The tutorial explains the basics of regression analysis and shows a few different ways to do linear regression in Excel. Show
Imagine this: you are provided with a whole lot of different data and are asked to predict next year's sales numbers for your company. You have discovered dozens, perhaps even hundreds, of factors that can possibly affect the numbers. But how do you know which ones are really important? Run regression analysis in Excel. It will give you an answer to this and many more questions: Which factors matter and which can be ignored? How closely are these factors related to each other? And how certain can you be about the predictions? Regression analysis in Excel - the basicsIn statistical modeling, regression analysis is used to estimate the relationships between two or more variables: Dependent variable (aka criterion variable) is the main factor you are trying to understand and predict. Independent variables (aka explanatory variables, or predictors) are the factors that might influence the dependent variable. Regression analysis helps you understand how the dependent variable changes when one of the independent variables varies and allows to mathematically determine which of those variables really has an impact. Technically, a regression analysis model is based on the sum of squares, which is a mathematical way to find the dispersion of data points. The goal of a model is to get the smallest possible sum of squares and draw a line that comes closest to the data. In statistics, they differentiate between a simple and multiple linear regression. Simple linear regression models the relationship between a dependent variable and one independent variables using a linear function. If you use two or more explanatory variables to predict the dependent variable, you deal with multiple linear regression. If the dependent variable is modeled as a non-linear function because the data relationships do not follow a straight line, use nonlinear regression instead. The focus of this tutorial will be on a simple linear regression. As an example, let's take sales numbers for umbrellas for the last 24 months and find out the average monthly rainfall for the same period. Plot this information on a chart, and the regression line will demonstrate the relationship between the independent variable (rainfall) and dependent variable (umbrella sales): Linear regression equationMathematically, a linear regression is defined by this equation: y = bx + a + ε Where:
The linear regression equation always has an error term because, in real life, predictors are never perfectly precise. However, some programs, including Excel, do the error term calculation behind the scenes. So, in Excel, you do linear regression using the least squares method and seek coefficients a and b such that: y = bx + a For our example, the linear regression equation takes the following shape:
There exist a handful of different ways to find a and b. The three main methods to perform linear regression analysis in Excel are:
Below you will find the detailed instructions on using each method. This example shows how to run regression in Excel by using a special tool included with the Analysis ToolPak add-in. Enable the Analysis ToolPak add-inAnalysis ToolPak is available in all versions of Excel 365 to 2003 but is not enabled by default. So, you need to turn it on manually. Here's how:
This will add the Data Analysis tools to the Data tab of your Excel ribbon. Run regression analysisIn this example, we are going to do a simple linear regression in Excel. What we have is a list of average monthly rainfall for the last 24 months in column B, which is our independent variable (predictor), and the number of umbrellas sold in column C, which is the dependent variable. Of course, there are many other factors that can affect sales, but for now we focus only on these two variables: With Analysis Toolpak added enabled, carry out these steps to perform regression analysis in Excel:
Interpret regression analysis outputAs you have just seen, running regression in Excel is easy because all calculations are preformed automatically. The interpretation of the results is a bit trickier because you need to know what is behind each number. Below you will find a breakdown of 4 major parts of the regression analysis output. Regression analysis output: Summary OutputThis part tells you how well the calculated linear regression equation fits your source data. Here's what each piece of information means: Multiple R. It is the Correlation Coefficient that measures the strength of a linear relationship between two variables. The correlation coefficient can be any value between -1 and 1, and its absolute value indicates the relationship strength. The larger the absolute value, the stronger the relationship:
R Square. It is the Coefficient of Determination, which is used as an indicator of the goodness of fit. It shows how many points fall on the regression line. The R2 value is calculated from the total sum of squares, more precisely, it is the sum of the squared deviations of the original data from the mean. In our example, R2 is 0.91 (rounded to 2 digits), which is fairy good. It means that 91% of our values fit the regression analysis model. In other words, 91% of the dependent variables (y-values) are explained by the independent variables (x-values). Generally, R Squared of 95% or more is considered a good fit. Adjusted R Square. It is the R square adjusted for the number of independent variable in the model. You will want to use this value instead of R square for multiple regression analysis. Standard Error. It is another goodness-of-fit measure that shows the precision of your regression analysis - the smaller the number, the more certain you can be about your regression equation. While R2 represents the percentage of the dependent variables variance that is explained by the model, Standard Error is an absolute measure that shows the average distance that the data points fall from the regression line. Observations. It is simply the number of observations in your model. Regression analysis output: ANOVAThe second part of the output is Analysis of Variance (ANOVA): Basically, it splits the sum of squares into individual components that give information about the levels of variability within your regression model:
The ANOVA part is rarely used for a simple linear regression analysis in Excel, but you should definitely have a close look at the last component. The Significance F value gives an idea of how reliable (statistically significant) your results are. If Significance F is less than 0.05 (5%), your model is OK. If it is greater than 0.05, you'd probably better choose another independent variable. Regression analysis output: coefficientsThis section provides specific information about the components of your analysis: The most useful component in this section is Coefficients. It enables you to build a linear regression equation in Excel: y = bx + a For our data set, where y is the number of umbrellas sold and x is an average monthly rainfall, our linear regression formula goes as follows:
Equipped with a and b values rounded to three decimal places, it turns into:
For example, with the average monthly rainfall equal to 82 mm, the umbrella sales would be approximately 17.8:
In a similar manner, you can find out how many umbrellas are going to be sold with any other monthly rainfall (x variable) you specify. Regression analysis output: residualsIf you compare the estimated and actual number of sold umbrellas corresponding to the monthly rainfall of 82 mm, you will see that these numbers are slightly different:
Why's the difference? Because independent variables are never perfect predictors of the dependent variables. And the residuals can help you understand how far away the actual values are from the predicted values: For the first data point (rainfall of 82 mm), the residual is approximately -2.8. So, we add this number to the predicted value, and get the actual value: 17.8 - 2.8 = 15. How to make a linear regression graph in ExcelIf you need to quickly visualize the relationship between the two variables, draw a linear regression chart. That's very easy! Here's how:
At this point, your chart already looks like a decent regression graph: Still, you may want to make a few more improvements:
How to do regression in Excel using formulasMicrosoft Excel has a few statistical functions that can help you to do linear regression analysis such as LINEST, SLOPE, INTERCEPT, and CORREL. The LINEST function uses the least squares regression method to calculate a straight line that best explains the relationship between your variables and returns an array describing that line. You can find the detailed explanation of the function's syntax in this tutorial. For now, let's just make a formula for our sample dataset:
Because the LINEST function returns an array of values, you must enter it as an array formula. Select two adjacent cells in the same row, E2:F2 in our case, type the formula, and press Ctrl + Shift + Enter to complete it. The formula returns the b coefficient (E1) and the a constant (F1) for the already familiar linear regression equation:
If you avoid using array formulas in your worksheets, you can calculate a and b individually with regular formulas: Get the Y-intercept (a):
Get the slope (b):
Additionally, you can find the correlation coefficient (Multiple R in the regression analysis summary output) that indicates how strongly the two variables are related to each other:
The following screenshot shows all these Excel regression formulas in action: Tip. If you'd like to get additional statistics for your regression analysis, use the LINEST function with the stats parameter set to TRUE as shown in this example. That's how you do linear regression in Excel. That said, please keep in mind that Microsoft Excel is not a statistical program. If you need to perform regression analysis at the professional level, you may want to use targeted software such as XLSTAT, RegressIt, etc. To have a closer look at our linear regression formulas and other techniques discussed in this tutorial, you are welcome to download our sample workbook below. Thank you for reading! Practice workbookRegression Analysis in Excel - examples (.xlsx file) You may also be interested inHow many types of reports are listed in the Solver Results dialog box?To record Solver's results and settings you can create reports. There are three types of reports: Answer, Sensitivity and Limits. Answer reports the original and final values for the target, as well as the settings.
Which of the following Solver reports identifies and lists each parameter?The answer report identifies and lists each parameter. It includes the original value and the value suggested by solver.
Which of the following analysis tools solves a problem in reverse?The Solver addin is an analysis tool that solves a problem in reverse. Solver determines the highest, the lowest, or a specific result for a formula by adjusting values in cells used in the formula. A Solver problem has three components known as parameters, identified in the Solver Parameters dialog box.
Which of the following add ins performs statistical and engineering functions?The add-in we are discussing today is the Analysis ToolPak which performs complex statistical and engineering analyses.
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