Two mean distributions feed one difference intervalUpdates with every input
Math & Statistics
Confidence Interval Graphing Calculator
Graph a Welch confidence interval for the difference between two independent means while keeping both sampling distributions, unequal variances, effective degrees of freedom, and the comparison threshold visible.
Mean difference -
Lower endpoint -
Upper endpoint -
Welch standard error -
Welch degrees of freedom -
t critical -
Margin of error -
Benchmark status -
TWO-GROUP UNCERTAINTY LANDSCAPE
Two mean distributions feed one difference interval
Separate group density hills retain unequal precision; a lower rail shows the resulting difference interval and benchmark.
ASSUMPTION PRESERVING SCENARIOS
Change one design lever at a time
Each row recomputes the Welch standard error and degrees of freedom without silently pooling variances.
| Scenario | n1 | n2 | SE difference | Welch df | Interval |
|---|
GRAPH SETUP
Define the comparison direction and group independence
- Enter group 1 and group 2 in the order used by the reported difference.
- Use sample SDs on the same outcome scale.
- Confirm the groups are independent rather than paired.
- Keep sample sizes tied to the observations summarized by each mean and SD.
- Use the benchmark for a meaningful comparison value, often but not always zero.
GRAPH INTERPRETATION
Overlapping raw distributions and an interval for means are not the same object
The density hills describe uncertainty in each sample mean under the entered summaries. Their visual overlap does not directly decide whether a difference is compatible with the data.
The lower difference rail is the inferential object: it combines both variance contributions and the Welch critical value.
WELCH INTERVAL
Keep unequal variance information instead of forcing a pooled SD
The variance of each sample mean contributes separately. Welch-Satterthwaite degrees of freedom reflect the imbalance in sample sizes and variances.
Detailed calculation process and general formulas
Delta = mean_1 - mean_2SE_Delta = sqrt(s1^2/n1 + s2^2/n2)df_W = v^2 / [(s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)]ME = t* x SE_DeltaCI_Delta = Delta +/- MESymbols, meanings, and units
- Delta
- difference in sample meansoutcome units
- s1, s2
- group sample standard deviationsoutcome units
- n1, n2
- independent sample sizesobservations
- df_W
- Welch effective degrees of freedomunitless
- SE_Delta
- standard error of the mean differenceoutcome units
COMPARISON LENSES
Read magnitude, precision, and imbalance separately
A useful two-group report does more than declare a direction.
Magnitude
-The center of the interval is the ordered mean difference.
Precision
-The interval width responds to both group variances and sample sizes.
Imbalance
The visual keeps the wider or smaller group from disappearing behind a pooled summary.
Decision takeaway: Name the difference direction as group 1 minus group 2 every time the interval is reported.
Applied decisions
Independent-group decisions
Two production lines
Independent runs from two lines have different variability and sample counts.
What the result clarifies: Welch preserves both variance contributions without an equal-variance assumption.
Two campaign cohorts
Average order values are compared across separately sampled cohorts.
What the result clarifies: The interval expresses uncertainty in the mean difference, not customer-level overlap.
Worked current scenario
Substitution, intermediate values, and reconciliation
Method references
Sources for this calculator's specific method
Scope and limitations
This summary-statistic interval assumes independent observations within and between groups and representative sampling. It does not correct for confounding, clustering, repeated measures, selective missingness, severe non-normality in very small samples, or multiple comparisons.
Confidence Interval Graphing Calculator | Welch Difference in Means FAQ
Why not pool the standard deviations?
Welch's method remains valid without assuming equal population variances and is generally a safer default.
Can I use this for before and after data?
Not if observations are matched; use a paired-difference interval instead.
Does overlap of the two hills decide significance?
No. Use the difference interval and its benchmark, not visual overlap of separate distributions.