Linear regression analysis in Excel

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

 Linear regression in Excel with Analysis ToolPak
 Draw a linear regression graph
 Regression analysis in Excel with formulas

Regression analysis in Excel - the basics

In 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 equation

Mathematically, a linear regression is defined by this equation:

y = bx + a + ε

Where:

 x is an independent variable.
 y is a dependent variable.
 a is the Y-intercept, which is the expected mean value of y when all x variables are equal to 0. On a regression graph, it's the point
where the line crosses the Y axis.
 b is the slope of a regression line, which is the rate of change for y as x changes.
 ε is the random error term, which is the difference between the actual value of a dependent variable and its predicted value.

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:

Umbrellas sold = b * rainfall + a

There exist a handful of different ways to find a and b. The three main methods to perform linear regression analysis in Excel are:

 Regression tool included with Analysis ToolPak

 Scatter chart with a trendline
 Linear regression formula

Below you will find the detailed instructions on using each method.

How to do linear regression in Excel with Analysis

ToolPak
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-in

Analysis 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:

1. In your Excel, click File > Options.

2. In the Excel Options dialog box, select Add-ins on the left sidebar, make sure Excel Add-ins is selected in the Manage box, and
click Go.
3.
4. In the Add-ins dialog box, tick off Analysis Toolpak, and click OK:

This will add the Data Analysis tools to the Data tab of your Excel ribbon.

Run regression analysis

In 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:

1. On the Data tab, in the Analysis group, click the Data Analysis button.

2. Select Regression and click OK.

3. In the Regression dialog box, configure the following settings:

o Select the Input Y Range, which is your dependent variable. In our case, it's umbrella sales (C1:C25).
o Select the Input X Range, i.e. your independent variable. In this example, it's the average monthly rainfall (B1:B25).

If you are building a multiple regression model, select two or more adjacent columns with different independent variables.

o Check the Labels box if there are headers at the top of your X and Y ranges.
o Choose your preferred Output option, a new worksheet in our case.
o Optionally, select the Residuals checkbox to get the difference between the predicted and actual values.
o

4. Click OK and observe the regression analysis output created by Excel.

Interpret regression analysis output

As 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 Output

 This 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:

 1 means a strong positive relationship

 -1 means a strong negative relationship
 0 means no relationship at all

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 R 2 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: ANOVA

The 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:

 df is the number of the degrees of freedom associated with the sources of variance.
 SS is the sum of squares. The smaller the Residual SS compared with the Total SS, the better your model fits the data.
 MS is the mean square.
 F is the F statistic, or F-test for the null hypothesis. It is used to test the overall significance of the model.
 Significance F is the P-value of F.

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: coefficients

This 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:

Y = Rainfall Coefficient * x + Intercept

Equipped with a and b values rounded to three decimal places, it turns into:

Y=0.45*x-19.074

For example, with the average monthly rainfall equal to 82 mm, the umbrella sales would be approximately 17.8:

0.45*82-19.074=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: residuals

If 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:

 Estimated: 17.8 (calculated above)

 Actual: 15 (row 2 of the source data)

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 Excel

If you need to quickly visualize the relationship between the two variables, draw a linear regression chart. That's very easy! Here's how:

1. Select the two columns with your data, including headers.

2. On the Inset tab, in the Chats group, click the Scatter chart icon, and select the Scatter thumbnail (the first one):

This will insert a scatter plot in your worksheet, which will resemble this one:
3. Now, we need to draw the least squares regression line. To have it done, right click on any point and choose Add
Trendline… from the context menu.
4. On the right pane, select the Linear trendline shape and, optionally, check Display Equation on Chart to get your regression
formula:

As you may notice, the regression equation Excel has created for us is the same as the linear regression formula we built based on
the Coefficients output.
5. Switch to the Fill & Line tab and customize the line to your liking. For example, you can choose a different line color and use a solid
line instead of a dashed line (select Solid line in the Dash
 graph:

Still, you may want to make a few more improvements:

 Drag the equation wherever you see fit.

 Add axes titles (Chart Elements button > Axis Titles).
 If your data points start in the middle of the horizontal and/or vertical axis like in this example, you may want to get rid of the excessive
white space. The following tip explains how to do this: Scale the chart axes to reduce white space.

And this is how our improved regression graph looks like:

How to do regression in Excel using formulas
Microsoft 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:

=LINEST(C2:C25, B2:B25)

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:

y = bx + a

If you avoid using array formulas in your worksheets, you can calculate a and b individually with regular formulas:

Get the Y-intercept (a):

=INTERCEPT(C2:C25, B2:B25)

Get the slope (b):

=SLOPE(C2:C25, B2:B25)

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:

=CORREL(B2:B25,C2:C25)
The following screenshot shows all these Excel regression formulas in action: