Analysis of Variance: One Way and Two Way Classifications

Analysis of Variance

Analysis of Variance (ANOVA) is a parametric statistical technique used to compare datasets. This technique was invented by R.A. Fisher, and is thus often referred to as Fisher’s ANOVA, as well. It is similar in application to techniques such as t-test and z-test, in that it is used to compare means and the relative variance between them. However, analysis of variance (ANOVA) is best applied where more than 2 populations or samples are meant to be compared.

Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the “variation” among and between groups) used to analyze the differences among group means in a sample. ANOVA was developed by statistician and evolutionary biologist Ronald Fisher. In the ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether the population means of several groups are equal, and therefore generalizes the t-test to more than two groups. ANOVA is useful for comparing (testing) three or more group means for statistical significance. It is conceptually similar to multiple two-sample t-tests, but is more conservative, resulting in fewer type I errors, and is therefore suited to a wide range of practical problems.

The Formula for ANOVA

The following formula represents a one-way ANOVA test:

ANOVA formula



F = ANOVA coefficient

MST = Mean sum of squares due to treatment

MSE = Mean sum of squares due to error.

Example of How to Use ANOVA

A researcher might, for example, test students from multiple colleges to see if students from one of the colleges consistently outperform students from the other schools. In a business application, an R&D researcher might test two different processes of creating a product to see if one process is better than the other in terms of cost efficiency.

The type of ANOVA run depends on a number of factors. It is applied when data needs to be experimental. Analysis of variance is employed if there is no access to statistical software resulting in computing ANOVA by hand. It is simple to use and best suited for small samples. With many experimental designs, the sample sizes have to be the same for the various factor level combinations.

Analysis of variances is helpful for testing three or more variables. It is similar to multiple two-sample t-tests. However, it results in fewer type I errors and is appropriate for a range of issues. ANOVA groups differences by comparing the means of each group, and includes spreading out the variance into diverse sources. It is employed with subjects, test groups, between groups and within groups.

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A one-way ANOVA is a type of statistical test that compares the variance in the group means within a sample whilst considering only one independent variable or factor. It is a hypothesis-based test, meaning that it aims to evaluate multiple mutually exclusive theories about our data. Before we can generate a hypothesis, we need to have a question about our data that we want an answer to. For example, adventurous researchers studying a population of walruses might ask “Do our walruses weigh more in early or late mating season?” Here, the independent variable or factor (the two terms mean the same thing) is “month of mating season”. In an ANOVA, our independent variables are organized in categorical groups. For example, if the researchers looked at walrus weight in December, January, February and March, there would be four months analyzed, and therefore four groups to the analysis.

A one-way ANOVA compares three or more than three categorical groups to establish whether there is a difference between them. Within each group there should be three or more observations (here, this means walruses), and the means of the samples are compared.

Hypotheses of One-Way ANOVA

In a one-way ANOVA there are two possible hypotheses.

  • The null hypothesis (H0) is that there is no difference between the groups and equality between means. (Walruses weigh the same in different months)
  • The alternative hypothesis (H1) is that there is a difference between the means and groups. (Walruses have different weights in different months)

Assumptions of One-Way ANOVA

  • Normality – That each sample is taken from a normally distributed population
  • Sample independence – that each sample has been drawn independently of the other samples
  • Variance Equality – That the variance of data in the different groups should be the same
  • Your dependent variable – here, “weight”, should be continuous – that is, measured on a scale which can be subdivided using increments (i.e. grams, milligrams)


Two-way ANOVA is, like a one-way ANOVA, a hypothesis-based test. However, in the two-way ANOVA each sample is defined in two ways, and resultingly put into two categorical groups. Thinking again of our walruses, researchers might use a two-way ANOVA if their question is: “Are walruses heavier in early or late mating season and does that depend on the gender of the walrus?” In this example, both “month in mating season” and “gender of walrus” are factors – meaning in total, there are two factors.  Once again, each factor’s number of groups must be considered – for “gender” there will only two groups “male” and “female”.

The two-way ANOVA therefore examines the effect of two factors (month and gender) on a dependent variable – in this case weight, and also examines whether the two factors affect each other to influence the continuous variable.

Assumptions of Two-Way ANOVA

  • Your dependent variable – Here, “weight”, should be continuous – that is, measured on a scale which can be subdivided using increments (i.e. grams, milligrams)
  • Your two independent variables – Here, “month” and “gender”, should be in categorical, independent groups.
  • Sample independence – That each sample has been drawn independently of the other samples.
  • Variance Equality – That the variance of data in the different groups should be the same.
  • Normality – That each sample is taken from a normally distributed population.

Hypotheses of Two-Way ANOVA

Because the two-way ANOVA consider the effect of two categorical factors, and the effect of the categorical factors on each other, there are three pairs of null or alternative hypotheses for the two-way ANOVA. Here, we present them for our walrus experiment, where month of mating season and gender are the two independent variables.

H0: The means of all month groups are equal

H1: The mean of at least one month group is different.

H0: The means of the gender groups are equal.

H1: The means of the gender groups are different.

H0: There is no interaction between the month and gender.

H1: There is interaction between the month and gender.

Summary: Differences between One-Way and Two-Way ANOVA

The key differences between one-way and two-way ANOVA are summarized clearly below.

  1. 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.
  2. A one-way ANOVA only involves one factor or independent variable, whereas there are two independent variables in a two-way ANOVA.
  3. In a one-way ANOVA, the one factor or independent variable analyzed has three or more categorical groups. A two-way ANOVA instead compares multiple groups of two factors.
  4. One-way ANOVA need to satisfy only two principles of design of experiments, i.e. replication and randomization. As opposed to Two-way ANOVA, which meets all three principles of design of experiments which are replication, randomization, and local control.

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