Idea, Solved Instance and Demonstration in Agri Analyze

October 21, 2025

3

The weblog talk about in particulars about concept of F take a look at, its use circumstances, solved instance (manually) and an illustration utilizing on-line software Agri Analyze (Studying time 10 min)

Introduction

The F-test is a statistical methodology used to match the variances of two samples or the ratio of variances throughout a number of samples. It assesses whether or not the information comply with an F-distribution underneath the null speculation, assuming customary situations for the error time period (ε). The take a look at statistic, denoted as F, is usually used to match fitted fashions to find out which finest represents the underlying inhabitants. F-tests are steadily employed in fashions fitted utilizing least squares. The take a look at is called after Ronald Fisher, who launched the idea because the “variance ratio” within the Nineteen Twenties, with George W. Snedecor later naming the take a look at in Fisher’s honor.

Definition

An F-test makes use of the F-statistic to judge whether or not the variances of two samples (or populations) are equal. The take a look at assumes that the inhabitants follows an F-distribution and that the samples are unbiased. If the F-test yields statistically vital outcomes, the null speculation of equal variances is rejected; in any other case, it’s not.

Use of F-Check in Statistics

The F-test is a statistical software used to match variances and decide if there are vital variations between two populations or samples. It’s generally utilized in regression evaluation, statistical inference, mannequin becoming, and evaluation of variance (ANOVA) to establish the best-fitting statistical mannequin or assess variations throughout teams.

Regression evaluation
Statistical inference
Mannequin becoming
Evaluation of variance (ANOVA)

Assumptions

Independence: The observations inside every group should be unbiased, which means there ought to be no relationship between observations throughout samples.
Normality: Knowledge in every group ought to comply with a standard distribution. For giant pattern sizes, this assumption might be relaxed based mostly on the Central Restrict Theorem.
Homogeneity of variances: The variances throughout teams being in contrast ought to be roughly equal.

Essential Notes on the F-Check

The F-test is used to evaluate whether or not the variances of two populations are equal by evaluating them utilizing an F distribution.
The F-test statistic is calculated as $F = frac{sigma_1^2}{sigma_2^2}$
The null speculation is evaluated utilizing a vital worth, which determines whether or not to reject the speculation.
A typical software of the F-test is the one-way ANOVA, which assesses variability between group means and inside group observations.

Choice Standards for σ₁² and σ₂² in an F-Check

For a right-tailed or two-tailed F-test, the variance with the better worth is positioned within the numerator, making the pattern equivalent to σ₁² the primary pattern. The smaller variance (σ₂²) is the denominator for the second pattern.
For a left-tailed take a look at, the smaller variance is within the numerator (pattern 1), whereas the bigger variance is within the denominator (pattern 2).

Hypotheses
Left-Tailed Check:

Null Speculation (H₀): σ₁² = σ₂²
Various Speculation (H₁): σ₁² < σ₂²
Choice Standards: Reject H₀ if the F-statistic < F-critical worth.

Proper-Tailed Check:

Null Speculation (H₀): σ₁² = σ₂²
Various Speculation (H₁): σ₁² > σ₂²
Choice Standards: Reject H₀ if the F-statistic > F-critical worth.

Two-Tailed Check:

Null Speculation (H₀): σ₁² = σ₂²
Various Speculation (H₁): σ₁² ≠ σ₂²

Process for Conducting an F-Check:

Outline Hypotheses
- Null Speculation (H₀): The variances of the teams are equal.
- Various Speculation (H₁): The variances of the teams usually are not equal.
Gather Knowledge
Collect pattern knowledge from the teams being in contrast.
Calculate Pattern Variances
For every group, compute the pattern variance (S²) utilizing the formulation:

$S^2 = frac{sum (x_i – bar{x})^2}{n – 1}$

the place $x_i$
Calculate F-Statistic
Compute the F-statistic as follows:

$F = frac{S_1^2}{S_2^2}$

the place $S_1^2$ is the bigger variance and $S_2^2$
Decide Levels of Freedom
Calculate levels of freedom for every group:

$textual content{df}_1 = n_1 – 1 quad textual content{(numerator)}$ $textual content{df}_2 = n_2 – 1 quad textual content{(denominator)}$
Discover the Important Worth
Utilizing an F-distribution desk, find the vital worth on your chosen significance stage (e.g., $α = 0.05$ ) based mostly on ${df}_{1}$
Make a Choice
- If $F > textual content{Important Worth}$ , reject the null speculation, indicating vital variations in variances.
- If $F leq textual content{Important Worth}$ , fail to reject the null speculation, suggesting no vital variance variations.
Conclusion

Reject the null speculation if $F$ exceeds the vital worth, indicating vital variance between teams.
Fail to reject the null speculation if $F$ is lower than or equal to the vital worth, implying inadequate proof for variance variations.

Instance: –

Life
expectancy in 9 areas of Brazil in 1900 and 11 areas of Brazil in 1970 was
as given within the desk under:

Area	Life expectancy (12 months)
Area	1900	1970
1	42.7	54.2
2	43.7	50.4
3	34.0	44.2
4	39.2	49.7
5	46.1	55.4
6	48.7	57.0
7	49.4	58.2
8	45.9	56.6
9	55.3	61.9
10		57.5
11		53.4

We goal to find out whether or not the variation in life expectancy throughout totally different areas in 1900 and 1970 is identical. Assuming the populations in 1900 and 1970 comply with regular distributions, $N(mu_1, sigma_1^2)$ and $N(mu_2, sigma_2^2)$ , the hypotheses might be formulated as:

Null Speculation $H_0$
Various Speculation $H_1$

The F-test is utilized to judge these hypotheses.

Calculate Pattern Variances:

$S_1^2 = frac{1}{8} left( sum_{i=1}^9 x_{1i}^2 – frac{left( sum_{i=1}^9 x_{1i} proper)^2}{9} proper)$ $S_1^2 = frac{1}{8} left( 18527.78 – frac{405^2}{9} proper) = frac{302.78}{8} = 37.848$ $S_2^2 = frac{1}{10} left( sum_{j=1}^{11} x_{2j}^2 – frac{left( sum_{j=1}^{11} x_{2j} proper)^2}{11} proper)$
Calculate the F-Statistic:

$F = frac{S_1^2}{S_2^2} = frac{37.848}{23.607} = 1.603$
Conclusion:
The vital values from the F-distribution desk at $alpha = 0.05$ for a two-tailed take a look at with levels of freedom (8, 10) are $F_{0.025} = 3.85$ and $F_{0.975} = 0.233$ . Because the calculated F-value (1.603) is lower than 3.85 and better than 0.233, we fail to reject the null speculation. This means that there isn’t a vital distinction within the variances of life expectancy between 1900 and 1970 throughout the areas of Brazil.

F take a look at Demonstration in Agri Analyze

Video Demo for F-test in Agri Analyze

Pattern Knowledge File: The info is identical as proven within the above instance. File Hyperlink

Step1: Put together knowledge file and save in a csv format

Step2: Register on Agri Analyze (Solely first time) Hyperlink

Step3: Go to Analytical Instrument -> Speculation testing -> F-test Hyperlink

Step4:

Add the file
Choose stage of significance (for five% write 0.05)
Variable identify: Life Expectancy
Class Kind: Time Body

Step5: Click on on Submit and Obtain

Output

Different Associated Matters:

The weblog is written with nice effort and due analysis by Uttam Baladaniya (PhD Scholar, Division of Agricultural Statistics, Anand Agricultural College)

Idea, Solved Instance and Demonstration in Agri Analyze

Process for Conducting an F-Check:

F take a look at Demonstration in Agri Analyze

Output

Different Associated Matters:

Related Articles

Quantum computing ‘lie detector’ lastly proves these machines faucet into Einstein’s spooky motion at a distance somewhat than simply faking it

Some Papua New Guinea information doodles

Saying StataNow – The Stata Weblog

Latest Articles

Quantum computing ‘lie detector’ lastly proves these machines faucet into Einstein’s spooky motion at a distance somewhat than simply faking it

Some Papua New Guinea information doodles

Saying StataNow – The Stata Weblog

Vibe Coding with GLM 4.6 Coding Plan

3 Quickest Methods To Study Energy BI In The UK & Progress Your Profession