One way Analysis of Variance ANOVA

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Definition: One way Analysis of Variance ANOVA

One way Analysis of Variance (ANOVA) is a statistical technique commonly used to compare means of two or more samples using variance. Here the null hypothesis is that the all population means are equal, while the alternative hypothesis is that at least one mean is different from the rest.


It is assumed that the populations are normally or approximately normally distributed, the samples are all independent and that the variances of the populations are equal. The summary table is obtained which gives the F- statistic:

SS

Df

MS

F

Between

SS(B)

k-1

SS(B)
-----------
k-1

MS(B)
--------------
MS(W)

Within

SS(W)

N-k

SS(W)
-----------
N-k

.

Total

SS(W) + SS(B)

N-1

.

.

Steps:

  • Grand mean of a set of samples: The total of all the data values divided by the total sample size.

, where N is total sample size.

  • The total variation: The sum of the squares of the differences of each mean with the grand mean.

 

  • Between Group Variation (SSB): Sum of squares between groups gives variation due to interaction between the samples.

 

  • Mean square between groups (MSB): Let there are k samples, hence k-1 degrees of freedom. Then, MSB is:

 

 

  • Within Group Variation (SSW): This gives differences within individual samples.

 

 

  • Mean square within groups (MSW): Total degrees of freedom are N-k. Then MSW is:

 

 

  • F-test statistic: This is found by dividing the between group variance (MSB) by the within group variance  (MSW)

 

  

If value of F is greater than 1, it signifies a greater chance of rejection of the null hypothesis.


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