Our innovative ANOVA calculator helps to quickly get the one-way and two-way ANOVA tables for up to 10 groups. These tables include all the relevant information from the observed data including the sum of squares, mean squares, degree of freedom, and test statistics.

## What is an ANOVA?

**“ANOVA stands for Analysis of Variance used to compare the means of three or more groups”.**

The one way ANOVA calculator functions by partitioning the total variance of sample data into two components: variance between groups and variance within groups.

It is distributed into parts:

- Systematic Factor
- Random Factor

### Types of ANOVA Test Statistic:

#### 1. One Way ANOVA

“This is used to compare the means of three or more groups based on a single independent variable”.

**Characteristics:**

- There is only one number of independent variables
- All groups must be independent and normally distributed
- The variances of the groups must be equal.

**Example:**

Comparing the mean height of men and women.

#### 2. Two Way ANOVA

“This is used to compare the means of three or more groups based on two independent variables”.

**Characteristics:**

- There are two numbers of independent variables
- All groups must be independent and normally distributed
- The variances of the groups must be equal within each level of the two independent variables.

**Example:**

Comparing the mean test scores of students who received different types of instruction and different levels of tutoring.

## Variance Analysis Formula:

The ANOVA test calculator uses the following formula to summarize the various components.

**F = MSB / MSW **

Where:

- F is used to test the equality of means among multiple groups.
- MSB is the mean square between groups, calculated as SSB / dfB
- MSW is the mean square within groups, calculated as SSW / df

The analysis of variance formula table summarizes the components to evaluate the F-statistic. It typically consists of the following components:

Source |
Sum of Squares |
Mean Squares |
Degrees of Freedom |
F Statistics |

Between Groups |
SSB = ∑i = 1k ni (X̄i - X̄)2 | MSG = SSG / (k - 1) | k - 1 | F = MSB/MSW |

Within Groups |
SSW = ∑i = 1K (ni – 1) Si2 | MSE = SSE / (n - k) | n - k | |

Total |
SST = SSB + SSW | Sample Variance = SS | n - 1 |

**Variance Between Groups: **The variance between groups is a measure of how different the means of the groups are.

**Variance Within Groups:** The variance within groups is a measure of how much the data points within each group vary around the group mean.

## How to do ANOVA?

The innovative ANOVA table calculator utilizes the test to determine the influence of independent variables on the dependent variable. There are a couple of steps that are important to consider as follows:

### Example of Anova in Use

A doctor wants to know to know the difference in mean effectiveness of three different drugs for treatment. The doctor assigned some random numbers to the patients to measure the mean effectiveness of three drugs.

Group # 1: 11, 3, 4, 7, 8

Group # 2: 0, 1, 12, 6, 3

Group # 3: 6, 13, 8, 7, 5

#### Solution:

Group 1 | Group 2 | Group 3 |
---|---|---|

11 | 0 | 6 |

3 | 1 | 13 |

4 | 12 | 8 |

7 | 6 | 7 |

8 | 3 | 5 |

∑Group 1 = 33 | ∑Group 2 = 22 | ∑Group 3 = 39 |

(Group 1)² | (Group 2)² | (Group 3)² |
---|---|---|

121 | 0 | 36 |

9 | 1 | 169 |

16 | 144 | 64 |

49 | 36 | 49 |

64 | 9 | 25 |

∑(Group1)² = 259 | ∑(Group2)² = 190 | ∑(Group3)² = 343 |

Data Summary | ||||||
---|---|---|---|---|---|---|

Groups | N | ∑x | Mean | ∑x² | Std. Dev. | Std. Error |

Group 1 | 5 | 33 | 6.6 | 259 | 3.2094 | 1.4353 |

Group 2 | 5 | 22 | 4.4 | 190 | 4.827 | 2.1587 |

Group 3 | 5 | 39 | 7.8 | 343 | 3.1145 | 1.3928 |

Total | 15 | 94 | 6.2666666666667 | 792 |

ANOVA Summary | |||||
---|---|---|---|---|---|

Source | Degrees of Freedom (DF) | Sum of Squares (SS) | Mean Square (MS) | F-Stat | P-Value |

Between Groups | 2 | 29.7333 | 14.8667 | 1.03 | |

Within Groups | 12 | 173.2 | 14.4333 | ||

Total | 14 | 202.9333 |

Step:1 - Sum of Squares Between Groups

$$ SS_B = \sum^k_{i=1} n_i(\bar x_i - \bar x)^2 $$

$$ SS_B = 5 * (6.6 - 6.2666666666667)^2 + 5 * (4.4 - 6.2666666666667)^2 + 5 * (7.8 - 6.2666666666667)^2 $$

$$ SS_B = 29.7333 $$

#### Step:2 - Sum of Squares Within Groups

$$ SS_W = \sum^k_{i=1} (n_i − 1)S_i^{\space 2} $$

$$ SS_W = (5 - 1) * (3.2094)^2 + (5 - 1) * (5.7966)^2 + 5 * (7.8 - 6.2666666666667)^2 $$

$$ SS_W = 173.2 $$

#### Step:3 - Total Sum of Squares

$$ SS_T = SS_B + SS_W $$

$$ SS_T = 29.7333 + 173.2 $$

$$ SS_T = 202.9333 $$

#### Step:4 - Mean Square Between Groups

$$ MS_B = \dfrac{SS_B}{k - 1} $$

$$ MS_B = \dfrac{29.7333}{3 - 1} $$

$$ MS_B = \dfrac{29.7333}{2} $$

$$ MS_B = 14.8667 $$

#### Step:5 - Mean Square Within Groups

$$ MS_W = \dfrac{SS_W}{N - k} $$

$$ MS_W = \dfrac{173.2}{15 - 3} $$

$$ MS_W = \dfrac{173.2}{12} $$

$$ MS_W = 14.4333 $$

#### Step:6 - One Way ANOVA Test Statistic

$$ F = \dfrac{MS_B}{MS_W} $$

$$ F = \dfrac{14.8667}{14.4333} $$

$$ F = 1.03 $$

- If F Test Result > Critical Value (Value in F-table), Reject null hypothesis
- If F Test Result < Critical Value (Value in F-table), Accept null hypothesis

## How Anova Calculator Functions?

Our online two way ANOVA calculator meets the data recovery where the hypothesis becomes insights so it functions well when you come with the below values:

### Begin Calculations With:

- Pick the method by which you want to get analysis
- Put the values for data sequences and you also add or del the treatment

### Calculation Outcomes:

**Test Statistics:**It is associated with the difference in mean among the various groups.**P-Value:**it shows the statistical significance of the difference between group means.**ANOVA Table Summary:**this table shows the various sources of variation in the data**Sum of Squares:**The ANOVA calculator shows the sum of the square value for both between and within the group's variation.**Mean Square:**for analysis of the variance it is essential to display the mean square values.

## Sources:

**Wikipedia:** Analysis of variance, Classes of models, Assumptions, Example, For a single factor.