# How to calculate F ratio

The F ratio, also known as the Fisher’s ratio, is a crucial test statistic used in ANOVA (Analysis of Variance), comparing the variance between groups with the variance within groups. Analyzing F ratios allows researchers to determine whether there are significant differences between groups, and if a relationship exists between independent variables. In this article, we will cover how to calculate the F ratio using a step-by-step approach.

**Step 1: Calculate the Variance Between Groups (SSB)**

First, calculate the mean for each group and the overall grand mean. The variance between groups (also known as Sum of Squares Between groups, or SSB) quantifies the variability among group means.

Formula:

SSB = Σn_i(m_i – GM)^2

Where:

n_i = number of observations in each group,

m_i = mean of each group,

GM = grand mean (the overall average),

Σ = sum.

**Step 2: Calculate the Variance Within Groups (SSW)**

Next, calculate the variance within each group (also known as Sum of Squares Within groups, or SSW) by considering individual data points relative to their respective group means.

Formula:

SSW = ΣΣ(x_ij – m_i)^2

Where:

x_ij = each individual data point,

m_i = mean of each respective group,

ΣΣ = sum over all data points and all groups.

**Step 3: Compute Mean Squares Between and Within Groups (MSB & MSW)**

Divide SSB by its degrees of freedom (k – 1), where k represents the number of groups being compared. This yields Mean Squares Between groups (MSB).

Formula:

MSB = SSB / (k – 1)

Do a similar calculation for SSW by dividing it by its degrees of freedom ((N – k)), where N is the total number of data points. This yields Mean Squares Within groups (MSW).

Formula:

MSW = SSW / (N – k)

**Step 4: Calculate the F Ratio**

With MSB and MSW computed, you can now determine the F ratio, which is the ratio of MSB to MSW.

Formula:

F ratio = MSB / MSW

**Step 5: Find the Critical F Value and Compare**

Lastly, find the critical F value from an F-distribution table, using the two degrees of freedom as parameters (k – 1) and (N – k). Compare the calculated F ratio to the critical value:

– If the F ratio is larger than the critical value, reject the null hypothesis and conclude that there is a significant difference between groups.

– If the F ratio is smaller than or equal to the critical value, accept the null hypothesis and assume no significant difference between groups.

**Conclusion**

The F ratio is a powerful method for determining statistical relevance among groups. By following these steps, you can calculate it efficiently and draw informed conclusions from your data. Keep in mind that larger sample sizes lead to more reliable results, so strive to collect as much data as feasible for your analysis.