Regression analyses in excel

R Square | Significance F and P-Values | Coefficients | Residuals

This example teaches you how to run a linear regression analysis in Excel and how to interpret the Summary Output.

Below you can find our data. The big question is: is there a relation between Quantity Sold (Output) and Price and Advertising (Input). In other words: can we predict Quantity Sold if we know Price and Advertising?

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

Note: can’t find the Data Analysis button? Click here to load the Analysis ToolPak add-in.

2. Select Regression and click OK.

3. Select the Y Range (A1:A8). This is the predictor variable (also called dependent variable).

4. Select the X Range(B1:C8). These are the explanatory variables (also called independent variables). These columns must be adjacent to each other.

5. Check Labels.

6. Click in the Output Range box and select cell A11.

7. Check Residuals.

8. Click OK.

Excel produces the following Summary Output (rounded to 3 decimal places).

R Square

R Square equals 0.962, which is a very good fit. 96% of the variation in Quantity Sold is explained by the independent variables Price and Advertising. The closer to 1, the better the regression line (read on) fits the data.

Significance F and P-values

To check if your results are reliable (statistically significant), look at Significance F (0.001). If this value is less than 0.05, you’re OK. If Significance F is greater than 0.05, it’s probably better to stop using this set of independent variables. Delete a variable with a high P-value (greater than 0.05) and rerun the regression until Significance F drops below 0.05.

Most or all P-values should be below below 0.05. In our example this is the case. (0.000, 0.001 and 0.005).

Coefficients

The regression line is: y = Quantity Sold = 8536.214 -835.722 * Price + 0.592 * Advertising. In other words, for each unit increase in price, Quantity Sold decreases with 835.722 units. For each unit increase in Advertising, Quantity Sold increases with 0.592 units. This is valuable information.

You can also use these coefficients to do a forecast. For example, if price equals $4 and Advertising equals $3000, you might be able to achieve a Quantity Sold of 8536.214 -835.722 * 4 + 0.592 * 3000 = 6970.

Residuals

The residuals show you how far away the actual data points are fom the predicted data points (using the equation). For example, the first data point equals 8500. Using the equation, the predicted data point equals 8536.214 -835.722 * 2 + 0.592 * 2800 = 8523.009, giving a residual of 8500 — 8523.009 = -23.009.

You can also create a scatter plot of these residuals.

Регрессионный анализ является одним из самых востребованных методов статистического исследования. С его помощью можно установить степень влияния независимых величин на зависимую переменную. В функционале Microsoft Excel имеются инструменты, предназначенные для проведения подобного вида анализа. Давайте разберем, что они собой представляют и как ими пользоваться.

Подключение пакета анализа

Но, для того, чтобы использовать функцию, позволяющую провести регрессионный анализ, прежде всего, нужно активировать Пакет анализа. Только тогда необходимые для этой процедуры инструменты появятся на ленте Эксель.

1. Перемещаемся во вкладку «Файл».



2. Переходим в раздел «Параметры».



3. Открывается окно параметров Excel. Переходим в подраздел «Надстройки».



4. В самой нижней части открывшегося окна переставляем переключатель в блоке «Управление» в позицию «Надстройки Excel», если он находится в другом положении. Жмем на кнопку «Перейти».



5. Открывается окно доступных надстроек Эксель. Ставим галочку около пункта «Пакет анализа». Жмем на кнопку «OK».

Теперь, когда мы перейдем во вкладку «Данные», на ленте в блоке инструментов «Анализ» мы увидим новую кнопку – «Анализ данных».

Виды регрессионного анализа

Существует несколько видов регрессий:

• параболическая;
• степенная;
• логарифмическая;
• экспоненциальная;
• показательная;
• гиперболическая;
• линейная регрессия.

О выполнении последнего вида регрессионного анализа в Экселе мы подробнее поговорим далее.

Внизу, в качестве примера, представлена таблица, в которой указана среднесуточная температура воздуха на улице, и количество покупателей магазина за соответствующий рабочий день. Давайте выясним при помощи регрессионного анализа, как именно погодные условия в виде температуры воздуха могут повлиять на посещаемость торгового заведения.

Общее уравнение регрессии линейного вида выглядит следующим образом: У = а0 + а1х1 +…+акхк. В этой формуле Y означает переменную, влияние факторов на которую мы пытаемся изучить. В нашем случае, это количество покупателей. Значение x – это различные факторы, влияющие на переменную. Параметры a являются коэффициентами регрессии. То есть, именно они определяют значимость того или иного фактора. Индекс k обозначает общее количество этих самых факторов.

1. Кликаем по кнопке «Анализ данных». Она размещена во вкладке «Главная» в блоке инструментов «Анализ».





2. Открывается небольшое окошко. В нём выбираем пункт «Регрессия». Жмем на кнопку «OK».



3. Открывается окно настроек регрессии. В нём обязательными для заполнения полями являются «Входной интервал Y» и «Входной интервал X». Все остальные настройки можно оставить по умолчанию.

   В поле «Входной интервал Y» указываем адрес диапазона ячеек, где расположены переменные данные, влияние факторов на которые мы пытаемся установить. В нашем случае это будут ячейки столбца «Количество покупателей». Адрес можно вписать вручную с клавиатуры, а можно, просто выделить требуемый столбец. Последний вариант намного проще и удобнее.

   В поле «Входной интервал X» вводим адрес диапазона ячеек, где находятся данные того фактора, влияние которого на переменную мы хотим установить. Как говорилось выше, нам нужно установить влияние температуры на количество покупателей магазина, а поэтому вводим адрес ячеек в столбце «Температура». Это можно сделать теми же способами, что и в поле «Количество покупателей».

   

   С помощью других настроек можно установить метки, уровень надёжности, константу-ноль, отобразить график нормальной вероятности, и выполнить другие действия. Но, в большинстве случаев, эти настройки изменять не нужно. Единственное на что следует обратить внимание, так это на параметры вывода. По умолчанию вывод результатов анализа осуществляется на другом листе, но переставив переключатель, вы можете установить вывод в указанном диапазоне на том же листе, где расположена таблица с исходными данными, или в отдельной книге, то есть в новом файле.

   

   После того, как все настройки установлены, жмем на кнопку «OK».

Разбор результатов анализа

Результаты регрессионного анализа выводятся в виде таблицы в том месте, которое указано в настройках.

Одним из основных показателей является R-квадрат. В нем указывается качество модели. В нашем случае данный коэффициент равен 0,705 или около 70,5%. Это приемлемый уровень качества. Зависимость менее 0,5 является плохой.

Ещё один важный показатель расположен в ячейке на пересечении строки «Y-пересечение» и столбца «Коэффициенты». Тут указывается какое значение будет у Y, а в нашем случае, это количество покупателей, при всех остальных факторах равных нулю. В этой таблице данное значение равно 58,04.

Значение на пересечении граф «Переменная X1» и «Коэффициенты» показывает уровень зависимости Y от X. В нашем случае — это уровень зависимости количества клиентов магазина от температуры. Коэффициент 1,31 считается довольно высоким показателем влияния.

Как видим, с помощью программы Microsoft Excel довольно просто составить таблицу регрессионного анализа. Но, работать с полученными на выходе данными, и понимать их суть, сможет только подготовленный человек.

In Excel for the web, you can view the results of a regression analysis (in statistics, a way to predict and forecast trends), but you can’t create one because the Regression tool isn’t available.

You also won’t be able to use a statistical worksheet function such as LINEST to do a meaningful analysis because it requires you enter it as an array formula, which isn’t supported in Excel for the web.

If you have the Excel desktop application, you can use the Open in Excel button to open your workbook and use either the Analysis ToolPak’s Regression tool or statistical functions to perform a regression analysis there.

Click Open in Excel and perform a regression analysis.

For news about the latest Excel for the web updates, visit the Microsoft Excel blog.

For the full suite of Office applications and services, try or buy it at Office.com.

Regression is an Analysis Tool, which we use for analyzing large amounts of data and making forecasts and predictions in Microsoft Excel.

Want to predict the future? No, we are not going to learn astrology. We are into numbers and we will learn regression analysis in Excel today.

To predict future estimates, we will study:

• REGRESSION ANALYSIS USING EXCEL FUNCTIONS (MANUAL REGRESSION FINDING)

• REGRESSION ANALYSIS USING EXCEL’S ANALYSIS TOOLPAK ADD-IN

• REGRESSION CHART IN EXCEL

Let’s do it…

Scenario:

Let’s assume you sell soft drinks. How cool will it be if you can predict:

• How many soft drinks will be sold next year based on previous year’s data?
• Which fields need to be focused?
• And how can you increase your sales by changing your strategy?

It will be profitably awesome. Right?… I know. So let’s get started.

You have 11 records of salesmen and soft drinks sold.

Now based on this data you want to predict the number of salesmen required to achieve 2000 sales of soft drinks.

The regression equation is a tool to make such close estimates. To do so, we need to know Regression first.

REGRESSION ANALYSIS USING EXCEL FUNCTIONS (MANUAL REGRESSION FINDING)

This part will make you understand regression better than just telling excel regression procedure.

Introduction:

Simple Linear Regression: 

The study of the relationship between two variables is called Simple Linear Regression. Where one variable depends on the other independent variable. The dependent variable is often called by names such as Driven, Response, and Target variable. And the independent variable is often pronounced as a Driving, Predictor or simply Independent variable. These names clearly describe them.

Now let’s compare this with your scenario. You want to know the number of salesmen required to achieve 2000 sales. So here, the dependent variable is the number of salesmen and the independent variable is sold soft drinks.

The independent variable is mostly denoted as x and dependent variable as y.

In our case, soft drinks are sold x and the number of salesmen is y.

If we want to know how many soft drinks will be sold if we appoint 200 salesmen, then the scenario will be vice-versa.

Moving On.

The “Simple” Math of Linear Regression Equation:

Well, it’s not simple. But Excel made it simple to do.

We need to predict the required number of salesmen for all 11 cases to get the 12th closest prediction.

Let’s say:

Soft Drink Sold is x

The number of Salesmen is y

The predicted y (number of salesmen) also called Regression Equation,  would be

Now you must be wondering where the stat will you get the slope and intercept. Don’t worry, excel has functions for them. You do not need to learn how to find the slope and intercept it manually.

If you want, I will prepare a separate tutorial for that. Let me know in the comments section. These are some important data analytics tools.

Now let’s step into our calculation:

Step1: Prepare this small table

Step 2: Find the slope of the regression line

Excel Function for slopes is

=SLOPE(known_y’s,known_x’s)

Your known_y’s are in range B2:B12 and known_x’s are in range C2:C12

In cell B16, write the formula below

(Note: Slope is also called coefficient of x in the regression equation)

You will get 0.058409. Round up to 2 decimal digits and you will get 0.06.

Step 3: Find the Intercept of Regression Line

Excel function for the intercept is

=INTERCEPT(known_y’s, known_x’s)

We know what our known x’s and y’s

In cell B17, write down this formula

=INTERCEPT(B2:B12, C2:C12)

You will get a value of -1.1118969. Roundup to 2 decimal digits. You will get -1.11.

Our Linear Regression Equation is = x*0.06 + (-1.11). Now we can predict possible y depending on the target x easily.

Step 4: In D2 write the formula below

=C2*$B$16+$B$17    (Regression Equation)

You will get a value of 13.55.

Select D2 to D13 and press CTRL+D to fill down the formula in the range D2:D13

In cell D13 you have your required number of salesmen.

Hence, to achieve the target of 2000 Soft Drink Sales, you need an estimate of 115.71 salesmen or say 116 since it is illegal to cut humans into pieces.

Now using this you can easily conduct What-If analysis in excel. Just change the number of sales and it will show you many salesmen will it take to get that sales target achieved.

Play around it to find out:

How much workforce do you need to increase sales?

How many sales will increase if you increase your salesmen?

Make Your Estimate More Reliable:

Now you know that you need 116 salesmen to get 2000 sales done.

In analytics, nothing is just said and believed. You must give a percentage of reliability on your estimate. It is like giving a certificate of your equation.

Correlation Coefficient Formula:

The next thing you will be asked is how much these two variables are related. In static terms, you need to tell the coefficient of correlation.

Excel function for correlation is

In your case, known_x’s and Know_y’s are array1 and array2 irrespectively.

In B18 enter this formula

You will have 0.919090. Formate cell B2 into the percentage. Now have 92% of correlation.

Now, what this 92% means. It means, there 92% of chances of sales increase if you increase the number of salesmen and 92% of sales decrease if you decrease the number of salesmen. It is called Positive Correlation Coefficient.

R Squire (R^2) :

R Squire value tells you, by what percentage your regression equation is not a fluke. How much it is accurate by the data provided.

The Excel function for R squire is RSQ.

RSQ(known_y’s, Known_x’s)

In our case, we will get R squire value in cell B19.

In B19 enter this formula

So we have 84% of r Square value. Which is a very good explanation of our regression. It says that 84% of our data is just not by chance. Y (number of salesmen) is very much dependent on X (sales of soft drinks).

There are many other tests we can do on this data to ensure our regression. But manually it will be a complex and lengthy procedure. That is why excel provides Analysis Toolpak. Using this tool we can do this regression analysis in seconds.

REGRESSION IN EXCEL USING EXCEL’S ANALYSIS TOOLPAK ADD-IN

If you already know what regression equations are, and you just want your results quickly then this part is for you. But if you want to understand regression equations easily then scroll up to REGRESSION ANALYSIS USING EXCEL FUNCTIONS (MANUAL REGRESSION FINDING).

Excel provides a whole bunch of tools for analysis in its Analysis Toolpak. By default, it is not available in the Data tab. You need to add it. So let’s add it first.

Adding Analysis Toolpak to Excel 2016

If you don’t know where is data analysis in excel follow these steps

Step 1: Go to Excel Options: File? Options? Add-Ins

Step 2: Click on Add-Ins. You will see a list of available add-Ins.

Select Analysis ToolPak and at the bottom of the window, find manage. In manage select Excel Add-Ins and Click on GO.

Add-ins window will open. Here, select Analysis ToolPak. Then click the ok button.

Now you can access all functions of data analysis ToolPak from Data Tab.

Using Analysis ToolPak for Regression

Step 1: Go to the Data tab, Locate Data Analysis. Then click on it.

A dialogue box will pop up.

Step 2: Find ‘Regression’ in Analysis Tools list and hit the OK button.

The regression input window will pop up. You will see a number of available input options. But for now, we will just concentrate on Y Range and X Range, leaving everything else to default.

Step 4: Provide Inputs:

No. of Salesmen is Y

Sales of soft drinks are X

Hence

• Y Range= B2:B11

And 

•  X Range = C2:C11

For the output range, I have selected E4 on the same sheet. You may select a new worksheet to get results on a new worksheet in the same workbook or a complete new workbook. When you are done with your inputs, hit the OK button.

Results:

You will be served with a variety of information from your data. Don’t get overwhelmed. You don’t need to consume all the dishes.

We will only deal with those results which will help us to estimate the required number of salesmen

Step 5: We know the regression equation for estimation of y, that is

x*Slope+Intercept

We just need to locate Slope and Intercept in results.

And here they are.

The intercept Coefficient is clearly mentioned.

The slope is written as ‘X Variable 1’, some times also mentioned as the coefficient of X. Round up them and we will get -1.11 as Intercept and 0.06 as Slope.

Step 6: From results, we can drive the Regression equation. And that would be

=x*(0.06) + (-1.11)

Prepare this table in excel.

For now, x is 2000, which is in cell E2.

In Cell F2 enter this formula

=E2*F21+F20

You will get a result of 115.7052757.

Rounding it up will give us 116 of Required Salesmen.

So we have learned how to form the regression equation manually and using Analysis ToolPak. How can you use this equation to estimate future stats?

Now let’s understand the regression output given by Analysis Toolpak.

Understanding the Regression Output:

There is no benefit, if you do regression analysis using analysis tool pack in excel and can’t interpret its meaning.

Summary Section:

As the name suggests, it is a summary of the data.

1. 1. Multiple R: It tells how fit the regression equation is to the data. It is also called the correlation coefficient. 

In our case, it is 0.919090619 or 0.92 (roundup). This means that there is a 92% chance of an increase in sales if we increase our salesmen count.

1. 1. R Square: It tells the reliability of found regression. It tells us how many observations are part of our line of regression. In our case, it is 0.844727566 or 0.85. It means that our regression is fit by 85%.
   2. Adjusted R Square: Theadjusted square is just a more testified version of R square. Mainly useful in Multiple Regression Analysis.
   3. Standard Error: While R. Squire tells you how many data points fall near the regression line, the standard error tells you how far a data point can go from the regression line. 

In our case, it is 6.74.

1. Observation: This is simply the number of observations, which is 11 in our example.

Anova Section:

This section is hardly used in linear regression.

1. df. It is a degree of freedom. It is used when calculating regression manually.
2. SS. Sum of squares. It is just a sum of squares of variances. Used to find R squire values.
3. MS. This means squared value.
4. And 5. F and Significance of F. If the significance of F (p-value of the slope) is less than the F test than you can discard the null hypothesis and prove your hypothesis. In simple language, you can conclude that there is some effect of x on y when changed. 

In our case, F is 48.96264 and Significance of F is 0.000063. It means our regression fits the data.

Regression Section:

In this section, we have the two most important values for our regression equation.

1. Intercept: We have an intercept here that tells where x-intercepts on Y. This is an important part of the regression equation. It is -1.11 in our case.
2. X variable 1 (Slope). Also called the coefficient of x. It defines the tangent of the regression line.

REGRESSION CHART IN EXCEL

In excel, it is easy to plot a regression chart. Just follow these steps. To add Regression Chart in Excel 2016, 2013, and 2010 follow these simple steps.

Step 1. Have your known x’s in the first column and know y’s in the second.

In our case, we know Known_ x’s are Soft Drinks Sold. And known_y’s are Salesmen.

Step 2. Select your known x’s and y’s range.

Step 3: Go to the Insert tab and click on the scatter chart.

You will have a chart that looks like this.

Step 4. Add the trend line: Goto layout and locate the trendline option in the analysis section.

Under the Trendline option, click on Linear Trendline.

You will have your graph looking like this.

This is your regression graph.

Now if you add the data below and extend the selected data. You will see a change in your graph.

For our example, we added 2000 to the Soft Drink Sold and left the Salesmen blank. And when we extend the range of the graph, this is what we will have.

It will give the required number of salesmen for doing 2000 sales of soft drinks in graphical form. Which is slightly below 120 in the graph. And from our regression equation, we know it is 116.

In this article, I tried to cover everything under Excel Regression Analysis. I explained regression in excel 2016. Regression in excel 2010 and excel 2013 is same as in excel 2016.

For any further query on this topic, use the comments section. Ask a question, give an opinion or just mention my grammatical mistakes. Everything is welcome. Just don’t hesitate to use the comment section.

Related Data:
How to Use STDEV Function in Excel

How To Calculate MODE function in Excel

How To Calculate Mean function in Excel

How to Create Standard Deviation Graph

Descriptive Statistics in Microsoft Excel 2016

How to Use Excel NORMDIST Function

How to use the Pareto Chart and Analysis

Popular Articles:

50 Excel Shortcut to Increase Your Productivity

How to use the VLOOKUP Function in Excel

How to use the COUNTIF function in Excel 2016

How to use the SUMIF Function in Excel