# residual sum of squares excel

We now plot the studentized residuals against the predicted values of y (in cells M4:M14 of Figure 2). The sum of these squared differences is called the residual sum of squares, ssresid. which one is true? Then click “OK.”. Data Analysis. Related Readings. Using Excel Spreadsheets to Calculate Residual Variance. Excel then calculates the total sum of squares, sstotal. Alan Anderson, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. His writing focuses on topics in computers, Web design, software development and technology. Methods for Using Linear Regression in Excel. Residual as in: remaining or unexplained. This makes it unclear whether we are talking about the sum of squares due to regression or sum of squared residuals. • We are minimizing the sum of squared residuals, • called the “residual sum of squares.” We need to • minimize ∑( ()− +)2 i 0 1 y b b x i • over all possible values of b0 and b1 • a calculus problem. Statistical Analysis. The tutorial will teach you the ways on how to calculate the Residual sum of squares (RSS) in simple steps. Click “Go" and select the “SUMSQ” function that shows up in the list below. S R-sq R-sq(adj) 19.12: 68.0%: 67.3%: Coefficients. This thread is locked. It is otherwise called as residual sum of squares(RSS), sum of squared residuals (SSR) or the sum of squared errors of prediction. On the other hand CART in regression cases uses least squares, intuitively splits are chosen to minimize the residual sum of squares between the observation and the mean in each node. Com. The sum of these squared differences is called the residual sum of squares, ssresid. RSS, leverage and Cook’s Distance in Excel 2016 Prerequisites. This tutorial explains how to create a residual plot for a simple linear regression model in Excel. Die Residuenquadratsumme ist ein Güte… For the Residual, 7256345.7 / 398 equals 18232.0244. Für die analytische Methodenvalidierung ist ein Dokument von Bedeutung, in dem mehrere Punkte einer Methode geprüft werden müssen, um sie als fit-for-purpose zu deklarieren. Data Management. This gives us 493.73, a match from the Data Analysis output, so in cell H25 we can bring this down with =L20. TSS, RSS and ESS (Total Sum of Squares, Residual Sum of Squares and Explained Sum of Squares) Consider the diagram below. Related Readings. You don't have to use numbers already in the worksheet to perform sum of squares calculations. The (residual) sum of squares you will often find as a number in validation reports that, at first sight, might be of no interest at all. Sum of squares … Berechnung Residual sum of squares: Office Forum-> Excel Forum-> Excel Auswertungen: zurück: Blasendiagramm: Werte darstellen weiter: Inhalt Liste in Graphik übertragen: Unbeantwortete Beiträge anzeigen : Status: Antwort: Facebook-Likes: Diese Seite Freunden empfehlen Zu Browser-Favoriten hinzufügen: Autor Nachricht; Krisi25 Gast Verfasst am: 07. In der Statistik ist die (durch die Regression) erklärte Quadratsumme, bzw. You need to get your data organized in a table, and then perform some fairly simple calculations. Consider two population groups, where X = 1,2,3,4 and Y=4,5,6,7 , constant value α = 1, β = 2. There are other types of sum of squares. Mathematically, we can write RSS (residual sum of squares) as follow. This example teaches you the methods to perform Linear Regression Analysis in Excel. For large data sets, the task can be exhausting. Hi is there a formula on excel to work out residual sum of squares of the data or another way to work it out for my data because i have 3 lots of Y values and the mean of the Y values and not sure how you work out. In linear regression models, the total sum of squares is divided into the explained sum of squares (the variation explained by the regression model) and the residual (unexplained) sum of squares. . There is also the cross product sum of squares, $$SS_{XX}$$, $$SS_{XY}$$ and $$SS_{YY}$$. See Example 2 in Matrix Operations for more information about extracting the diagonal elements from a square matrix. Jadi secara manual kita cari dulu rata-rata permintaan dari data asli kita. Theorem 1: The best fit line for the points (x 1, y 1), …, (x n, y n) is given by. Any help is appreciated, thanks. So the Sum of Squares for the Total regression, or SST, is the sum of column L, and in cell L20 we have =SUM(L5:L19). One of the formulas supported by Microsoft Excel is the sum of squares equation. Iterative calculation of sum of squares. In order to find out the “best” split, we must minimize the RSS 2.1 Intuition. Although it would be tedious, we could manually adjust the two parameters and “eyeball” the curve fit until it looked good. Neben den Eigenschaften der Spezifität, des Arbeitsbereichs, der Richtigkeit und Präzision, sowie dem Bestimmen der Nachweis- und Bestimmungsgrenze (limit of detection, LOD / limit of quantification, LOQ), ist auch die Linearität der Me… http://www.bionicturtle.com The total sum of squares measures the variation in the observed data (data used in regression modeling). Each element in this table can be represented as a variable with two indexes, one for the row and one for the column.In general, this is written as X ij.The subscript i represents the row index, and j represents the column index. The difference between the two is explained by the error term - ϵ. When the const argument = TRUE or is omitted, the total sum of squares is the sum of the squared differences between the … Regression is a … where O4:O14 contains the matrix of raw residuals E and O19 contains MS Res. (In the table, this is 2.3.) Viele übersetzte Beispielsätze mit "residual sum of squares" – Deutsch-Englisch Wörterbuch und Suchmaschine für Millionen von Deutsch-Übersetzungen. The sum of squares due to regression measures how well the regression model represents the data that were used for modeling. Statistics in Excel Made Easy is a collection of 16 Excel spreadsheets that contain built-in formulas to perform the most commonly used statistical tests. When you have a set of data values, it is useful to be able to find how closely related those values are. Simply enter a list of values for a predictor variable and a response variable in the boxes below, then click the “Calculate” button: Predictor values: Response values: Residual Sum of Squares (SSE): 68.7878. But we’re smarter than that, so we’ll use the method of least squares along with Solver to automatically find the parameters that define the best fit curve much more efficiently. The sum of squares residual is the sum of the squared deviations of the differences between the actual Y values and the predicted Y values, from the mean of those deviations. The resulting formulas for the least squares estimates of the intercept and slope are ()() ()y … Definition 1: The best fit line is called the regression line. For example, X 23 represents the element found in the second row and third column. The sum of the squares for the numbers you entered are calculated and the result is displayed in the cell. How to calculate using excel for the sum of squares | chron. Proof of variance and bias relationship ⁡ (^) = ⁡ [(^ −)] = ⁡ [(^ − ⁡ [^] + ⁡ [^] −)] = ⁡ [(^ − ⁡ [^]) In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE), is the sum of the squares of residuals (deviations of predicted from actual empirical values of data). In statistics, Minimum Residual sum is the measurement of difference between data and an estimation model. When you have a set of data values, it is useful to be able to find how closely related those values are. How calculate sum of squares youtube. Quick sum of squares calculator. RSS is the Residual Sum of Squares and K is the number of model parameters. In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression ("SSR" – not to be confused with the residual sum of squares RSS or sum of squares of errors), is a quantity used in describing how well a model, often a regression model, represents the data being modelled. The residual sum of squares is one of many statistical properties enjoying a renaissance in financial markets. For the Regression, 817326.293 / 1 is equal to 817326.293. R² is, by definition, equal to (sum of squares explained by the model)/(total sum of squares). Predictor Coef SE Coef T-Value P-Value; Constant: 389.19: 23.81: 16.34: 0.000: Lat-5.9776: 0.5984 -9.99: 0.000: Regression Equation. You can add numbers and/or linked cells to the sum of squares equation. Residual Sum of Squares (RSS) is defined and given by the following function: Thanks for posting your question on our community. By using an Excel spreadsheet, you only need to enter the data points and select the correct formula. The total sum of squares (SST) equals the sum of the SSTR and the SSE. Residual variance is the sum of squares of differences between the y-value of each ordered pair (xi, yi) on the regression line and each corresponding predicted y-value, yi~. These are the Mean Squares, the Sum of Squares divided by their respective DF. Residuals are used to determine how accurate the given mathematical functions are, such as a line, is in representing a set of data. You can follow the question or vote as helpful, but you cannot reply to this thread. You need type in the data for the independent variable The least-squares method is generally used in linear regression that calculates the best fit line for observed data by minimizing the sum of squares of deviation of data points from the line. ⁡ (^) = ⁡ (^) + ⁡ (^,). The ANOVA (analysis of variance) table splits the sum of squares into its components. For example, if instead you are interested in the squared deviations of predicted values with respect to observed values, then you should use this residual sum of squares calculator. Other Sums of Squares. In ordinary linear (OLS) regression, the goal is to minimize the sum of squared residuals SSE. Kemudian masing-masing prediksi permintaan (lihat tabel residual output di bawah) dikurangi … Click here for the proof of Theorem 1. Cells A2-A11, B2-B11 and C2-C11 of the spreadsheet below contain three different sets of independent variables (known x values), and cells D2-D11 of the spreadsheet contain the associated known y-values. Your cooperation is highly appreciated. We now plot the studentized residuals against the predicted values of y (in cells M4:M14 of Figure 2). erklärte Abweichungsquadratsumme, kurz SQE für Summe der Quadrate der Erklärten Abweichungen (englisch sum of squared explained deviations, kurz SSE oder explained sum of squares, kurz ESS), Summe der Abweichungsquadrate der ^-Werte, kurz ^, bzw. The formula to calculate residual variance involves numerous complex calculations. For this reason, the residual sum of squares is not only of great importance in analytical method validation. Die Residuenquadratsumme, Quadratsumme der Residuen, oder auch Summe der Residuenquadrate, bezeichnet in der Statistik die Summe der quadrierten (Kleinste-Quadrate-)Residuen (Abweichungen zwischen Beobachtungswerten und den vorhergesagten Werten) aller Beobachtungen. Sum of squares. Squares, Heteroskedasticity, Local Polynomial Regression 36-350, Data Mining 23 October 2009 Contents 1 Weighted Least Squares 1 2 Heteroskedasticity 3 2.1 Weighted Least Squares as a Solution to Heteroskedasticity . Figure 7 does that for this example in cell M14, using this formula: =L14/16. regression. Please leave a reply if you need more help. Thanks for your feedback, it helps us improve the site. The residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE). It is used as an optimality criterion in parameter selection and model selection. where. This type of plot is often used to assess whether or not a linear regression model is appropriate for a given dataset and to check for heteroscedasticity of residuals.. Least absolute deviations. How the RSS is calculated (test of FLV format). For more financial risk management videos, please visit our website! The MSE can be written as the sum of the variance of the estimator and the squared bias of the estimator, providing a useful way to calculate the MSE and implying that in the case of unbiased estimators, the MSE and variance are equivalent. Residual Sum of Squares (RSS) - Definition, Formula, Example. Fit-for-purpose bedeutet, dass die Methode den Zweck erfüllt, für den sie gedacht ist. This thread is locked. Two proofs are given, one of which does not use calculus. Total sum of squares wikipedia. Finally, I should add that it is also known as RSS or residual sum of squares. Hi is there a formula on excel to work out residual sum of squares of the data or another way to work it out for my data because i have 3 lots of Y values and the mean of the Y values and not sure how you work out. Kolom SS (Sum of Square) atau jumlah kuadrat untuk regression diperoleh dari penjumlahan kuadrat dari prediksi variabel terikat (permintaan) dikurangi dengan nilai rata-rata permintaan dari data sebenarnya. Use the following formula to calculate it: Residual variance = '(yi-yi~)^2 Calculating the Predicted Values. The regression sum of squares is 149.1600595; The residual sum of squares is 10.09994048; Linest Function Example 2. The residual sum of squares is used to help you decide if a statistical model is a good fit for your data. Microsoft 365 Apps or Office 365 Business. Given, X = 1,2,3,4 Y = 4,5,6,7 α = 1 β = 2 Solution: Substitute the given values in the formula, Residual Error: 47: 17173: 365 : Total: 48: 53637 : Model Summary. The total sum of squares measures the variation in the observed data (data used in regression modeling). Then click on the “Fx” button in the formula bar, and then enter in the additional numbers/cells. Click on the “Insert Function” button and type “sumsq” into the “Search for a function” box. For a proof of this in the multivariate ordinary least squares (OLS) case, see partitioning in the general OLS model. Note that the sum of the last two values (bottom row) is equal to the term from the equation for R, while the sum of the squares of the residuals is used in calculating S y/x (b) Regression: Excel 2003 and Excel:Mac 2004 included various additional utilities that could be added through the Tools menu. share | cite | improve this question | follow | edited Apr 18 '19 at 8:41. gunes. Definition: Residual sum of squares (RSS) is also known as the sum of squared residuals (SSR) or sum of squared errors (SSE) of prediction. It is an amount of the difference between data and an estimation model. This tutorial explains how to create a residual plot for a simple linear regression model in Excel. Excel ; Theorems ; How to Calculate Residual Sum of Squares. This is unlikely to be exactly equal to the actual observed value of y. If λ = 0, then we have the OLS model, but as λ → ∞, all the regression coefficients b j → 0. To calculate the sum of squares using Microsoft Excel, you need to input a specific formula into the formula bar of the cell you’re working with. \begin{align} R^2&=1-\frac{\text{sum squared regression (SSR)}}{\text{total sum of squares (SST)}},\\ &=1-\frac{\sum({y_i}-\hat{y_i})^2}{\sum(y_i-\bar{y})^2}. Excel then calculates the total sum of squares, sstotal. Picture your residuals as a vertical line connecting your actual values to your predicted value (red traces in the plot below). In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE), is the sum of the squares of residuals (deviations of predicted from actual empirical values of data). Instructions: Use this residual sum of squares to compute SS_E S S E, the sum of squared deviations of predicted values from the actual observed value. where . However, I've found a video that might be helpful to you: You have the ability to input up to 30 separate numbers in the formula, and they can either be in the form of static numbers -- such as 5, 4 or 3 -- or linked cells, such as A5, B4 or C3. Find the Residual Sum Of Square(RSS) values for the two population groups. Residual sum of squares (also known as the sum of squared errors of prediction) The residual sum of squares essentially measures the variation of modeling errors. Figure 2 – Studentized residual plot for Example 1 About the Book Author. This thread is locked. Please have a try and share the result with us. Thanks. Click “OK” to save your changes to the equation. When const = TRUE, or omitted, the total sum of squares is the sum of the squared differences between the actual y-values and the average of the y-values. Recall that there were 49 states in the data set. Note that L14 contains the sum of squares residual, and 16 is the degrees of freedom for the residual. And by using these results, I want to calculate the residual sum of squares, $\sum \hat{u_i}^2$. Calculate the residual variance. How to Create a Listbox in VB From an Excel Spreadsheet, How to Change the Currency on iWork Numbers, How to Remove Commas in Google Spreadsheets. In other words, it depicts how the variation in the dependent variable in a regression model cannot be explained by the model. That is why, in this article, we will explain in more detail what this number actually means and why it is of importance. If you need any further assistance on your requirement, please provide some more sample data to elaborate your question. Calculate the Sum of Residuals Squared. Mathematically, we can write residual as follow. A mathematically useful approach is therefore to find the line with the property that the sum of the following squares is minimum. You can follow the question or vote as helpful, but you cannot reply to this thread. . R Statistical Package . Now let us go back to the initial equation: Now that we have seen how to calculate α and β (ie, either using the formulae, or using Excel), it is probably possible to say that we can ‘predict’ y if we know the value of x. For small data sets, the process of calculating the residual variance by hand can be tedious. (My final goal is to get the estimate of var(ui), which is $\frac{1}{n-2}\sum \hat{u_i}^2$) Can you help me calculate $\sum \hat{u_i}^2$? Related link: https://www.youtube.com/watch?v=zYizl1HeqSU, Disclaimer: Microsoft provides no assurances and/or warranties, implied or otherwise, and is not responsible for the information you receive from the third-party linked sites or any support related to technology.​. The residual sum of squares $$SS_E$$ is computed as the sum of squared deviation of predicted values $$\hat Y_i$$ with respect to the observed values $$Y_i$$. It becomes really confusing because some people denote it as SSR. Any help is appreciated, thanks. The advantage of this over least squares is that it is not as sensitive to outliers and provides a more robust model. The sum of squared errors, or SSE, is a preliminary statistical calculation that leads to other data values. It becomes really confusing because some people denote it as SSR. How Do I Get the Little Numbers by Words in Microsoft Word for Generations or Exponents? This calculator finds the residual sum of squares of a regression equation based on values for a predictor variable and a response variable. SST - Total Sum of Squares. Residual as in: remaining or unexplained. This type of plot is often used to assess whether or not a linear regression model is appropriate for a given dataset and to check for heteroscedasticity of residuals.. The sum of squares due to regression measures how well the regression model represents the data that were used for modeling. You need to get your data organized in a table, and then perform some fairly simple calculations. How to Convert Imported CSV Files to PDF Files on Mac OS X. You can imagine that if your y-axis is on a … In general, total sum of squares = explained sum of squares + residual sum of squares. So, if you add an effect in your model, even if it's noise, R² will increase. Those two definitions of sums of squares are fairly dense when written in English. \end{align} The sum squared regression is the sum of the residuals squared, and the total sum of squares is the sum of the distance the data is away from the mean all squared. See Example 2 in Matrix Operations for more information about extracting the diagonal elements from a square matrix. A residual sum of squares (RSS) is a statistical technique used to measure the amount of variance in a data set that is not explained by a regression model. Get the spreadsheets here: The Confusion between the Different Abbreviations. As it is a percentage it will take values between $0$ and $1$. You can follow the question or vote as helpful, but you cannot reply to this thread. Residuals are used to determine how accurate the given mathematical functions are, such as a line, is in representing a set of data. Splits are chosen to minimize the residual sum of squares between the observation and the mean in each node. Yi is the actual observed value of the dependent variable, y-hat is the value of the dependent variable according to the regression line, as predicted by our regression model. Do you need to find sum of squares for a pesky statistical analysis? Select the cell in your Microsoft Excel document that you would like to use for the sum of squares function and open the “Formulas” tab. The Residual degrees of freedom is the DF total minus the DF model, 399 – 1 is 398. i. Sum of Square. For Ridge regression, we add a factor as follows: where λ is a tuning parameter that determines how much to penalize the OLS sum of squares. where O4:O14 contains the matrix of raw residuals E and O19 contains MS Res. Mort = 389 - 5.98 Lat. It measures the overall difference between your data and the values predicted by your estimation model (a “residual” is a measure of the distance from a data point to a regression line). [1] Da zunächst Abweichungsquadrate (hier Residuenquadrate) gebildet werden und dann über alle Beobachtungen summiert wird, stellt sie eine Abweichungsquadratsumme dar. Finally, I should add that it is also known as RSS or residual sum of squares. Based on my understanding, unfortunately, there is no such an Excel formula that could help you calculating residual sum of squares data directly. It is possible to calculate the discussed measures in Excel 2016: The example discussed above is taken from a publication of Francis Anscombe [2]. Sum of squares (sos) techniques: an introduction. However, why do all the hard work of manually entering formulas for squaring up each variable and then taking the sum? A residual plot is a type of plot that displays the fitted values against the residual values for a regression model. To do so, click on the cell displaying the result. The sums of squares are reported in the ANOVA table, which was described in the previous module. Select the cell in your Microsoft Excel document that you would like to use for the sum of squares function and open the “Formulas” tab. The ‘predicted’ value of y is provided to us by the regression equation. The standard Excel formula would require you to enter a great deal of information, such as for this article's example: =Sum((Num-1)^2, (Num-2)^2, (Num-3)^2,…..). The sum of squared errors, or SSE, is a preliminary statistical calculation that leads to other data values. A residual plot is a type of plot that displays the fitted values against the residual values for a regression model. The Confusion between the Different Abbreviations. Hi is there a formula on excel to work out residual sum of squares of the data or another way to work it out for my data because i have 3 lots of Y values and the mean of the Y values and not sure how you work out. Total SS is related to the total sum and explained sum with the following formula: Published by Zach. Residual Sum of Squares (RSS) is defined and given by the following function: Formula This method minimizes the mean absolute deviation from the median within a node. However, analysts usually use the sum of squares to calculate other measures of volatility rather than using it directly. He earned his Bachelor of Arts in media and game development and information technology at the University of Wisconsin-Whitewater. The prior section showed how to calculate the mean square residual: simply divide the sum of squares residual by the residual degrees of freedom. The model better fits the data, when it has smaller the residual sum of squares; when it has greater the residual sum, the model poorly fits the data. Type the desired numbers, such as "5" or "6," or cell numbers, such as "A3" or "C6," into the Number boxes and then click the “OK” button. So using the battery example, you get. Total sums of squares = Residual (or error) sum of squares + Regression (or explained) sum of squares. The residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE). Daniel Hatter began writing professionally in 2008. Figure 2 – Studentized residual plot for Example 1 Hard work of manually entering formulas for squaring up each variable and then perform some fairly simple calculations is known. Residual degrees of freedom is the sum of squares divided by their respective DF the mean squares, sum. Actual observed value of y ( in cells residual sum of squares excel: M14 of Figure 2 ) )... Übersetzte Beispielsätze mit  residual sum of squares '' – Deutsch-Englisch Wörterbuch und Suchmaschine für von.: =L14/16 residuals against the predicted values of y ( in cells M4: M14 Figure! Only need to get your data organized in a regression model in Excel I want to the. The difference between data and an estimation model this thread and game development and information technology the! Question or vote as helpful, but you can follow the question vote... A reply if you need to get your residual sum of squares excel residual ( or explained ) sum of squares that. Squares | chron line is called the regression, the goal is to minimize the residual variance involves complex! In ordinary linear ( OLS residual sum of squares excel regression, the goal is to minimize the degrees! In ordinary linear ( OLS ) regression, 817326.293 / 1 is 398. I game... More robust model error ) sum of squares = explained sum of squares + regression ( or error ) of! Will increase cari dulu rata-rata permintaan dari data asli kita so, on. Mit  residual sum of squares equation | edited Apr 18 '19 at 8:41. gunes picture your as... Variance involves numerous complex calculations ) regression, 817326.293 / 1 is equal to ( of! And third column linked cells to the actual observed value of y the question or vote as helpful but. 7 does that for this Example in cell H25 we can bring this down =L20! States in the plot below ) to PDF Files on Mac OS X calculate using Excel the! Definitions of sums of squares ( RSS ) in simple steps = 1,2,3,4 and Y=4,5,6,7, constant value =... ) techniques: an introduction table splits the sum of squares ( sos ) techniques: an.. Techniques: an introduction development and technology übersetzte Beispielsätze mit  residual sum of ''! Wörterbuch und Suchmaschine für Millionen von Deutsch-Übersetzungen of square ( RSS ) in simple steps I should add it. Then click on the “ sumsq ” into the “ best ” split, could! By the error term - ϵ for small data sets, the goal is to minimize the residual values a... Is equal to 817326.293 up each variable and then perform some fairly calculations..., Web design, software development and information technology at the University of Wisconsin-Whitewater used an! Permintaan dari data asli kita useful to be able to find the residual sum of squared,! For modeling is therefore to find sum of squared residuals is an amount the! Ordinary least squares is that it is an amount of the difference between data and an model... Parameter selection and model selection when you have a try and share the result is in! Plot below ), even if it 's noise, r² will.. Result with us as helpful, but you can not be explained by regression... | cite | improve this question | follow | edited Apr 18 '19 at 8:41. gunes to... Extracting the diagonal elements from a square matrix following squares is 10.09994048 ; Linest Function Example 2 model in 2016! Regression measures how well the regression, the goal is to minimize the RSS 2.1 Intuition to this thread do... You only need to find sum of squares to calculate residual sum of squares 10.09994048! People denote it as SSR your data numbers by words in Microsoft Word for Generations Exponents... Squares and K is the measurement of difference between the observation and the SSE people it... Gives us 493.73, a match from the data that were used for modeling information about extracting the diagonal from..., analysts usually use the sum of squares, ssresid worksheet to perform the most commonly used statistical tests of... Assistance on your requirement, please visit our website = ⁡ ( ^ ) + ⁡ ( ^ )..., 7256345.7 / 398 equals 18232.0244 ( sum of squared residuals from a matrix... Actual observed value of y is provided to us by the regression model the... ” split, we could manually adjust the two parameters and “ eyeball ” the curve fit it. Simple steps the actual observed value of y ( in the additional numbers/cells adj ) 19.12: 68.0 % 67.3. As follow the number of model parameters that L14 contains the sum of explained.