Research Methodology - Dr. Krishan K. Pandey
Research Methodology - Dr. Krishan K. Pandey Research Methodology - Dr. Krishan K. Pandey
Analysis of Variance and Co-variance 259 MS SS within within = ( n – k) where (n – k) represents degrees of freedom within samples, n = total number of items in all the samples i.e., n 1 + n 2 + … + n k k = number of samples. (vii) For a check, the sum of squares of deviations for total variance can also be worked out by adding the squares of deviations when the deviations for the individual items in all the samples have been taken from the mean of the sample means. Symbolically, this can be written: F H SS for total variance = ∑ X − X ij I K 2 i = 1, 2, 3, … j = 1, 2, 3, … This total should be equal to the total of the result of the (iii) and (v) steps explained above i.e., SS for total variance = SS between + SS within. The degrees of freedom for total variance will be equal to the number of items in all samples minus one i.e., (n – 1). The degrees of freedom for between and within must add up to the degrees of freedom for total variance i.e., (n – 1) = (k – 1) + (n – k) This fact explains the additive property of the ANOVA technique. (viii) Finally, F-ratio may be worked out as under: F-ratio = MS between MS within This ratio is used to judge whether the difference among several sample means is significant or is just a matter of sampling fluctuations. For this purpose we look into the table * , giving the values of F for given degrees of freedom at different levels of significance. If the worked out value of F, as stated above, is less than the table value of F, the difference is taken as insignificant i.e., due to chance and the null-hypothesis of no difference between sample means stands. In case the calculated value of F happens to be either equal or more than its table value, the difference is considered as significant (which means the samples could not have come from the same universe) and accordingly the conclusion may be drawn. The higher the calculated value of F is above the table value, the more definite and sure one can be about his conclusions. SETTING UP ANALYSIS OF VARIANCE TABLE For the sake of convenience the information obtained through various steps stated above can be put as under: * An extract of table giving F-values has been given in Appendix at the end of the book in Tables 4 (a) and 4 (b).
260 Research Methodology Table 11.1: Analysis of Variance Table for One-way Anova (There are k samples having in all n items) Source of Sum of squares Degrees of Mean Square (MS) F-ratio variation (SS) freedom (d.f.) (This is SS divided by d.f.) and is an estimation of variance to be used in F-ratio Between samples or categories Within samples or categories F H 2 I K + ... (k – 1) F kH k 2 I K dX1i 2 X1i ... (n – k) dXki Xki 2 n X − X 1 1 + n X − X ∑ − + +∑ − i = 1, 2, 3, … F H Total ∑ X − X ij I K i = 1, 2, … j = 1, 2, … 2 (n –1) SHORT-CUT METHOD FOR ONE-WAY ANOVA SS between ( k – 1) SS within ( n – k) MS between MS within ANOVA can be performed by following the short-cut method which is usually used in practice since the same happens to be a very convenient method, particularly when means of the samples and/or mean of the sample means happen to be non-integer values. The various steps involved in the shortcut method are as under: (i) Take the total of the values of individual items in all the samples i.e., work out ∑X ij i = 1, 2, 3, … j = 1, 2, 3, … and call it as T. (ii) Work out the correction factor as under: bg 2 Correction factor = T n
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Analysis of Variance and Co-variance 259<br />
MS<br />
SS within<br />
within =<br />
( n – k)<br />
where (n – k) represents degrees of freedom within samples,<br />
n = total number of items in all the samples i.e., n 1 + n 2 + … + n k<br />
k = number of samples.<br />
(vii) For a check, the sum of squares of deviations for total variance can also be worked out by<br />
adding the squares of deviations when the deviations for the individual items in all the<br />
samples have been taken from the mean of the sample means. Symbolically, this can be<br />
written:<br />
F<br />
H<br />
SS for total variance = ∑ X − X<br />
ij<br />
I<br />
K 2<br />
i = 1, 2, 3, …<br />
j = 1, 2, 3, …<br />
This total should be equal to the total of the result of the (iii) and (v) steps explained above<br />
i.e.,<br />
SS for total variance = SS between + SS within.<br />
The degrees of freedom for total variance will be equal to the number of items in all<br />
samples minus one i.e., (n – 1). The degrees of freedom for between and within must add<br />
up to the degrees of freedom for total variance i.e.,<br />
(n – 1) = (k – 1) + (n – k)<br />
This fact explains the additive property of the ANOVA technique.<br />
(viii) Finally, F-ratio may be worked out as under:<br />
F-ratio<br />
=<br />
MS between<br />
MS within<br />
This ratio is used to judge whether the difference among several sample means is significant<br />
or is just a matter of sampling fluctuations. For this purpose we look into the table * , giving<br />
the values of F for given degrees of freedom at different levels of significance. If the<br />
worked out value of F, as stated above, is less than the table value of F, the difference is<br />
taken as insignificant i.e., due to chance and the null-hypothesis of no difference between<br />
sample means stands. In case the calculated value of F happens to be either equal or more<br />
than its table value, the difference is considered as significant (which means the samples<br />
could not have come from the same universe) and accordingly the conclusion may be<br />
drawn. The higher the calculated value of F is above the table value, the more definite and<br />
sure one can be about his conclusions.<br />
SETTING UP ANALYSIS OF VARIANCE TABLE<br />
For the sake of convenience the information obtained through various steps stated above can be put<br />
as under:<br />
* An extract of table giving F-values has been given in Appendix at the end of the book in Tables 4 (a) and 4 (b).
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