Anova is a statistical test which analyzes variance. It is helpful in making comparison of two or more means which enables a researcher to draw various results and predictions about two or more sets of data. Anova test includes one-way anova, two-way anova or multiple anova depending upon the type and arrangement of the data. One-way anova has the following test statistics:

Where,

$F$ = Anova Coefficient

$MST$ = Mean sum of squares due to treatment

$MSE$ = Mean sum of squares due to error.

Where,

$SST$ = Sum of squares due to treatment

$p$ = Total number of populations

$n$ = Total number of samples in a population.

Where,

$SSE$ = Sum of squares due to error

$S$ = Standard deviation of the samples

$N$ = Total number of observations.

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Anova Calculator | anova effect size calculator |

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Few problems based on Anova formula are given below: ### Solved Examples

**Question 1: **Following data is given about cricket teams of three countries:

Find Anova coefficient?

** Solution: **

Construct the following table:

n = 11

p = 3

N = 33

$\bar{x}$ = $\frac{60+50+70}{3}$ = 60

$SST=\sum n(x-\bar{x})^{2}$

$SST=11(60-60)^{2}+11(50-60)^{2}+11(70-60)^{2}$

= 2200

$MST$ = $\frac{SST}{p-1}$

$MST$ = $\frac{2200}{3-1}$

= 1100

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

SSE = 10*225 + 10*100 + 10*144

= 4690

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

$MSE$ = $\frac{4690}{33-3}$

MSE = 156.33

$F$ = $\frac{MST}{MSE}$

$F$ = $\frac{1100}{156.33}$

= 7.036

**Question 2: **The following data is given:

Calculate the Anova coefficient.

** Solution: **

Construct the following table:

p = 3

n = 5

N = 15

$\bar{x}$ = 16

$SST$ = $\sum n(x-\bar{x})^{2}$

$SST$ = $5(12-16)^{2}+5(16-16)^{2}+11(20-16)^{2}$

= 160

$MST$ = $\frac{SST}{p-1}$

$MST$ = $\frac{160}{3-1}$

= 80

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

SSE = 4*4 + 4*1 + 4*16

= 84

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

$MSE$ = $\frac{84}{15-3}$

MSE = 7

$F$ = $\frac{MST}{MSE}$

$F$ = $\frac{80}{7}$

= 11.429

Countries |
Number of Players |
Average Runs | Standard Deviations |

India | 11 |
60 |
15 |

New Zealand |
11 |
50 |
10 |

South Africa |
11 |
70 |
12 |

Find Anova coefficient?

Construct the following table:

Cricket Teams | n |
x |
S |
S^{2 } |

India |
11 |
60 |
15 |
225 |

New Zealand |
11 |
50 |
10 |
100 |

South Africa |
11 |
70 |
12 |
144 |

n = 11

p = 3

N = 33

$\bar{x}$ = $\frac{60+50+70}{3}$ = 60

$SST=\sum n(x-\bar{x})^{2}$

$SST=11(60-60)^{2}+11(50-60)^{2}+11(70-60)^{2}$

= 2200

$MST$ = $\frac{SST}{p-1}$

$MST$ = $\frac{2200}{3-1}$

= 1100

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

SSE = 10*225 + 10*100 + 10*144

= 4690

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

$MSE$ = $\frac{4690}{33-3}$

MSE = 156.33

$F$ = $\frac{MST}{MSE}$

$F$ = $\frac{1100}{156.33}$

= 7.036

Plant Name | Number of plants | Average Flowers | Standard Deviation |

Rose |
5 |
12 |
2 |

Marigold | 5 |
16 | 1 |

Lily |
5 |
20 |
4 |

Calculate the Anova coefficient.

Construct the following table:

Plant name | n |
x |
S |
S^{2 } |

Rose |
5 |
12 | 2 | 4 |

Marigold |
5 |
16 |
1 |
1 |

Lily |
5 |
20 | 4 |
16 |

p = 3

n = 5

N = 15

$\bar{x}$ = 16

$SST$ = $\sum n(x-\bar{x})^{2}$

$SST$ = $5(12-16)^{2}+5(16-16)^{2}+11(20-16)^{2}$

= 160

$MST$ = $\frac{SST}{p-1}$

$MST$ = $\frac{160}{3-1}$

= 80

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

SSE = 4*4 + 4*1 + 4*16

= 84

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

$MSE$ = $\frac{84}{15-3}$

MSE = 7

$F$ = $\frac{MST}{MSE}$

$F$ = $\frac{80}{7}$

= 11.429