KEVOS® MAKEOS · Quality
Two-Variance F Test (Compare the Spread of Two Machines, Batches or Methods)
Two machines, batches or methods give the same average but one looks more variable: from their sample standard deviations, is the difference in spread real at my significance level?
F test for the equality of two variances from summary statistics: the ratio of the sample variances, the critical values of the F distribution for your significance level and test direction, the p-value, a confidence interval for the ratio of the two standard deviations, and whether the spreads differ.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Alternative | choice | Two-sided asks whether the spreads differ either way; one-sided asks whether sample 1 is more (or less) variable. |
| Significance level | % | Chance of finding a difference when there is none (5 % is common). |
| Sample 1 | ||
| Sample 1 size | values | Number of values in sample 1. |
| Sample 1 standard deviation | Standard deviation of sample 1 (divisor n − 1). | |
| Sample 2 | ||
| Sample 2 size | values | Number of values in sample 2. |
| Sample 2 standard deviation | Standard deviation of sample 2 (divisor n − 1). | |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Alternative
- What it is
- Which question the test answers: whether the two spreads differ in either direction, or whether sample 1 is specifically larger or smaller.
- Units and format
- Choose one of 3 options from the list: Two-sided: spreads differ; One-sided: sample 1 spread larger; One-sided: sample 1 spread smaller.
- Example
- The form opens with “Two-sided: spreads differ” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- A one-sided test puts the whole significance level in one tail, so it detects a difference in that direction sooner.
- Get it right
- Choose one-sided only if you decided the direction before seeing the data, such as a new machine claimed to be tighter.
What this means: Significance level
- What it is
- The significance level: the chance you accept of declaring that the spreads differ when there is no real difference.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0.1% to 20%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A smaller level widens the acceptance region, so a difference must be larger to show.
- Get it right
- Use 5 % unless your procedure says otherwise; 1 % for decisions where a false alarm is costly.
Sample 1
What this means: Sample 1 size
- What it is
- The number of values in sample 1, from the first machine, batch or method.
- Units and format
- Enter a number in values: values. The form accepts 2 values to 1000000 values. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 240 values as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More values add degrees of freedom and narrow the critical values around 1.
- Get it right
- Count only values from a stable period; samples of 20 or more give a useful test.
What this means: Sample 1 standard deviation
- What it is
- The standard deviation of sample 1, calculated with the divisor n − 1.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 65.54909 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The variance ratio F rises with its square.
- Get it right
- Calculate it from individual values, not from subgroup means.
Sample 2
What this means: Sample 2 size
- What it is
- The number of values in sample 2, from the second machine, batch or method.
- Units and format
- Enter a number in values: values. The form accepts 2 values to 1000000 values. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 240 values as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More values add degrees of freedom and narrow the critical values around 1.
- Get it right
- Count only values from a stable period; the two samples need not be the same size.
What this means: Sample 2 standard deviation
- What it is
- The standard deviation of sample 2, calculated with the divisor n − 1.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 61.85425 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The variance ratio F falls with its square.
- Get it right
- Calculate it in the same units and the same way as sample 1.
Method and formulas
- F = s₁² ÷ s₂² with N₁ − 1 and N₂ − 1 degrees of freedom (NIST e-Handbook 1.3.5.9).
- Two-sided: reject equal variances if F is below the α/2 point or above the 1 − α/2 point of the F distribution; one-sided: compare with the 1 − α point (larger) or the α point (smaller).
- Confidence interval for σ₁ ÷ σ₂: √(F ÷ F₁₋α/₂) to √(F ÷ F_α/₂), at confidence 1 − α.
- Example: F = 1.123 against critical values 0.77556 to 1.2894 (p = 0.37038).
Limitations
- The F test is sensitive to non-normal data: with skewed data or outliers use Levene’s test (NIST 1.3.5.10) instead.
- Standard deviations must come from independent samples, each with the divisor n − 1.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check whether a new machine holds a tighter spread
- Compare the scatter of two measurement methods
- Check the equal-spread assumption before a t test or ANOVA
Who it is for
- Quality and process engineers
- Laboratory staff
- Test engineers
What you get
- Variance ratio F
- Critical values and p-value
- Confidence interval for the ratio of standard deviations
- Whether the spreads differ
Interpreting the result
- Equal sample standard deviations are not needed: a ratio near 1 within the critical values only means the data cannot show a difference.
References
- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.9 F-Test for Equality of Two Variances — https://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Measuring mixedness: variance, Lacey's index and comparing blenders
In Blending Bulk Solids by the Numbers
What it covers: Turn sample results into a measure of mixedness, compare it with the random limit, see why Lacey's index can mislead, and compare blend times or blenders with confidence intervals and an F test.
Method explained here · KEVOS® original book
Sorting steel parts by permeability and hardness
In Eddy-Current Inspection by the Numbers
What it covers: Comparator sorting of steel parts with an encircling coil, separating hardened from unhardened shafts, what overlapping populations do to a threshold, and a new bar supplier's shifted signals.
Useful alongside · KEVOS® original book
Averaging and the square-root rule
In Random Error and Drift by the Numbers
What it covers: Apply the square-root rule to CW-5's 200-sample window, see why its measured scatter is nearly three times what the rule predicts, and use window length to cancel conveyor vibration.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Two-Variance F Test (Compare the Spread of Two Machines, Batches or Methods) calculator