## Introduction to One-Way ANOVA

Many times we need to compare more than two groups' means. Instead of multiple pairwise t-tests (which result in high Type I error), ANOVA is one procedure that tests if there is at least one mean that cannot be considered as coming from the same population than other means.

The One-Way ANOVA technique consists in decomposing the total variation of data into: (a) the internal or “natural” or “within” groups variation, and (b) the “between” groups variation in such a way that when these two types of variation are compared, it’s possible to determine if there is a statistically significant difference between their means being analyzed. One of the principle advantages of the ANOVA technique is that the number of observations need not be the same in each group.^{1}

The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:

- The populations from which the samples were obtained are normally or approximately normally distributed.
- The samples are independent.
- The variances of the populations are equal.

The null hypothesis states that all groups come from the same population (or from populations that have the same mean).

\[ H_0 : \text{All means are equal}. \]

The alternate hypothesis states that at least one of the groups come from a population with different mean.

\[ H_1 : \text{Not all means are equal}. \]

## Example

In a company that produces alloys for aircrafts, four prototypes have been produced to determine the effects of a fixed high altitude on their density. Several tests were performed and the following data have been obtained. Perform an ANOVA to determine the effect of high altitude on the density of these four prototypes.

PROTOTYPE | DENSITY | ||
---|---|---|---|

1 | 4.5 | 7.8 | 6.7 |

2 | 3.8 | 5.6 | 9.1 |

3 | 7.6 | 4.6 | 7.6 |

4 | 3.5 | 3.5 | 4.8 |

The equality of density for the four prototypes is not rejected.

## Using Excel

Under Data, Data Analysis, select ANOVA-SINGLE FACTOR and enter the input range (select all the table) and hit OK.

Since P-value is less than alpha (5%), the null hypothesis is rejected.

Or since F = 22.07 is greater than F_{crit} = 5.14, the null hypothesis is rejected.

Although it’s vital to check model assumptions, these are not going to be covered here.

## Running a One-Way ANOVA test

## Do it yourself

Step 1: Go to "Attachments" on the bottom of this page and dowload the Excel spreadsheet.

Step 2: Follow the steps teached in the video above and solve the exercises in the Excel spredsheet dowloaded.

Step 3: Answer the questions in the Interactive Content below.

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