GUM ANOVA Measurement Uncertainty Calculator
Enter the mean and standard deviation of each group from a balanced nested measurement series. The calculator follows the one-way analysis of variance of JCGM 100:2008 Annex H.5 and reports the variance components, the F ratio and the standard uncertainty of the overall mean under both assumptions about the between-group effect.
Inputs
Results
- Overall mean x̄ (average of group means)
- —µV
- Sample variance of group means v
- —µV²
- Standard deviation of group means √v
- —µV
- Between-group mean square a² = K·v
- —µV²
- Within-group variance b² (mean of squared SDs)
- —µV²
- F = a² / b² (dimensionless)
- —
- Numerator degrees of freedom J − 1
- —
- Denominator degrees of freedom J(K − 1)
- —
- Estimated between-group variance (a² − b²)/K
- —µV²
- Between-group standard deviation √((a² − b²)/K)
- —µV
Assuming no between-group effect
All J·K observations treated as one population: pooled variance divided by J·K.
Assuming a random between-group effect
Only the scatter of the J group means: √(v / J).
Both results are shown side by side. The calculator does not test F against a critical value, does not declare a between-group effect significant and does not choose between the two standard uncertainties.
What this calculator does
Annex H.5 of JCGM 100:2008, the Guide to the Expression of Uncertainty in Measurement, shows how a one-way analysis of variance separates two sources of random scatter in a repeated measurement. In its example a Zener voltage standard is compared with a reference on J = 10 days, with K = 5 independent readings each day. The readings within one day scatter, and the daily means may scatter more than that within-day noise alone would explain.
This calculator works from the group summaries only: the J group means and the J sample standard deviations, each based on the same number K of readings. No individual readings are needed and none are assumed. From these summaries it estimates the within-group variance, the between-group variance component, the F ratio and the standard uncertainty of the overall mean.
Random effects only. These calculations describe random variability within and between groups. They do not detect systematic effects. An offset that is the same in every group, such as an error in the value assigned to the reference standard, shifts all the data together and leaves no trace in the analysis of variance. Systematic effects must be evaluated by other means and combined separately in the uncertainty budget.
How to use it
- Choose the measurement unit in which all group means and standard deviations are expressed. Use one unit for every value.
- Enter the number of groups J. The list of group rows grows or shrinks to match; values you typed in a removed row come back if you increase J again.
- Enter K, the number of readings in each group. The design must be balanced, so K is the same for all groups.
- For every group, enter its mean and its sample standard deviation (calculated with divisor K − 1). A standard deviation may be zero but not negative.
- Read the results. They update live; the Calculate button runs the same computation.
- If you want to test for a between-group effect, compare F with a critical value of the F distribution at the two degrees of freedom shown, taken from a statistical table. Then decide which standard uncertainty of the mean applies to your measurement.
Method and formulas
With group means x̄1 … x̄J and group standard deviations s1 … sJ, each from K readings, the calculator evaluates the following, following equations H.25 to H.32 and Table H.9 of JCGM 100:2008.
- x̄ = (x̄₁ + x̄₂ + … + x̄J) / J — overall mean
- v = Σ (x̄ᵢ − x̄)² / (J − 1) — sample variance of the group means
- a² = K · v — between-group mean square, J − 1 degrees of freedom
- b² = (s₁² + s₂² + … + sJ²) / J — pooled within-group variance, J(K − 1) degrees of freedom
- F = a² / b² — undefined when b² = 0
- sB² = (a² − b²) / K — estimated between-group variance; its square root is shown only when the estimate is zero or positive
- u(x̄) = √( ((J − 1)·a² + J(K − 1)·b²) / (J·K·(J·K − 1)) ) — no between-group effect, J·K − 1 degrees of freedom
- u(x̄) = √( v / J ) — random between-group effect, J − 1 degrees of freedom
All arithmetic keeps full double-precision values internally; results are rounded only for display. The overall mean is formed by summing deviations from the first group mean, which preserves precision when the means are large numbers with small differences, as in the Zener example.
Check against the GUM example
The values pre-filled above are those of Table H.9 in microvolts (J = 10, K = 5). They give an overall mean of 10,000,097.1 µV, a² ≈ 16,296.06 µV², b² = 7,205.8 µV², F ≈ 2.2615, a between-group variance of about 1,818.05 µV², a pooled standard uncertainty of about 13.32 µV with 49 degrees of freedom and a random-effect standard uncertainty of about 18.05 µV with 9 degrees of freedom.
Frequently asked questions
What does the F ratio tell me?
F = a²/b² compares the variance estimated from the spread of the group means with the pooled within-group variance. If there is no between-group effect, F is expected to be near 1. Testing F means comparing it with a critical value of the F distribution at J − 1 and J(K − 1) degrees of freedom. This calculator reports F and both degrees of freedom but does not supply critical values or declare significance.
Why can the between-group variance be negative?
The estimate (a² − b²)/K is a difference of two variance estimates that both contain random error. When the group means agree more closely than the within-group scatter predicts, a² is smaller than b² and the difference is negative. A true variance cannot be negative, so the calculator reports the signed estimate and does not take its square root.
Which standard uncertainty of the mean should I use?
The calculator shows both. The pooled result assumes no between-group effect and has J·K − 1 degrees of freedom. The random-effect result, the standard deviation of the group means divided by the square root of J, allows for a between-group effect and has J − 1 degrees of freedom. The choice depends on your knowledge of the measurement and on any statistical test you apply; the calculator does not choose for you.
Must every group have the same number of observations?
Yes. The equations of JCGM 100:2008 Annex H.5 used here are for a balanced design, so the same K applies to every group. Unequal group sizes need different formulas.
Does this analysis detect systematic effects?
No. The analysis of variance only separates random variability within groups from random variability between groups. An effect that is the same in every group, such as an offset of the reference standard, leaves all the data shifted together and cannot be seen. Such effects must be evaluated separately.
What happens if every group standard deviation is zero?
Then b² = 0 and F = a²/b² is undefined, which the calculator states. The other quantities are still calculated: the between-group variance becomes a²/K and both standard uncertainties of the mean remain defined.
This calculator is provided for general information and education. It implements the balanced one-way analysis of variance described in JCGM 100:2008 Annex H.5 (pp. 98–103) and does not replace a full uncertainty evaluation, professional judgment or the requirements of your quality system. Verify results independently before using them in calibration certificates, conformity decisions or other critical work.