Research Methodology - Dr. Krishan K. Pandey

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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 <strong>Research</strong> <strong>Methodology</strong> 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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In loving memory of my revered fath
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Preface to the Second Edition vii P
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Preface to the First Edition ix Pre
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Contents xi Contents Preface to the
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Contents xiii Selection of Appropri
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Contents xv ANOVA in Latin-Square D
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Research Methodology: An Introducti
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Define research problem I Review th
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Defining the Research Problem 25 Ov
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Defining the Research Problem 27 so
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Defining the Research Problem 29 (a
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Research Design 31 MEANING OF RESEA
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Research Design 33 FEATURES OF A GO
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Research Design 35 of students and
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Research Design 37 Now, what sort o
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Research Design 39 The difference b
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Research Design 41 Important Experi
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Research Design 43 extraneous facto
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Research Design 45 From the diagram
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Research Design 47 equal. This redu
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Research Design 49 The graph relati
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Research Design 51 The dotted line
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Appendix: Developing a Research Pla
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Sampling Design 55 CENSUS AND SAMPL
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Sampling Design 57 would like to ma
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Sampling Design 59 Element selectio
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Sampling Design 61 successive drawi
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Sampling Design 63 The following th
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Sampling Design 65 where C 1 = Cost
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Sampling Design 67 Table 4.1 City n
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Measurement and Scaling Techniques
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Methods of Data Collection 95 6 Met
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Methods of Data Collection 97 There
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Methods of Data Collection 99 (viii
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Methods of Data Collection 101 2. I
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Methods of Data Collection 103 inst
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Methods of Data Collection 105 is g
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Methods of Data Collection 107 amon
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Methods of Data Collection 109 Holt
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Methods of Data Collection 111 COLL
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Methods of Data Collection 113 usin
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Methods of Data Collection 115 (v)
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Methods of Data Collection 117 2.
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Appendix (ii): Guidelines for Succe
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Appendix (iii): Difference Between
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Processing and Analysis of Data 123
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Editing Coding Processing of Data (
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Sampling Fundamentals 153 1. Univer
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Sampling Fundamentals 155 should no
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Sampling Fundamentals 157 certain l
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Sampling Fundamentals 159 (i) Stati
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Sampling Fundamentals 161 (ii) To t
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Sampling Fundamentals 163 2 (ii) Sq
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Sampling Fundamentals 165 (ii) Stan
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Sampling Fundamentals 167 σ s 1⋅
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Sampling Fundamentals 169 mean is c
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Sampling Fundamentals 171 08 . = ×
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Sampling Fundamentals 173 We now il
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Sampling Fundamentals 175 (v) Stand
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Sampling Fundamentals 177 where N =
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Sampling Fundamentals 179 Since $p
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Sampling Fundamentals 181 (i) Find
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Sampling Fundamentals 183 25. A tea
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Testing of Hypotheses I 185 Charact
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Testing of Hypotheses I 187 when th
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Testing of Hypotheses I 189 Mathema
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Testing of Hypotheses I 191 PROCEDU
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Testing of Hypotheses I 193 MEASURI
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Testing of Hypotheses I 195 We can
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Testing of Hypotheses I 197 HYPOTHE
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1 2 3 4 5 z OR X − X 1 2 2 2 p p
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1 2 3 4 5 z = p q 0 0 p$ − p$ F H
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Testing of Hypotheses I 203 to have
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Testing of Hypotheses I 205 S. No.
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Testing of Hypotheses I 207 S. No.
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Testing of Hypotheses-II 309 (ii) I
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Testing of Hypotheses-II 311 all po
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Testing of Hypotheses-II 313 CONCLU
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Multivariate Analysis Techniques 31
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Appendix Summary Chart: Showing the
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Interpretation and Report Writing 3
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The Computer: Its Role in Research
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Appendix 375 Appendix (Selected Sta
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Appendix 377 Table 2: Critical Valu
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Appendix 379 v 2 Table 4(a): Critic
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Appendix 381 Table 5: Values for Sp
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s l Min Max 0 1 2 3 4 5 6 7 8 9 10
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Appendix 385 Table 8: Cumulative Bi
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Appendix 387 Table 9: Selected Crit
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Appendix 389 1 2 3 4 5 6 40 0.368 0
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Selected References and Recommended
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Selected References and Recommended
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Author Index 395 Ackoff, R.L., 25,
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Author Index 397 Murphy, James L.,
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Subject Index 399 Content analysis,
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Subject Index 401 Research Plan, 53
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Research Methodology - Dr. Krishan K. Pandey

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