In statistical computations it is desirable to have a simplified system of notation to avoid complicated formulas describing mathematical operations.

Transcription

1 Chapte 1 STATISTICAL NOTATION AND ORGANIZATION 11 Summation Notation fo a One-Way Classification In statistical computations it is desiable to have a simplified system of notation to avoid complicated fomulas descibing mathematical opeations A measuable chaacteistic of an expeimental unit is called a vaiable Geneally, X, Y, o Z ae symbols used to epesent vaiables Individual obsevations of a vaiable ae called vaiances A vaiable can be discete (discontinuous) and only have a limited numbe of values, o it can be continuous and assume any value within a given ange Fo example, a discete vaiable would be the numbe of leaves on a tee A continuous vaiable would be the height o weight of a plant Suppose that we have vaiates, then thei sum may be witten Y1 + Y + Y+ + Y = Σ Yj Y 1, Y, Y,, Y whee the Geek lette signifies the sum of all tems The summation index indicates the vaiates j Σ to be summed; j anges fom 1 to 5 Often, when it is obvious what vaiates ae to be summed, the subject j and its limits ae omitted and the opeation simplified to Y The following examples illustate the use of summation notation Σ Yj = Y1 + Y + Y 5 Σ Yj = Y + Y4 + Y5 j = Σ Yj( Yj 1) = Y1( Y1 1) + Y( Y 1) If constants such as a and b ae involved, then Σ ( Yj + ayj + b) = Y1 + a Y1 + b

2 Y + ay +b Y + ay + b Y + ay + b = ΣY + a ΣY + b j Fom this example the following ules can be stated: 1 The summation of a constant is multiplied by the constant j The summation of a constant multiplied by a vaiable is the same as the constant multiplied by the summation of the vaiable The summation of the sum of two o moe tems is equal to the sum of the summations of the sepaate tems Note the diffeence between ΣY and (ΣY) Example 11 If Y 1 =, Y = 4, and Y = 6, then ΣY = = 61 (ΣY) = ( ) 69 1 Summation Notation Fo a Two-Way Classification To facilitate computations, data ae often aanged in a two-way table and symbolized as in Table 1-1 Table 1-1 Vaiates in a two-way table Columns Rows 1 Totals Means 1 Y 11 Y 1 Y 1 Y 1 Y 1 Y 1 Y 1 Y Y Y Y Y Y 1 Y Y Y Y Y k Y k1 Y k Y k Y k Y k Y k Totals Y 1 Y Y Y Y Means Y 1 Y Y Y Y

3 The symbol fo a geneal vaiate in this two-way table is Y ij, whee i epesents a ow fom 1 to k and j epesents a column fom 1 to Fo example, the vaiate in ow and column is denoted by Y The dot subscipt indicates an opeation ove all the vaiates in a ow o column Y 1 theefoe epesents the total of the vaiates of ow one ( Σ Y j ), and Y 1 epesents the total of all vaiates k ( Σ Σ Y ij ) The mean of ow one is Y1 = ( Y1 / ), and the gand mean of all vaiates is i j Y = ( Y/ k) Table 1- Numeical data classified in two ways Columns ( = ) Rows (k=4) 1 Y i Y i Y j Y j k The double summation index Σ Σ is often shotened to a single summation index ( ) if it is i j clea which vaiates ae to be summed Thus, k k Σ Σ Yij = ΣYij and Σ Σ Yij = ΣYij i j i j Fo the values in Table 1-: Σ Y ij = = 40, and Σ Y ij = In these expessions note that "" indicates the continuation of the opeation ove all values ending with the last two vaiates 1 Factoials It is convenient in a geat many poblems to have a symbol to epesent the poduct of the fist n positive integes Thus, = 70

4 is called "6 factoial" and is usually denoted by the symbol 6! In geneal, n! = n(n-1) (n-) 1 Fo consistency 0! is defined as equal to 1 The following factoial foms ae useful in calculations: 8! = = 8 7! = 8 7 6! = ! etc and n!/! = [n(n - 1) (n - ) ( + 1)!] /! = n(n - 1) ( n 1 ) ( + 1) whee is an intege less than n 1 A vaiable is symbolized by X, Y, o Z SUMMARY Σ Y j epesents the sum of values of Y fom Y1 to Y inclusive and can be simplified to ΣY j o ΣY Y epesents the aveage of Y fom Y 1 to Y, Y/ k 4 Σ Σ Y ij epesents the sum of the vaiates in columns 1 to and ows fom 1 to k and can be j simplified to ΣYij o Y 5 Y epesents the aveage of all vaiates in a two way table, Y/k 6 Summation ules: a) c = c, whee c is a constant j Σ b) Σ c Yj = c ΣY j c) Σ ( Yj + ayj + b) = ΣYj + a ΣYj + b

5 EXERCISES 1 Given the following set of measuements, 1,, 5, 7, 9, what is: a) b ) Σ Yi i c) Σ ( i+ Yi) i 5 d) Σ Yi i 5 e) Σ ( i)( Yi) i Given the following set of measuements,, 4, 6, 8, 10, show that: Algebaically show that Σ(Y- Y ) = 0 Use the fist column of Table 1- to illustate k 4 Algebaically show that Σ Y j = Σ Yifom a two-way table such as Table 11 Use the data i in Table 1- to demonstate this 5 Fo Table 1- compute the following tems: 4 a) Σ Σ ( Y Y) ( 56 68) i 4 4 ij b) Σ Σ Y ( Σ Σ Y ) / k ij i i 4 ij (56 68) c) Σ ( Y Y) ( 14 94) i i d) k Σ ( Y Y) ( 468 ) 4 e) Σ Σ ( Y Y Y + Y) ( 7 5) i j ij i j

6 6 Suppose students in a class wee counted and classified as: Yea/Sex Male Female Total J S G Total a) Identify the values fo the following symbols: Y 1, Y 1, Y, and Y (50, 50, 90, 10) b) Calculate Y1, Y, and Y (45, 0, 5) c) Calculate Σ Y ij (7900) d) Calculate Σ( Y Y ) (550) ij