# In statistics what is R-squared?

Updated: Mar 7

I came across an article the other day talking about R-Squared. Since I haven't been in a stats class in a long time I had no clue what R-squared was, so I kept reading and barely understood anything that was being said. Because of this I dug deeper and wanted to write up an article explaining R-Squared and some practical applications for personal finance.

## In statistics what is R-squared?

R-squared, also known as the coefficient of determination, is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a linear regression model. In other words, it is a measure of how well the data points fit a linear regression line.

R-squared is a value between 0 and 1, with values closer to 1 indicating a stronger linear relationship between the independent and dependent variables. A value of 0 means that the model explains none of the variability of the response data around its mean, while a value of 1 means that the model explains all the variability of the response data around its mean.

R-squared is used to evaluate the goodness of fit of a linear regression model. A higher R-squared value indicates a better fit and a stronger linear relationship between the variables. However, it is important to note that a high R-squared value does not necessarily mean that the model is a good fit for the data, as other factors such as outliers, non-linearity, and model assumptions should also be considered.

It is also important to note that R-squared is not a measure of prediction accuracy and it should not be used as a sole criterion for model selection. It can be affected by the number of predictor variables in the model, so adjusted R-squared is often used instead which adjusts for the number of predictor variables in the model.

## R-Squared in more basic terms

R-squared is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. It ranges from 0 to 1, where 0 means no variance is explained and 1 means all variance is explained. So, it's a measure of how well the model fits the data.

## How about a real-world example

One example of a real-world scenario with a high R-squared value is the relationship between a person's income and their level of education. In this case, the dependent variable would be income and the independent variable would be the level of education. There is a strong positive relationship between income and education, with an R-squared value close to 1. This indicates that the level of education explains a large proportion of the variation in a person's income.

## How can this knowledge be applied to my personal finances

R-squared can be applied to personal finances in a number of ways. One way is to use it to analyze the performance of a portfolio of investments. For example, if you want to see how well a particular stock or mutual fund is performing compared to a benchmark index, you can calculate the R-squared value between the stock or fund and the index. A high R-squared value between the stock or fund and the index would indicate that the stock or fund's performance is closely tied to the performance of the index, while a low R-squared value would indicate that the stock or fund's performance is not closely tied to the performance of the index.

Another way R-squared can be applied to personal finance is in understanding the relationship between saving and expenditure. For example, a person could use the R-squared value to understand how much of their saving is explained by the amount of money they spend. A high R-squared value would indicate that a large proportion of their saving is explained by their expenditure, while a low R-squared value would indicate that their saving is not closely tied to their expenditure. Put another way, if you are still saving the same amount of money each month and the amount of money you spend fluctuates up and down a lot then the R-squared will be low. If your saving rate fluctuates a lot as you spend more money then your R-squared will be closer to 1.

In both cases, it can be used as a tool to help make more informed decisions about managing personal finance.

As a statistician, you may have come across the term "R-squared" or "R²" in your work. R-squared is a statistical measure that represents the proportion of variation in a dependent variable that is explained by an independent variable(s). It is commonly used in regression analysis to evaluate the goodness of fit of a model. In this article, we will provide a comprehensive guide to understanding R-squared, including what it means, how it is calculated, and its importance in statistical analysis.

## What does R-squared mean?

R-squared is a measure of the goodness of fit of a regression model. It is a value between 0 and 1, where 0 indicates that the model explains none of the variability of the response data around its mean, while 1 indicates that the model explains all the variability of the response data around its mean.

In other words, the R-squared value indicates the proportion of the total variation in the response variable that is explained by the independent variable(s) in the regression model. A higher R-squared value implies that the model is a better fit for the data.

## How is R-squared calculated?

The calculation of R-squared involves comparing the variation in the dependent variable explained by the regression model to the total variation in the dependent variable. The formula for calculating R-squared is as follows:

R-squared = 1 - (SSres / SStot)

where SSres is the sum of squares of residuals (the differences between the predicted values and the actual values) and SStot is the total sum of squares (the differences between the actual values and the mean value of the dependent variable).

To calculate R-squared, we first calculate the sum of squares of residuals (SSres) by taking the sum of the squared differences between the predicted values and the actual values. Next, we calculate the total sum of squares (SStot) by taking the sum of the squared differences between the actual values and the mean value of the dependent variable. Finally, we subtract the ratio of SSres to SStot from 1 to obtain the R-squared value.

## Importance of R-squared in statistical analysis

R-squared is a commonly used measure of goodness of fit in regression analysis. It helps in evaluating the accuracy of the regression model and provides insight into the relationship between the independent and dependent variables. A higher R-squared value indicates a better fit of the model to the data, while a lower R-squared value indicates a poor fit of the model.

R-squared is useful in identifying the proportion of variability in the response variable that is explained by the independent variable(s). It helps in determining the significance of the independent variable(s) in the model and in identifying any outliers or influential points that may affect the model's performance.

## Conclusion

R-squared is a pretty technical statistical term that doesn't have a ton of practical application. In reality, I was just curious and wanted to write something summarizing R-squared in a really easy way to read with some real-world examples to make it easier to understand for you and for me.