# Anova Formula

Analysis of variance, or ANOVA, is strong statistical technique that is used to show difference between two or more means or components through significance tests. It also shows us a way to make multiple comparisons of several population means. The Anova test is performed by comparing two types of variation, the variation between the sample means, as well as the variation within each of the samples. Below mentioned formula represents one way Anova test statistics:

Where,

F = Anova Coefficient

MST = Mean sum of squares due to treatment

MSE = Mean sum of squares due to error.

F = Anova Coefficient

MST = Mean sum of squares due to treatment

MSE = Mean sum of squares due to error.

Formula for MST is given below:

Where,

SST = Sum of squares due to treatment

p = Total number of populations

n = The total number of samples in a population.

SST = Sum of squares due to treatment

p = Total number of populations

n = The total number of samples in a population.

Formula for MSE is given below:Where,

SSE = Sum of squares due to error

S = Standard deviation of the samples

N = Total number of observations.

SSE = Sum of squares due to error

S = Standard deviation of the samples

N = Total number of observations.

### Solved Examples

**For reference please go through the below-mentioned example:**

**Question 1: **The following data is given:

Types of Animals | Number of animals | Average Domestic animals | Standard Deviation |

Dogs | 5 | 12 | 2 |

Cats | 5 | 16 | 1 |

Hamsters | 5 | 20 | 4 |

Calculate the Anova coefficient.

**Solution:**

Construct the following table:

Animal name | n | x | S | $S^{2}$ |

Dogs | 5 | 12 | 2 | 4 |

Cats | 5 | 16 | 1 | 1 |

Hamster | 5 | 20 | 4 | 16 |

p = 3

n = 5

N = 15

x̄ = 16

SST = ∑n (x−x̄)^{2}

SST= 5(12−16)^{2}+5(16−16)^{2}+11(20−16)^{2
}= 160

MST = $\frac{SST}{p−1}$

MST = $\frac{160}{3−1}$

MST = 80

SSE = ∑ (n−1)$S^{2}$

SSE = 4$\times$4 + 4$\times$1 + 4$\times$16

SSE = 84

MSE= $\frac{SSE}{N−p}$

MSE=$\frac{8415}{38415−3}$

MSE = 7

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

F = $\frac{80}{84}$

F = 11.429