# ANOVA Calculator

One-Way ANOVA Calculator and Tukey HSD

## Enter sample data directly

**Header**: You may change groups' names to the real names.

**Data**: When entering data, press Enter or , (comma) after each value.

You may paste full column from excel.

The calculator ignores empty cells or non-numeric cells.

## ANOVA

### What is the ANOVA?

The ANOVA test checks if the difference between the averages of two or more groups is significant, using sample data.

ANOVA is usually used when there are at least three groups since for two groups, the *two-tailed* pooled variance t-test and the *right-tailed* ANOVA test have the same result.

The basic ANOVA test contains only one categorical value, one-way ANOVA. For example, if you compare the performence of three schools, the categorical variable is school, and the possible values of the categorical variable are School-A, School-B, School-C. There are more complex ANOVA tests that contain two categorical variables ( Two-way ANOVA calculator ), or more. When performing a one-way ANOVA test, we try to determine if the difference between the averages reflects a real difference between the groups, or is due to the random noise inside each group.

The F statistic represents the ratio of the variance between the groups and the variance inside the groups. Unlike many other statistic tests, the smaller the F statistic the more likely the averages are equal.**Example**: Compare four fertilizers used in four fields

H_{0}: The average weight of crops per square meter is equal in all fields.

H_{1}: At least one field yields a different average per square meter.

**Right-tailed**, for ANOVA test you can use only the right tail. Why?

_{0}: μ

_{1}= ... = μ

_{k}

_{1}: not(μ

_{1}= ... = μ

_{k})

## Assumptions

- Independent samples
- Normal distribution of the analyzed population
- Equal standard deviation, σ
_{1}=σ_{2}=...=σ_{k}

The assumption is more important when the groups' sizes not similar

## Required Sample Data

- Sample data from all compared groups

## Parameters

**k**- Number of groups.**n**- Sample side of group i_{i}**n**- Overall sample side, includes all the groups (Σn_{i}, i=1 to k)**x̄**- Average of group i._{i}**x̄**- Overall average (Σx_{i,j}/ n, i=1 to k, j=1 to n_{i})**S**- Standard deviation of group i_{i}

## Results calculations

Source | Degrees of Freedom | Sum of Squares | Mean Square | F statistic | p-value |
---|---|---|---|---|---|

Groups(between groups) | k - 1 | MSG = SSG / (k - 1) | F = MSG / MSE | P(x > F) | |

Error(within groups) | n - k | MSE = SSE / (n - k) | |||

Total | n - 1 | SS(total) = SSG + SSE | Sample Variance = SS(total) / (n - 1) |