Example Data Table
| Case | Mean 1 | SD 1 | N 1 | Mean 2 | SD 2 | N 2 | Method | Confidence | Expected Reading |
|---|---|---|---|---|---|---|---|---|---|
| Training scores | 72.4 | 8.5 | 36 | 78.9 | 9.1 | 42 | Welch | 95% | Second mean is likely higher |
| Machine output | 101.2 | 4.8 | 25 | 99.7 | 4.6 | 27 | Pooled | 90% | Small negative difference |
| Known standard | 15.2 | 2.1 | 50 | 16.0 | 2.4 | 55 | Known | 99% | Use supplied deviations |
Formula Used
The calculator reports the interval for the population mean difference u2 - u1.
Observed difference: d = x̄2 - x̄1
Confidence interval: d ± critical value × standard error
Welch standard error: SE = √(s1² / n1 + s2² / n2)
Pooled standard error: SE = sp × √(1 / n1 + 1 / n2)
Pooled variance: sp² = ((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2)
Known deviation standard error: SE = √(σ1² / n1 + σ2² / n2)
How to Use This Calculator
- Enter the mean, deviation, and sample size for group one.
- Enter the same values for group two.
- Select the confidence level for the interval.
- Choose Welch for unequal variance in most independent sample work.
- Choose pooled only when equal variance is a reasonable assumption.
- Choose known deviation when population deviations are trusted.
- Press the calculate button and read the interval above the form.
- Download the CSV or PDF file when you need a record.
Reliable Difference Intervals
A confidence interval for u2 minus u1 estimates the likely range for the difference between two population means. The calculator uses group two minus group one, so a positive result means the second group has the larger estimated mean. A negative result means the first group is higher. This direction matters when you write reports, compare treatments, or review process changes.
Why This Calculator Helps
Manual interval work can be slow because each method uses a different standard error and critical value. Welch’s method is useful when spreads are different. The pooled method is useful when equal variance is a fair assumption. The known deviation option supports cases where population deviations are supplied from a trusted standard or long historical record.
Inputs That Matter
Enter each sample mean, sample deviation, and sample size with care. Larger sample sizes usually shrink the margin of error. Larger deviations usually widen it. The confidence level controls how conservative the interval becomes. A higher level gives wider limits because it asks for more certainty. The claimed difference field lets you compare the observed result with a practical target or null value.
Reading The Output
The center is the observed difference, mean two minus mean one. The standard error measures estimated sampling variation. The critical value sets the interval width. The margin of error is the critical value times the standard error. The lower and upper limits describe plausible values for the population difference. If zero is inside the interval, the data do not show a clear directional difference at that confidence level.
Good Statistical Practice
Use independent samples for this version. Use paired methods when the same subjects are measured twice. Check for data entry mistakes before interpreting results. Extreme outliers can distort means and deviations. Small samples need more caution, especially when distributions are strongly skewed. The interval is not a guarantee. It is a repeatable procedure that captures the true difference at the chosen rate over many similar studies.
Export And Share
The CSV file is helpful for spreadsheets. The PDF file is useful for notes, audits, or client summaries. Keep the method, confidence level, and assumptions with every exported result. Store source data beside the final interpretation too.
FAQs
What does u2 minus u1 mean?
It means population mean two minus population mean one. A positive interval suggests group two may be higher. A negative interval suggests group one may be higher.
When should I use Welch method?
Use Welch method when the two samples are independent and the variances may be different. It is a safe default for many practical comparisons.
When is the pooled method suitable?
Use the pooled method only when equal variance is a reasonable assumption. It combines both sample deviations into one shared estimate.
What if I know population deviations?
Select the known deviation method. Then enter trusted population deviations. The calculator will use a normal critical value instead of a t value.
What does the margin of error show?
It shows how far the interval extends from the observed difference. Wider margins usually come from smaller samples, larger deviations, or higher confidence.
Can zero be inside the interval?
Yes. If zero is inside, the interval includes no difference. That means the evidence may not show a clear directional difference.
Does this calculator handle paired data?
No. This calculator is for independent samples. Paired data needs a different interval based on differences within matched observations.
Why are CSV and PDF exports included?
CSV is useful for spreadsheets and data logs. PDF is useful for reports, summaries, and sharing the result without changing the layout.