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Statistics · Sampling & confidence

Confidence Interval for Two Means Calculator

Estimate the difference between two independent population means using Welch’s t interval.

Statistics · Sampling & confidence

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Private in-browser calculation · explicit assumptions
A confidence interval pairs a point estimate with model-based uncertainty. Repeated samples produce different intervals.
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
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Understand sampling uncertainty

How confidence intervals express uncertainty

Confidence Interval for Two Means. A confidence procedure adds model-based uncertainty around an estimate. Wider confidence levels and noisier or smaller samples generally produce wider intervals.

Method used

(x̄₁−x̄₂)±t*√(s₁²/n₁+s₂²/n₂).

Intervals containing zero do not demonstrate equality; they indicate the data are compatible with zero difference under the method.

What the result cannot establish

A 95% confidence interval does not mean there is a 95% probability that a fixed parameter lies inside this realized interval. The 95% describes long-run coverage of the procedure under its assumptions.

Assumption: Samples should be independent within and between groups; Welch’s method does not assume equal variances.

Quick guide

How to use this calculator

  1. Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
  2. Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
  3. Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.

Calculation method

How the confidence interval for two means calculator works

(x̄₁−x̄₂)±t*√(s₁²/n₁+s₂²/n₂).

Intervals containing zero do not demonstrate equality; they indicate the data are compatible with zero difference under the method.

Worked example

Confidence Interval for Two Means example

Means 12 and 10 with standard errors 1 and 1 yield a difference centered at 2.

(x̄₁−x̄₂)±t*√(s₁²/n₁+s₂²/n₂).

Supported inputs

Precision and limits

Model and design

Samples should be independent within and between groups; Welch’s method does not assume equal variances.

Numerical scope

Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.

Interpretation

Intervals containing zero do not demonstrate equality; they indicate the data are compatible with zero difference under the method.

Decision boundary

The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.

Privacy

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