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# Anova Calculator

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ANOVA is one-way analysis of variance that is used to determine any differences between the means of two or more groups. ANOVA calculator gives  statistical test that calculates SST, MST, SSE, MSE and Anova Coefficient. Plug in the number of samples(n), mean(x) of observation and standard deviation(S) to get those values.

## Steps

Step 1 : From the problem, note the values and create the table which contains $n$, $x$, $S$ and $S^{2}$.

Step 2 : Find $\bar{x}$ and Sum of square due to treatment(SST) by using the formula,
SST = $\sum n(x-\bar{x})^{2}$

Where, $\bar{x}$ are the mean of all the observation.

Step 3 : Find the mean sum of square due to tretment (MST), Sum of square due to error(SSE) and Mean sum of square due to error(MSE) by using the formula,
MST = $\frac{SST}{p-1}$

SSE = $\sum(n-1)S^2$

MSE = $\frac{SSE}{N-p}$

Where,
p = total number of populations ,
n = number of samples,
N = total number of observations,

Step 4 : At last find the Anova coefficient(F) by using the formula,
F = $\frac{MST}{MSE}$

## Problems

Based on ANOVA, some of the problems are given below.

### Solved Examples

Question 1: Match was played between three groups and scores were noted. Following data is given about football teams of three countries:

 Countries Number of Players Scores Standard Deviations India 11 6 20 New Zealand 11 5 18 South Africa 11 7 22

Find Anova coefficient.

Solution:

Given data :
 Countries Number of Players Scores Standard Deviations India 11 6 20 New Zealand 11 5 18 South Africa 11 7 22

n = 11
p = 3
N = 33

$\bar{x}$ = $\frac{6+5+7}{3}$ = 6

Sum of square due to treatment, $SST=\sum n(x-\bar{x})^{2}$
$SST=11(6-6)^{2}+11(5-6)^{2}+11(7-6)^{2}$
= 22

Sum of square due to tretment, $MST$ = $\frac{SST}{p-1}$
$MST$ = $\frac{22}{3-1}$
= 11

Sum of square due to error, $SSE=\sum (n-1)S^{2}$
SSE = 10 $\times$ 400 + 10 $\times$ 324 + 10 $\times$ 484
= 12080

Mean sum of square due to error, $MSE$ = $\frac{SSE}{N-p}$
$MSE$ = $\frac{12080}{33-3}$
MSE = 402.667

Anova coefficient, $F$ = $\frac{MST}{MSE}$
$F$ = $\frac{11}{402.667}$
= 0.027

Question 2: Work was assigned to 3 group of employees in a railway factory. Following are the data that contains different potential.

 Group Number of employees Potential Standard Deviations 1 6 10 15 2 6 14 11 3 6 11 20

Find Anova coefficient.

Solution:

Given data :
 Group Number of employees Potential Standard Deviations 1 6 10 15 2 6 14 11 3 6 11 20

n = 6
p = 3
N = 18

$\bar{x}$ = $\frac{10+14+11}{3}$ = 11.66

Sum of square due to treatment, $SST=\sum n(x-\bar{x})^{2}$
$SST=6(10-11.66)^{2}+6(14-11.66)^{2}+6(11-11.66)^{2}$
= 52

Sum of square due to tretment, $MST$ = $\frac{SST}{p-1}$
$MST$ = $\frac{52}{3-1}$
= 26

Sum of square due to error, $SSE=\sum (n-1)S^{2}$
SSE = 5 $\times$ 225 + 5 $\times$ 121 + 5 $\times$ 400
= 3730

Mean sum of square due to error, $MSE$ = $\frac{SSE}{N-p}$
$MSE$ = $\frac{3730}{18-3}$
MSE = 248.667

Anova coefficient, $F$ = $\frac{MST}{MSE}$
$F$ = $\frac{26}{248.667}$
= 0.105