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barcode font reporting services We shall denote by xj. the mean of the measurements in the jth row. We have # xj. # 1 x b ka jk 1 in Software
We shall denote by xj. the mean of the measurements in the jth row. We have # xj. # 1 x b ka jk 1 Decode QR Code 2d Barcode In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Painting QR Code 2d Barcode In None Using Barcode maker for Software Control to generate, create QR image in Software applications. 1, 2, c, a
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Barcode Encoder In None Using Barcode maker for Software Control to generate, create barcode image in Software applications. GS1  13 Drawer In None Using Barcode maker for Software Control to generate, create EAN13 Supplement 5 image in Software applications. We define the total variation, denoted by v, as the sum of the squares of the deviations of each measurement from the grand mean x, i.e., # Total variation By writing the identity, xjk x # (xjk xj.) # (xj. # x) # (4) v a (xjk UCC  12 Maker In None Using Barcode printer for Software Control to generate, create UPCA image in Software applications. Draw Code 3 Of 9 In None Using Barcode creator for Software Control to generate, create Code 39 Full ASCII image in Software applications. x)2 # Identcode Generator In None Using Barcode creation for Software Control to generate, create Identcode image in Software applications. Paint EAN13 In Java Using Barcode encoder for BIRT reports Control to generate, create UPC  13 image in BIRT reports applications. and then squaring and summing over j and k, we can show (see Problem 9.1) that a (xjk
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x)2 # or a (xjk
x)2 # a (xjk
xj.)2 # b a (xj. # x)2 # We call the first summation on the right of (5) or (6) the variation within treatments (since it involves the squares of the deviations of xjk from the treatment means xj.) and denote it by vw. Therefore, # vw a (xjk xj.)2 # The second summation on the right of (5) or (6) is called the variation between treatments (since it involves the squares of the deviations of the various treatment means xj. from the grand mean x) and is denoted by vb. Therefore, # # vb Equations (5) or (6) can thus be written v vw vb (9) a (xj. b a (xj.
Shortcut Methods for Obtaining Variations
To minimize the labor in computing the above variations, the following forms are convenient: v vb vw a x2 jk t2 ab t2 ab
(10) (11) (12) 1 t2 b a j. j v vb
where t is the total of all values xjk and tj. is the total of all values in the jth treatment, i.e., t a xjk a xjk
(13) In practice it is convenient to subtract some fixed value from all the data in the table; this has no effect on the final results. Linear Mathematical Model for Analysis of Variance
We can consider each row of Table 91 as representing a random sample of size b from the population for that particular treatment. Therefore, for treatment j we have the independent, identically distributed random variables Xj1, Xj2, . . . , Xjb, which, respectively, take on the values xj1, xj2, . . . , xjb. Each of the Xjk (k 1, 2, . . . , b) CHAPTER 9 Analysis of Variance
can be expressed as the sum of its expected value and a chance or error term: Xjk mj
(14) The jk can be taken as independent (relative to j as well as to k), normally distributed random variables with mean zero and variance s2. This is equivalent to assuming that the Xjk ( j 1, 2, . . . , a; k l, 2, . . . , b) are mutually independent, normal variables with means mj and common variance s2. Let us define the constant m by m 1 a a mj We can think of m as the mean for a sort of grand population comprising all the treatment populations. Then (14) can be rewritten as (see Problem 9.18) Xjk m aj where
a aj
(15) The constant aj can be viewed as the special effect of the jth treatment. The null hypothesis that all treatment means are equal is given by (H0: aj 0; j 1, 2, . . . , a) or equivalently by (H0: mj m; j 1, 2, . . . , a). If H0 is true, the treatment populations, which by assumption are normal, have a common mean as well as a common variance. Then there is just one treatment population, and all treatments are statistically identical. Expected Values of the Variations
The betweentreatments variation Vb, the withintreatments variation Vw, and the total variation V are random variables that, respectively, assume the values vb, vw, and v as defined in (8), (7), and (3). We can show (Problem 9.19) that E(Vb) E(Vw) E(V) From (17) it follows that EB (a a(b (ab 1)s2 1)s2 1)s2 b a a2 j b a a2 j (16) (17) (18)

