# One-Way ANOVA Calculator

This One-way ANOVA Test Calculator helps you to quickly and easily produce a one-way analysis of variance (ANOVA) table that includes all relevant information from the observation data set including sums of squares, mean squares, degrees of freedom, F- and P-values.

One-Way ANOVA Calculator

Related

## What is One-Way ANOVA

One-Way ANOVA (Analysis of Variance) is a statistical technique used to compare the means of three or more groups or treatments. It assesses whether there are any statistically significant differences between the means of the groups based on the variances within and between the groups.

Here's an overview of how One-Way ANOVA works:

1. Formulate Hypotheses: Begin by stating the null hypothesis (H0) and alternative hypothesis (Ha). The null hypothesis assumes that there are no significant differences between the means of the groups, while the alternative hypothesis suggests that at least one group mean differs significantly from the others.

2. Collect Data: Gather the data for each group or treatment under consideration. There should be one independent variable with three or more categories and a corresponding dependent variable.

3. Calculate Variability: Determine the variability between the groups (explained variability) and within the groups (unexplained variability). The goal is to determine if the differences in the means of the groups are larger than the differences within each group.

4. Compute F-statistic: Calculate the F-statistic, which is the ratio of the explained variability to the unexplained variability. The F-statistic follows an F-distribution under the assumption that the null hypothesis is true.

5. Determine Significance: Compare the calculated F-statistic with the critical F-value from the F-distribution table or use software to obtain the p-value associated with the F-statistic. If the p-value is smaller than the chosen significance level (usually 0.05), then the null hypothesis is rejected, indicating that at least one group mean differs significantly from the others.

6. Post-hoc Analysis: If the overall test is statistically significant, perform post-hoc tests such as Tukey's Honestly Significant Difference (HSD), Bonferroni correction, or others, to identify which specific groups differ significantly from each other.

One-Way ANOVA assumes several assumptions, including normality (the data should follow a normal distribution within each group), independence (observations within each group should be independent), and homogeneity of variances (variances within each group should be equal).

One-Way ANOVA is widely used in various fields, including social sciences, biology, business, and manufacturing, to compare means across multiple groups or treatments.

If you provide me with the specific data and groups, I can help you further by performing a One-Way ANOVA analysis or answering any related questions.

## ANOVA Formulas

One-Way ANOVA Table
Source Degrees of Freedom
DF
Sum of Squares
SS
Mean Square
MS
F-Stat P-Value
Between Groups k − 1 SSB MSB = SSB / (k − 1) F = MSB / MSW Right tail of
F(k-1,N-k)
Within Groups N − k SSW MSW = SSW / (N − k)
Total: N − 1 SST = SSB+SSW

Between Groups Degrees of Freedom: DF = k − 1 , where k is the number of groups

Within Groups Degrees of Freedom: DF = N − k , where N is the total number of subjects

Total Degrees of Freedom: DF = N − 1

Sum of Squares Between Groups: SSB = Ski=1ni (xix)2 , where ni is the number of subjects in the i-th group

Sum of Squares Within Groups: SSW = Ski=1(ni − 1) Si2 , where Si is the standard deviation of the i-th group

Total Sum of Squares: SST = SSB + SSW

Mean Square Between Groups: MSB = SSB / (k − 1)

Mean Square Within Groups: MSW = SSW / (N − k)

F-Statistic (or F-ratio): F = MSB / MSW

## One-Way ANOVA Example

Certainly! Here's an example of how to perform a one-way ANOVA (analysis of variance) using a dataset:

Suppose we have three groups, each with its own set of observations. Let's denote them as Group 1, Group 2, and Group 3.

Group 1: [5, 8, 7, 6, 10] Group 2: [12, 9, 11, 13, 8] Group 3: [6, 7, 9, 11, 10]

To perform a one-way ANOVA, you can follow these steps:

Step 1: Calculate the mean of each group. - Mean(Group 1) = (5 + 8 + 7 + 6 + 10) / 5 = 7.2 - Mean(Group 2) = (12 + 9 + 11 + 13 + 8) / 5 = 10.6 - Mean(Group 3) = (6 + 7 + 9 + 11 + 10) / 5 = 8.6

Step 2: Calculate the grand mean by averaging the means of all groups. - Grand Mean = (Mean(Group 1) + Mean(Group 2) + Mean(Group 3)) / 3 = (7.2 + 10.6 + 8.6) / 3 = 8.8

Step 3: Calculate the sum of squares between groups (SSB). - SSB = (n1 * (Mean(Group 1) - Grand Mean)^2) + (n2 * (Mean(Group 2) - Grand Mean)^2) + (n3 * (Mean(Group 3) - Grand Mean)^2) = (5 * (7.2 - 8.8)^2) + (5 * (10.6 - 8.8)^2) + (5 * (8.6 - 8.8)^2) = 19.2 + 5.76 + 0.08 = 25.04

Step 4: Calculate the sum of squares within groups (SSW). - SSW = (n1 - 1) * Variance(Group 1) + (n2 - 1) * Variance(Group 2) + (n3 - 1) * Variance(Group 3) = (4 * Variance(Group 1)) + (4 * Variance(Group 2)) + (4 * Variance(Group 3))

To calculate the variances, compute the squared differences from each observation to its group mean:

• Group 1:

• Squared differences: (5 - 7.2)^2, (8 - 7.2)^2, (7 - 7.2)^2, (6 - 7.2)^2, (10 - 7.2)^2
• Sum of squared differences: 12.8
• Group 2:

• Squared differences: (12 - 10.6)^2, (9 - 10.6)^2, (11 - 10.6)^2, (13 - 10.6)^2, (8 - 10.6)^2
• Sum of squared differences: 10.4
• Group 3:

• Squared differences: (6 - 8.6)^2, (7 - 8.6)^2, (9 - 8.6)^2, (11 - 8.6)^2, (10 - 8.6)^2
• Sum of squared differences: 9.6

Plugging these values into the SSW equation:

SSW = (4 * 12.8) + (4 * 10.4) + (4 * 9.6) = 51.2 + 41.6 + 38.4 = 131.2

Step 5: Calculate the degrees of freedom between groups (dfB) and within groups (dfW). - dfB = Number of groups - 1 = 3 - 1 = 2 - dfW = Total number of observations - Number of groups = 15 - 3 = 12

Step 6: Calculate the mean squares between groups (MSB) and within groups (MSW). - MSB = SSB / dfB = 25.04 / 2 = 12.52 - MSW = SSW / dfW = 131.2 / 12 = 10.93

Step 7: Calculate the F-statistic. - F = MSB / MSW = 12.52 / 10.93 ≈ 1.14

Step 8: Determine the critical value for the chosen significance level (e.g., α=0.05) and compare it with the calculated F-statistic to determine statistical significance.

Step 9: If the calculated F-statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

In this example, you would need to consult an F-distribution table or use statistical software to determine the critical value and assess statistical significance.

Please note that this example assumes independent and identically distributed data with equal variances across groups. Additionally, the specific steps and formulas presented here may vary depending on the statistical software or programming environment used.