Total sum of squares

In <a href="/facts/Statistics/DNPsKGYU">statistical data analysis</a> the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. For a set of observations, 
 
 
 
 
 y
 
 i
 
 
 ,
 i
 ≤
 n
 
 
 {\displaystyle y_{i},i\leq n}
 
, it is defined as the sum over all squared differences between the observations and their overall <a href="/facts/Mean/swcEd4Pg">mean</a> 
 
 
 
 
 
 
 y
 ¯
 
 
 
 
 
 {\displaystyle {\bar {y}}}
 
.:

T
          S
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            =
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                y
                
                  i
                
              
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    {\displaystyle \mathrm {TSS} =\sum _{i=1}^{n}\left(y_{i}-{\bar {y}}\right)^{2}}

For wide classes of <a href="/facts/Linear_regression/5n998IhK">linear models</a>, the total sum of squares equals the <a href="/facts/Explained_sum_of_squares/97KEl3dj">explained sum of squares</a> plus the <a href="/facts/Residual_sum_of_squares/vbGsrB3x">residual sum of squares</a>. For proof of this in the multivariate OLS case, see <a href="/facts/Explained_sum_of_squares/97KEl3dj">partitioning in the general OLS model</a>.
In <a href="/facts/Analysis_of_variance/HHR2jkDk">analysis of variance</a> (ANOVA) the total sum of squares is the sum of the so-called "within-samples" sum of squares and "between-samples" sum of squares, i.e., partitioning of the sum of squares.
In <a href="/facts/Multivariate_analysis_of_variance/YacRVgl9">multivariate analysis of variance</a> (MANOVA) the following equation applies

T
 
 =
 
 W
 
 +
 
 B
 
 ,
 
 
 {\displaystyle \mathbf {T} =\mathbf {W} +\mathbf {B} ,}
 
 g
where T is the total sum of squares and products (SSP) <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>, W is the within-samples SSP matrix and B is the between-samples SSP matrix.
Similar terminology may also be used in <a href="/facts/Linear_discriminant_analysis/Mj6KQhRN">linear discriminant analysis</a>, where W and B are respectively referred to as the within-groups and between-groups SSP matrices.

Total sum of squares open-in-new

Total sum of squares