The F-test is used to check whether the variances of two groups differ statistically significantly. It is used to assess whether an observed difference in variability goes beyond random fluctuations.
The decision is made by comparing the p-value with the predetermined significance level (usually α = 0.05):
- p ≤ α → Accept H₁ (reject H₀)
- p > α → Retain H₀
In product development, a new recipe for tomato sauce is being tested. The goal is to check whether the spread of the viscosity of the new recipe differs from the previous recipe.
Interpretation of the results:
The determined p-value is significantly below the significance level of 0.05, so the null hypothesis is rejected. The spread of the viscosity of the old and new recipe is not the same. The new recipe spreads significantly more than the old one.
Explanations of the graphic:
- The points mark the standard deviations of the viscosity for the old and new recipe.
- The error bars represent the 95% confidence interval for the respective standard deviation.
- The non-overlapping confidence intervals show that there is a difference in the standard deviation.
Preparation
- Select a suitable measure of dispersion to be compared.
- Define two groups whose variances are to be compared.
- Set the significance level (usually α = 0.05).
- Check if the data shows no strong deviations from the normal distribution.
Usage in AlphadiTab
- Select the 2-Sample-F tool in the Analyze phase.
- Select the first group for Sample 1.
- Select the second group for Sample 2.
- Conduct the analysis by Create New.
Interpretation
- Check if the p-value is less than or equal to the significance level.
- p ≤ α → statistically significant difference in variances.
- p > α → no statistically significant difference in variances.
- The interpretation refers exclusively to the dispersion, not the mean.
The F-test offers two main settings: the type of data input and the direction of the hypothesis.
Manual input: The comparison is based on manually entered standard deviations and sample sizes of two samples.
H₀: σ₁² / σ₂² = 1 | H₁: σ₁² / σ₂² ≠ 1
Select two-sided if you want to test whether the variances of the two samples differ without specifying a particular direction. Useful when there is no specific expectation about the direction of the difference. Example: Do the variances of reaction times of group A and group B differ?
H₀: σ₁² / σ₂² ≤ 1 | H₁: σ₁² / σ₂² > 1
Select greater if you want to test whether the variance of the first sample is greater than that of the second. Differences in the opposite direction are not considered. Example: Is the variance of delivery time before a measure greater than after?
H₀: σ₁² / σ₂² ≥ 1 | H₁: σ₁² / σ₂² < 1
Select smaller if you want to test whether the variance of the first sample is smaller than that of the second. Differences in the opposite direction are not considered. Example: Is the variance of the viscosity of the old formulation smaller than that of the new one?
Filling Quantity Tomato Sauce – Machine A vs. Machine B
In production, two filling machines are used. It is to be investigated whether the dispersion of the filling quantity differs between Machine A and Machine B. The measurement data for both machines are available in summarized form:
- Machine A: n = 25, Mean = 500.2 ml, Standard deviation = 1.1 ml
- Machine B: n = 25, Mean = 498.9 ml, Standard deviation = 1.0 ml
The comparison of the dispersions is carried out using an F-test (2 samples).
Interpretation
The F-test shows no statistically significant difference in the dispersion of the filling quantity between the two machines. The p-value is 0.644, which is above 0.05, so the null hypothesis is retained.
→ Both machines operate equally consistently in terms of filling quantity.
Response time
In the IT service desk, tickets are processed at multiple locations. Response times are regularly evaluated to identify differences in process stability. In the example, data from three locations is available. The F-test (2 samples) is basically only suitable for comparing two groups. For more than two locations, there are two approaches:
Pairwise comparisons with the F-test: Each location can be compared pairwise with the others (e.g., A vs. B, A vs. C, B vs. C).
Alternative – Levene's test over multiple groups: If all locations are to be considered simultaneously, a robust variance comparison over multiple groups is usually the more suitable tool.
Note: With multiple pairwise F-tests, the risk of random hits increases.
Interpretation
The F-test shows a statistically significant difference between the variances of the locations DLZ North and DLZ East (p = 0.000). The lead times at the East location vary significantly more than at the North location.
→ For more than two locations, a procedure for multiple groups is preferable.
Lead time by team
In sales, customer offers are processed by two teams. It should be investigated whether the variation in lead time differs between Team A and Team B.
Interpretation
The F-test shows a statistically significant difference in the variation of lead time between the two teams. The p-value is 0.047, just below 0.05, so the null hypothesis is rejected.
→ Team A varies more and works less consistently than Team B.
Delivery time after logistics center
In the logistics department, customer orders are picked and shipped. To increase efficiency, new forklifts have been introduced. It should be investigated whether the dispersion of delivery time (in hours) has decreased after the introduction. The analysis is carried out as a one-sided F-test (direction “greater”).
H₀: σ²Before / σ²After ≤ 1 | H₁: σ²Before / σ²After > 1
Interpretation
The one-sided F-test shows a statistically significant difference (F = 3.5934; p = 0.000). The delivery times before the introduction scatter significantly more than afterwards.
→ The new forklifts have not only shortened the delivery time but also made it more stable.
Supplier Comparison
In purchasing, components are sourced from two suppliers. It should be investigated whether the variation in the rejection rate per delivery differs between Supplier A and Supplier B (measured in %).
Note: The F-test assumes approximately normally distributed, metric data. Percentage values can be discrete; with small delivery quantities, the normal distribution assumption may be violated. If in doubt, a more robust method should be used.
Interpretation
The F-test shows a statistically significant difference in the variation of the rejection rate between the suppliers (p = 0.005). The null hypothesis is rejected.
→ Supplier B operates more consistently; Supplier A shows greater variation in the rejection rate.
Forecast Deviation
In production planning, demand forecasts are created for different planning periods. To assess the quality of the forecast, the forecast deviation is calculated. It is examined whether the dispersion between short- and long-term planning periods differs:
- Short-term Horizon: n = 30, standard deviation = 1.5 %
- Long-term Horizon: n = 30, standard deviation = 3.8 %
Interpretation
The F-test shows that the dispersions of the short- and long-term planning periods differ statistically significantly (p = 0.000). The null hypothesis is rejected.
→ Long-term forecasts vary significantly more than short-term ones.
F-value: Test statistic of the F-test. Ratio of the two sample variances.
s₁², s₂²: Sample variances of groups 1 and 2.
σ₁², σ₂²: Population variances of groups 1 and 2 (unknown, estimated).
α = Significance level: Preset error probability (usually 0.05).
H₀ (Null hypothesis): The variances of the two groups are equal (σ₁² = σ₂²).
H₁ (Alternative hypothesis): The variances of the two groups differ.
df = Degrees of freedom: Derived from the sample sizes of both groups (n₁ − 1 and n₂ − 1).
Confidence interval: Range of values that contains the true variance ratio with the chosen confidence level.
F-Test Statistic
Degrees of Freedom
df₂ = n₂ − 1
Confidence Interval (Lower Bound)
Confidence Interval (Upper Bound)
Ratio of Standard Deviations
Note: For the calculation of sample standard deviations as well as the distribution and quantile values of the F and Chi-square distribution, NMath was used.