F-Test Calculator

What is an F-Test?

An F-test is a statistical test used to compare the variances of two populations. It helps determine if two population variances are equal. This test is particularly useful in comparing the fits of different models, analyzing the equality of variances in ANOVA, and in various experimental designs.

Formula and Its Components

The F-test statistic is calculated as:

\[F = \frac{s_1^2}{s_2^2}\]

Where:

• \(s_1^2\) is the larger sample variance
• \(s_2^2\) is the smaller sample variance

The sample variance is calculated as:

\[s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}\]

Where:

• \(x_i\) are individual values
• \(\bar{x}\) is the mean of the sample
• \(n\) is the sample size

Calculation Steps

1. Calculate the mean for each dataset.
2. Calculate the variance for each dataset using the formula above.
3. Divide the larger variance by the smaller variance to get the F-value.

Example Calculation

Let's calculate the F-value for two datasets:

Dataset 1: 2, 4, 4, 4, 5, 5, 7, 9

Dataset 2: 1, 3, 3, 3, 5, 6, 8, 8

1. Calculate means:
   \(\bar{x}_1 = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5\)
   \(\bar{x}_2 = \frac{1 + 3 + 3 + 3 + 5 + 6 + 8 + 8}{8} = 4.625\)
2. Calculate variances:
   \(s_1^2 = \frac{(2-5)^2 + (4-5)^2 + ... + (9-5)^2}{7} = 4.57\)
   \(s_2^2 = \frac{(1-4.625)^2 + (3-4.625)^2 + ... + (8-4.625)^2}{7} = 6.27\)
3. Calculate F-value:
   \(F = \frac{6.27}{4.57} = 1.37\)

Visual Representation

This scatter plot represents both datasets. Blue points are from Dataset 1, and red points are from Dataset 2. The spread of points illustrates the variance in each dataset.