Welch’s Two-Sample t-Test & Welch-Satterthwaite Degrees of Freedom Calculator

Compute Welch’s t-statistic, Welch-Satterthwaite effective df, and 95% mean difference CI.

Welch’s Two-Sample t-Statistic
2.19 t (df = 51.2)
Welch-Satterthwaite Effective df 51.239 df
Standard Error of Difference 2.831 SE
95% CI Half-Width (≈ 2.0 × SE) 5.662 ±units
Inputs
⚡ Scenarios:
x̄₁
Sample mean of Group 1 (x̄₁).
s₁
Sample standard deviation of Group 1 (s₁).
n₁
Sample size of Group 1 (n₁).
x̄₂
Sample mean of Group 2 (x̄₂).
s₂
Sample standard deviation of Group 2 (s₂).
n₂
Sample size of Group 2 (n₂).
Results & Graph
Welch’s Two-Sample t-Statistic
2.19 t (df = 51.2)
Welch-Satterthwaite Effective df
51.239 df
Standard Error of Difference
2.831 SE
95% CI Half-Width (≈ 2.0 × SE)
5.662 ±units

Welch’s Two-Sample t-Statistic

Graph 1 • Welch’s t-Test & Satterthwaite df • Domain Architecture & Live Telemetry Maps Mean Group1, Sd Group1, N Group1 through Welch’s t-Test & Satterthwaite df Formula.
Welch’s Two-Sample t-Statistic: 2.19 t (df = 51.2) Welch’s t-Test & Satterthwaite df • Invariants & Metrics Welch’s Two-Sample t-Statist 2.19 t (df = 51.2) Welch-Satterthwaite Effectiv 51.239 df Standard Error of Difference 2.831 SE 95% CI Half-Width (≈ 2.0 × S 5.662 ±units
Graph 2 • Welch’s Two-Sample t-Statistic Across Mean Group1 Sweep (40% → 160%) Evaluates t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂); df = (s₁²/n₁ + s₂²/n₂)² / [ (s₁² across 7 operating points (current input highlighted).
-9.30 t (df 2.87 t (df 15.04 t (df -9.30 21.68x̄₁ -5.47 32.52x̄₁ -1.64 43.36x̄₁ 2.19 54.20x̄₁ 6.02 65.04x̄₁ 9.85 75.88x̄₁ 13.68 86.72x̄₁
Graph 3 • Joint Sensitivity: Mean Group1 × Sd Group1 Compares Welch’s Two-Sample t-Statistic when Sd Group1 is shifted +25% (solid) vs. −25% (dashed).
-3.87 t (df 2.82 t (df 9.51 t (df --- Sd Group1 @ 75% -3.24 -30% -0.621 -15% 2.00 +0% 4.61 +15% 7.23 +30%
Summary: Mean difference is 6.2 (SE = 2.831), yielding Welch’s t = 2.19 with Welch-Satterthwaite df = 51.24.
Summary: Mean difference is 6.2 (SE = 2.831), yielding Welch’s t = 2.19 with Welch-Satterthwaite df = 51.24.
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Formula & Mathematical Methodology: Welch’s t-Test & Satterthwaite df Formula

t = (x̄₁ − x̄₂) / s₁²/n₁ + s₂²/n₂
df = (s₁²/n₁ + s₂²/n₂)² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ]
t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂); df = (s₁²/n₁ + s₂²/n₂)² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ]
  • meanGroup1: Mean Group1 (x̄₁)
  • sdGroup1: Sd Group1 (s₁)
  • nGroup1: N Group1 (n₁)
  • meanGroup2: Mean Group2 (x̄₂)
  • sdGroup2: Sd Group2 (s₂)
  • nGroup2: N Group2 (n₂)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Welch’s t-Test & Satterthwaite df

Compute Welch’s t-statistic, Welch-Satterthwaite effective df, and 95% mean difference CI.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂); df = (s₁²/n₁ + s₂²/n₂)² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ].
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Welch’s t-Test & Satterthwaite df via t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂); df = (s₁²/n₁ + s₂²/n₂)² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ].

Frequently Asked Questions

How does the Welch’s Two-Sample t-Test & Welch-Satterthwaite Degrees of Freedom Calculator compute the solution?

It evaluates t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂); df = (s₁²/n₁ + s₂²/n₂)² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ] directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Non-Parametric & Hypothesis Testing

Welch’s Two-Sample t-Test & Welch-Satterthwaite Degrees of Freedom Calculator: inputs, method, and how to read the result

Compute Welch’s t-statistic, Welch-Satterthwaite effective df, and 95% mean difference CI.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Mean Group1: Sample mean of Group 1 (x̄₁).
  • Sd Group1: Sample standard deviation of Group 1 (s₁).
  • N Group1: Sample size of Group 1 (n₁).
  • Mean Group2: Sample mean of Group 2 (x̄₂).

What the calculation includes

With the default values shown, Welch’s Two-Sample t-Statistic is 2.19 t (df = 51.2). The calculation follows Welch’s t-Test & Satterthwaite df Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Mean Group1 from 54.2 x̄₁ to 59.62 x̄₁ changes Welch’s Two-Sample t-Statistic from 2.19 t (df = 51.2) to 4.104 t (df = 51.2). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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