Anova
ANOVA is a study about the variances of two or more series. It was merely used in Agricultural Research. But now the technique finds useful applications in the field of both natural and social sciences.
What is ANOVA?
The variation present in a set of observations under study may be caused by known and unknown factors. Variations on account of unknown factors are known as random variations. In
analysis of variance, an attempt is made to separate the variation due to known factors from the variations due to unknown factors. For this, the total variance present in the whole set of observations is divided into a number of component variances caused by each set of independent factors.
If the component variances do not differ significantly it is concluded that the effects of all the factors are equal.
ANOVA Definition
ANOVA (Analysis of Variance is defined as a technique which analyses the variances of two or more comparable series for determining the significance of differences in their arithmetic means and for determining whether different samples under study are drawn from same population of not, with the help if the statistical technique called F test.
ANOVA Analysis
In ANOVA Analysis, we make study of variances to compare two or more series. The aim of the analysis is to determine whether the means of the different sample differ significantly or not. We use the F test in the analysis of variance. The null hypothesis assumed is
" all population means are equal" or
" all the samples belong to the same population having same variance".
ANOVA Assumptions
 Populations from which samples have been drawn are normally distributed.
 Populations from which samples have been drawn have same variance.
 The observations are non correlated random variables.
 Any observation is the sum of the effect of the factors influencing it
 The observations in the sample are randomly selected from the population.
 The random errors are normally distributed with mean 0 and variance $\sigma^{2}$.
One Way ANOVA
One way ANOVA is a method of testing the equality population means by analyzing sample variances. It is used with data categorized with one factor (treatment), which is a characteristic that allows us to distinguish the different populations from one another.
The term treatment is used because of the early applications of analysis of variance involved agricultural experiments in which different plots of farmland were treated with different fertilizers, seed types, insecticides etc. It is also known as single factor ANOVA. The following is the procedure for carrying out analysis of variance in one way classification: 
 The null hypothesis is taken as "all population means are equal"
 Compute the mean square between the samples MSC and MSE
 Calculate the F ratio
 Obtain the F table value with k – 1 and N – k degrees of freedom.
 If the calculated F value is less than the table value, we accept the hypothesis that the samples are equal.
Two Way ANOVA
The following is the procedure for carrying out analysis of variance in two way classification: 
 The null hypothesis is taken; as "all means of the columns are equal" Also another null hypothesis is "all means of the rows are equal".
 Compute the mean square between the samples MSC, MSE and MSR
 Calculate the F ratios
 Obtain the F table values.
 If the calculated F values are less than the table value, we accept the hypothesis that the samples are equal.
Three Way ANOVA
It is also known as factorial ANOVA. It is a statistical test which is used to determine the effect of three variables. A threeway ANOVA checks the effect of the independent variables on the expected outcome along with their relation with the outcomes. Random factors are considered to have no statistical influence on a data set, while systematic factors are considered to have statistical significance.
ANOVA Table
ANOVA table presents the various results obtained while carrying out the Analysis of Variance. A specimen of an ANOVA table in one way classification is given below.
Sources of Variation 
Sum of squares
 Degree of freedom 
Mean squares

F ratio 
P value

Between samples

SSC 
k  1 
MSC 


Within samples 
SSE 
N – k 
MSE 


Total 
SST 
N  1 



Similarly, A specimen of an ANOVA table in two way classification is given below.
Sources of Variation

Sum of squares 
Degree of freedom

Mean squares 
F ratio

P value 
Between columns 
SSC 
c  1 
MSC 


Between rows 
SSR 
r  1 
MSR 


Residual 
SSE 
(c – 1) $\times$ (r – 1)

MSE 


Total 
SST 
N  1 



ANOVA Table Explained
If the computed F value in the ANOVA table is less than the table F value or if the P value is greater than the alpha level of significance, then there is no evidence to reject the null hypothesis. That is we conclude that the all the population means are equal.
ANOVA F Formula
The formula to find the F statistic is given by
F = $\frac{Unexplained\ variance}{explained\ variance}$
ANOVA P Value
The P value is obtained by finding the area from the F distribution table, which represents the area to the right of the test statistic. If this value is greater than the level of significance, then we accept the null hypothesis and we conclude that "all the population means are same".