## What Is One-Way ANOVA?

**One-way analysis of variance** (abbreviated **one-way ANOVA**) is a technique that can be used to compare means of two or more samples (using the F distribution). This technique can be used only for numerical response data, the “Y”, usually one variable, and numerical or (usually) categorical input data, the “X”, always one variable, hence “one-way”.

The ANOVA tests the null hypothesis, which states that samples in all groups are drawn from populations with the same mean values. To do this, two estimates are made of the population variance.

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

- Response variable residuals are normally distributed (or approximately normally distributed).
- Variances of populations are equal.
- Responses for a given group are independent and identically distributed normal random variables (not a simple random sample (SRS)).

If data are ordinal, a non-parametric alternative to this test should be used such as *Kruskal–Wallis one-way analysis of variance*. If the variances are not known to be equal, a generalization of 2-sample *Welch’s t-test *can be used

The ANOVA produces an F-statistic, the ratio of the variance calculated among the means to the variance within the samples. If the group means are drawn from populations with the same mean values, the variance between the group means should be lower than the variance of the samples, following the *central limit theorem*. A higher ratio therefore implies that the samples were drawn from populations with different mean values.

### What You Need To Know About One Way ANOVA

- One-way ANOVA is a hypothesis test that allows one to make comparisons between the means of three or more group of data.
- A one-way ANOVA only involves one factor or independent variable.
- In the one-way ANOVA, the factor or independent variable analyzed has three or more categorical groups.
- A one-way ANOVA is primarily designed to enable the equality testing between three or more means.
- One-way ANOVA need to satisfy only two principles of design of experiments i.e replication and randomization.
- In one-way ANOVA, the number of observations need not be same in each group.

## What Is Two-Way ANOVA?

The two-way analysis of variance (ANOVA) test is an extension of the one-way ANOVA test that examines the influence of different categorical independent variables on one dependent variable. While the one-way ANOVA measures the significant effect of one independent variable (IV), the two-way ANOVA is used when there is more than one Independent variable and multiple observations for each Independent variable. The two-way ANOVA can not only determine the main effect of contributions of each Independent variable but also identifies if there is a significant interaction effect between the Independent variables.

### Assumptions of the Two-Way ANOVA

As with other parametric tests, we make the following assumptions when using two-way ANOVA:

- The populations from which the samples are obtained must be normally distributed.
- Sampling is done correctly. Observations for within and between groups must be independent.
- The variances among populations must be equal (homoscedastic).
- Data are interval or nominal.

### What You Need To Know About One Way ANOVA

- Two-way ANOVA is a hypothesis test that allows one to make comparisons between the means of three or more groups of data, where two independent variables are considered.
- A two-way ANOVA involves two independent variables.
- A two-way ANOVA compares multiple groups of two factors.
- A two-way ANOVA is designed to assess the interrelationship of two independent variables on a dependent variable.
- Two-way ANOVA meets all three principles of design of experiments which are replication, randomization and local control.
- In two-way ANOVA, the number of observations need to be same in each group.

**Also Read:** *Difference Between Parameters And Statistics *

## Difference Between One Way And Two Way ANOVA In Tabular Form

BASIS OF COMPARISON |
ONE-WAY ANOVA |
TWO-WAY ANOVA |

Description | One-way ANOVA is a hypothesis test that allows one to make comparisons between the means of three or more group of data. | Two-way ANOVA is a hypothesis test that allows one to make comparisons between the means of three or more groups of data, where two independent variables are considered. |

Number of Independent
Variables | A one-way ANOVA only involves one factor or independent variable. | A two-way ANOVA involves two independent variables. |

Number of Groups Of Sample | In the one-way ANOVA, the factor or independent variable analyzed has three or more categorical groups. | A two-way ANOVA compares multiple groups of two factors. |

Use | A one-way ANOVA is primarily designed to enable the equality testing between three or more means. | A two-way ANOVA is designed to assess the interrelationship of two independent variables on a dependent variable. |

Principles of Design | One-way ANOVA need to satisfy only two principles of design of experiments i.e replication and randomization. | Two-way ANOVA meets all three principles of design of experiments which are replication, randomization and local control. |

Number Of Observations | In one-way ANOVA, the number of observations need not be same in each group. | In two-way ANOVA, the number of observations need to be same in each group. |