# Population Samples

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Introduction

Analysis of variance is a statistical tool used to test the hypotheses postulated about means drawn from different population samples. It simply detects whether the means between the two or more population samples are equal. This is done with assumptions that the involved populations are under a normal distribution (Bhattacharya & Johnson, 2009). The test identifies whether there is significant variable relations between populations. Question-How can ANOVA be compared to regression?

Variance portioning is a method used in the analysis of variance to determine the factors that account for variances within various groups of samples within a population. This is used because any statistical analysis process has variations in mean within groups and among the groups that constitute the whole population. It is a refinement of the process of NOVA and it tells the potential sources of various differences in variance within and among groups which make the population. The partitioning is meant to account for within group and between group variability (Bhattacharya & Johnson, 2009).

Question: How does partitioning help in identifying sources of variability? The f-statistic is random variable alternatively termed as the f-value and it has an f-distribution. The computation of an f-statistic takes the following form: a random sample of size n1 is selected from a population that is normally distributed and with a standard deviation of Ïƒ1. Another independent sample that is random is also selected. The samples size of this sample can be denoted by n2. This selection from a normally distributed population too and having a standard deviation of Ïƒ2. Finally, the f-statistic can be determined by this ratio s12/Ïƒ12 and s22/Ïƒ22 (Bhattacharya & Johnson, 2009).

The three major assumptions of ANOVA are independence, normal distribution and homogeneity of variances. Questions-What are the consequences of non-normality in an ANOVA test? How can one handle a case where non-normality has been detected so as to have correct results?

One-way ANOVA between groups is used to analyze the variability that exists between various groups with relation to one factor. This can be exemplified by an experiment conducted to test the effect of temperature on various groups of resistors heated in different ovens placed at different temperature levels. The variability is tested with regard to resistance changes between the groups. This is a one-way ANOVA because there is only a single factor that is varied-the temperature. The temperature is the main source of variation (Bhattacharya & Johnson, 2009).

One-way ANOVA within Groups is used to analyze the variability that exists within a sampled group with relation to one factor. This can be exemplified by an experiment conducted to test the effect of temperature on various components such as a sample of resistors heated in one oven to a uniform temperature level. The variability in resistance among the resistors in the group is tested with regard to resistance changes between within each resistor that makes up the group. This is a one-way ANOVA within a group because there is only a single factor that varies- the resistor. Each resistors difference is the source of variation (Bhattacharya & Johnson, 2009).

Two-way ANOVA between groups is akin to one-way ANOVA between groups, but with only difference being that a second factor or source of variation is introduced. In the examples offered on the resistors above, a second factor introduced would be the position of the resistor in the oven or the length of time that the resistor was heated in the oven. This second factor introduces a new source of variation (Bhattacharya & Johnson, 2009).

In some randomized and controlled experiments measurements are made prior to and after a test. The best analysis for data from such experiments would be the comparison of the results from pre-test and post-test activities. In such cases the post-test results are the findings, design factor is the treatment and the pre-test is the covariate. Thus giving us the ANCOVA tests (Analysis of covariance).  It takes a generally linear model and is actually a merger of regression and ANOVA for variables that are continuous (Bhattacharya & Johnson, 2009).

References

Bhattacharya, K.G. and Johnson, A. R. (2009),. Statistics: Principles and Methods, 6th edition, John Wiley and Sons
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