} can be used to include a **more appropriate** number of variables, thwarting your temptation to keep on adding variables to your data set. The adjusted R^{2} will increase only if a new data point improves the regression more than you would expect by chance. R^{2} doesn’t include all data points, is always lower than R^{2} and can be negative (although it’s usually positive). Negative values will likely happen if R^{2} is close to zero — after the adjustment, the value will dip below zero a little.

## What is the Coefficient of Determination Formula?

In statistics, coefficient of determination, also termed as R^{2} 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 R^{2 }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 R^{2} lies between 0 and 1 and higher the value of R^{2}, better will be the prediction and strength of the model. R^{2} is very similar to the correlation coefficient since the correlation coefficient measures the direct association of two variables. R^{2} is basically a square of a correlation coefficient.

Formula For Coefficient of Determination:

There are multiple Formulas to calculate the coefficient of determination:

**Using Correlation Coefficient :**

**Correlation Coefficient = Σ [(X – X _{m}) * (Y – Y_{m})] / √ [Σ (X – X_{m})^{2} * Σ (Y – Y_{m})^{2}]**

Where:

**X**– Data points in Data set X**Y**– Data points in Data set Y**X**– Mean of Data set X_{m }**Y**– Mean of Data set Y_{m }

So

**Coefficient of Determination(R ^{2}) = (Correlation Coefficient)^{2}**

**Using Regression outputs**

Coefficient of Determination (R^{2}) = Explained Variation / Total Variation

Coefficient of Determination (R^{2}) = MSS / TSS**Coefficient of Determination (R ^{2}) = (TSS – RSS) / 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.

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 R^{2} 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 R^{2} does not always imply that the 2 variables have strong relationships and it can be a fluke. For example: Let’s say R^{2} 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 R^{2} and understand the data first and then apply the method

### Relevance and Use of Coefficient of Determination Formula

There are many practical applications of R^{2}. 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 R^{2} 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 R^{2 }makes sense in real life or not.

## Coefficient of Determination Definition

The **coefficient of determination** or R squared method is the proportion of the variance in the dependent variable that is predicted from the independent variable. It indicates the level of variation in the given data set.

- The coefficient of determination is the square of the correlation(r), thus it ranges from 0 to 1.
- With linear regression, the coefficient of determination is equal to the square of the correlation between the x and y variables.
- If R
^{2}is equal to 0, then the dependent variable cannot be predicted from the independent variable. - If R
^{2}is equal to 1, then the dependent variable can be predicted from the independent variable without any error. - If R
^{2}is between 0 and 1, then it indicates the extent that the dependent variable can be predictable. If R^{2 }of 0.10 means, it is 10 percent of the variance in the y variable is predicted from the x variable. If 0.20 means, 20 percent of the variance in the y variable is predicted from the x variable, and so on.

The value of R^{2} shows whether the model would be a good fit for the given data set. In the context of analysis, for any given per cent of the variation, it(good fit) would be different. For instance, in a few fields like rocket science, R^{2} is expected to be nearer to 100 %. But R^{2} = 0(minimum theoretical value), which might not be true as R^{2 }is always greater than 0( by Linear Regression).

The value of R^{2} increases after adding a new variable predictor. Note that it might not be associated with the result or outcome. The R^{2 }which was adjusted will include the same information as the original one. The number of predictor variables in the model gets penalized. When in a multiple linear regression model, new predictors are added, it would increase R^{2}. Only an increase in R^{2} which is greater than the expected(chance alone), will increase the adjusted R^{2}.

** Try Out**: Coefficient of Determination Calculator

Following is the Regression line equation

**p’ = aq + r**

Where ‘p’ is the predicted function value of q. So, the method of checking how good the least-squares equation p̂ = aq + r will make a prediction of how p will be made.

## Coefficient of Determination Formula

We can give the formula to find the coefficient of determination in two ways; one using correlation coefficient and the other one with sum of squares.

**Formula 1:**

As we know the formula of correlation coefficient is,

Where

n = Total number of observations

Σx = Total of the First Variable Value

Σy = Total of the Second Variable Value

Σxy = Sum of the Product of first & Second Value

Σx^{2} = Sum of the Squares of the First Value

Σy^{2} = Sum of the Squares of the Second Value

Thus, the coefficient of of determination = (correlation coefficient)^{2} = r^{2}

**Formula 2:**

The formula of coefficient of determination is given by:

R^{2} = 1 – (RSS/TSS)

Where,

R^{2} = Coefficient of Determination

RSS = Residuals sum of squares

TSS = Total sum of squares

### Properties of Coefficient of Determination

- It helps to get the ratio of how a variable which can be predicted from the other one, varies.
- If we want to check how clear it is to make predictions from the data given, we can determine the same by this measurement.
- It helps to find Explained variation / Total Variation
- It also lets us know the strength of the association(linear) between the variables.
- If the value of r
^{2}gets close to 1, The values of y become close to the regression line and similarly if it goes close to 0, the values get away from the regression line. - It helps in determining the strength of association between different variables.

### Steps to Find the Coefficient of Determination

- Find r, Correlation Coefficient
- Square ‘r’.
- Change the above value to a percentage.

## Understanding the Coefficient of Determination

The coefficient of determination is a measurement used to explain how much variability of one factor can be caused by its relationship to another related factor. This correlation, known as the “goodness of fit,” is represented as a value between 0.0 and 1.0. A value of 1.0 indicates a perfect fit, and is thus a highly reliable model for future forecasts, while a value of 0.0 would indicate that the calculation fails to accurately model the data at all. But a value of 0.20, for example, suggests that 20% of the dependent variable is predicted by the independent variable, while a value of 0.50 suggests that 50% of the dependent variable is predicted by the independent variable, and so forth.

## Graphing the Coefficient of Determination

On a graph, the goodness of fit measures the distance between a fitted line and all of the data points that are scattered throughout the diagram. The tight set of data will have a regression line that’s close to the points and have a high level of fit, meaning that the distance between the line and the data is small. Although a good fit has an R^{2} close to 1.0, this number alone cannot determine whether the data points or predictions are biased. It also doesn’t tell analysts whether the coefficient of determination value is intrinsically good or bad. It is at the discretion of the user to evaluate the meaning of this correlation, and how it may be applied in the context of future trend analyses.

## The Coefficient of Determination

## Frequently Asked Questions – FAQs

### How is R^2 calculated?

The value of R^2 is calculated using the below formula.

R^2 = 1 – (RSS/TSS)

Here,

RSS = Residuals sum of squares

TSS = Total sum of squares

### How is the coefficient of determination calculated?

Using the correlation coefficient formula, the coefficient of determination can be calculated in three steps.

Step 1: Find r, the correlation coefficient

Step 2: Square the value of ‘r’

Step 3: Change the obtained value to a percentage

### What is a good coefficient of determination?

Generally, the coefficient of determination with about 70% is considered good. Also, we can say that 50% of this is considered a moderate fit for the given model.

### Is the coefficient of determination the same as R^2?

Yes, the coefficient of determination is denoted by R^2.

### What does R^2 tell us?

R^2 or R-squared is a statistical measure of how close the data are to the fitted regression line. It is also called the coefficient of determination.

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