New PDF release: A first course in calculus

By Serge Lang

This 5th version of Lang's booklet covers all of the themes often taught within the first-year calculus series. Divided into 5 components, each one component of a primary path IN CALCULUS includes examples and functions when it comes to the subject coated. furthermore, the rear of the publication comprises exact options to a lot of the routines, permitting them to be used as worked-out examples -- one of many major advancements over past variations.

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00 This is a universal rule for computing the within cells sum of squares; subtract T 2/N and all substantive sources from ΣX 2. It is not difficult to verify the intuitively obvious idea that if there is only one score per cell, the within-cells sum of squares must be zero. It is worthy of notice that the sum of the sums of squares for the three substantive sources, SSA, SSB, and SSAB, is equal to what would have been computed as Σ(tj2/nj) − T 2/N had the six cells in the design simply been treated as though they were groups in a one-way ANOVA.

1. For example, 5 C2 = 5 ؒ 4 ؒ 3 ؒ 2 ؒ1 ( 2 ؒ 1) ؒ ( 3 ؒ 2 ؒ 1) = 10 For a two-factor design with r rows, c columns, and n scores in each cell, the row and column sums of squares may be expressed in terms of the effects as follows: SS ROWS = SSCOLS = nؒc r nؒr c Σ( ∆2 ROWS ) (5-1) Σ( ∆2 COLS ) (5-2) TWO-WAY ANOVA 55 In these equations, Σ(∆2ROWS) is the sum of the squares of all of the differences between pairings of the row marginal means. Σ(∆2COLS) is a similar summation using the column marginal means.

2COLS) is a similar summation using the column marginal means. Of course, equations 5-1 and 5-2 are not practical computational expressions for calculating sums of squares. The arithmetic using them is much more tedious than for the simpler expressions given previously. The point is to show that sums of squares can be given an intuitive interpretation in terms of differences between mean responses to the various levels of a factor. ANOVA looks at the square of what you see on the graph (plus variability).

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