SS, DF, MS - Sum of Squares, Mean sum of squares and Degrees of freedom recap and equations PDF

Preview

Summary

Sum of Squares, Mean sum of squares and Degrees of freedom recap and equations for mean sum of squares...

Description

Sum of Squares   



Sum of Squares (SS): the sum of squared differences between individual elements and a mean average. Total SS: the total fluctuation (variability) in the data around the grand mean (i.e. treating it as one large group). Within Groups SS: the fluctuation that occurs within each group around the group mean, due to unexplained factors (aka Error SS, because it is variance that is not due to the presence of different groups). Between Groups SS: the variance that occurs due to the fact that there are different groups within the data (aka Treatment SS, because different groups may receive different treatment).

Degrees of Freedom    

Degrees of Freedom (df): the number of opportunities for independent variation in a given data set given that we know the mean or variance. Total df: the degrees of freedom of the entire data set (i.e. treating it as one large group). Within Groups df: the degrees of freedom within each group, summed across the groups. Between Groups df: the degrees of freedom across the groups themselves.

Mean sum of squares  

Mean Sum of Squares (MS): the SS divided by the degrees of freedom, to give the amount of variance explained by each component. Total MS: the variance in the data as a whole (i.e. treating it as one large group).

M S to ta l o 

SS 1 = to ta l = d f to ta l n - 1

å

(xi - x )2

i =1

Within Groups MS: the variance within the groups, due to unexplained factors (aka Error MS, because it is variance that is not due to the presence of different groups).

M S w ith i n o 

n

S S w i th i n 1 = = d f w ith i n n - k

k

n

j

åå

( x ji - x j )

2

j =1 i =1

Between Groups MS: the variance that occurs due to the fact that there are different groups within the data (aka Treatment MS, because different groups may receive different treatment).

M S b e tw e e n o

1 SS = b e tw e e n = d f b e tw e e n k -1

k

å j =1

n j (x j - x )2...