Anova Formula
Anova Formula is used to compare the results of two or more experiments or samples. The mean of the results is compared with the help of the ANOVA Formula, and it helps us to conclude if we can accept the null hypothesis or the alternate hypothesis. A manufacturer wishing to compare two methods of manufacturing soap, a Principal of a college trying to check if there is any appreciable difference in the performance of students of two classes across an exam, a researcher willing to check the results of testing of a dug with two or more group of mice: All of these situations would need the Anova Formula.
What is Anova Formula?
Anova refers to the analysis of variance or the Ftest, and F is called the Anova Coefficient. Anova checks for variations of the mean between the samples, with respect to the variation of each value from the mean value, Anova oneway test is with respect to one variable across the samples, and Anova twoway test is with respect to two variables across the samples.
F = MSB/MSW
Mean Sum of Squares Between Samples(MSB) = \(\dfrac{(\bar x_1  \bar x)^2+ (\bar x_2  \bar x)^2 + (\bar x_3  \bar x)^2 + .......(\bar x_k  \bar x)^2}{k  1}\)
Mean Sum of Squares within the Samples(MSW) = \(\dfrac{( x_1  \bar x)^2+ ( x_2  \bar x)^2 + ( x_3  \bar x)^2 + .......(x_n  \bar x)^2}{n  k}\)
Here in the above formula we have:
 n = number of individual elements in the population
 k = number of samples taken from the population
 x_{1}, x_{2}, x_{3}, ........x_{n} are the individual elements of the population.
 \(\bar x_1\), \(\bar x_2 \), \(\bar x_3\), .....\(\bar x_k \) are the sample mean
Let us check a few examples to more clearly understand how to use the Anova formula.
Solved Examples on Anova Formula

Example 1: A bakery employs two different machines and the time is taken to make five loaves of bread from each machine is noted. Find the F score.
I 2 3 4 5 Baking Machine  I 2 mins 40 sec 2min 10 sec 2min 30 sec 2 min 25 sec 2min 15 sec Baking Machine  II 2min 50 sec 3min 15 sec 2min 35 sec 2min 45sec 3 min 5 sec Solution:
From the above information, we find the overall average time for each loaf of bread.
\(\bar x \)= (2min 40 sec + 2min 10 sec + 2min 30 sec + 2min25 sec + 2min 15 sec + 2min 50 sec + 3min 15 sec + 2min 35 sec + 2min 45sec + 3min 5 sec)/10 = 25min/10 = 2min 30 sec
Average of Baking Machine  I = \(\bar x_1 \) = (2min 40 sec + 2min 10 sec + 2min 30 sec + 2min25 sec + 2min 15 sec)/5 = 10min 30 sec/5 = 2min 6 sec
Average of Baking Machine  II = \(\bar x_2 \) = (2min 50 sec + 3min 15 sec + 2min 35 sec + 2min 45sec + 3min 5 sec)/5 = 14min 30 sec/5 = 2min 54 sec
Mean Sum of Squares Between Samples(MSB) = \(\dfrac{(\bar x_1  \bar x)^2+ (\bar x_2  \bar x)^2 + (\bar x_3  \bar x)^2 + .......(\bar x_k  \bar x)^2}{k  1}\)
= [(2min 6 sec  2min 30 sec)^{2} + (2min 54 sec  2min 30 sec)^{2}]/(2  1)
= (24^{2} + 24^{2} )/1
= 576 + 576
= 1152
Mean Sum of Squares within the Samples(MSW) = \(\dfrac{( x_1  \bar x)^2+ ( x_2  \bar x)^2 + ( x_3  \bar x)^2 + .......(x_n  \bar x)^2}{n  k}\)
= {(2min 40 sec  2min 30 sec)^{2} + (2min 10 sec  2min 30 sec)^{2} +( 2min 30 sec  2min 30 sec)^{2} + (2min25 sec  2min 30 sec)^{2} + (2min 15 sec  2min 30 sec)^{2} + (2min 50 sec  2min 30 sec)^{2} + (3min 15 sec  2min 30 sec)^{2} + (2min 35 sec  2min 30 sec)^{2} + (2min 45sec  2min 30 sec)^{2} + (3min 5 sec  2min 30 sec)^{2}}/(10  2)
= {10^{2} + 20^{2} + 0^{2} + 5^{2} + 15^{2} + 20^{2} + 45^{2} + 5^{2} + 15^{2} + 35^{2} )/8
= (100 + 400 + 0 + 25 + 225 + 400 + 2025 + 25 + 225 + 1225)/8
= (4650)/8 = 581.25
F Score = MSB/MSW = 1152/581.25 = 1.982
Answer: F Score = 1.082