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Confidence Band Calculator

Calculate a confidence band around a fitted regression line.

About Confidence Band Calculator

The confidence band is narrowest at the mean of x and widens toward the extremes, producing the characteristic bow-tie shape. It describes uncertainty in the fitted line, not in individual observations. Confusing a confidence band with a prediction interval understates the spread of actual observations substantially.

Formula

Confidence Band

SE(fit) = s √(1/n + (x − x̄)² ÷ Σ(x − x̄)²)

confidence band
Uncertainty in the mean response, narrower than a prediction interval

Worked example

With the default values below, Confidence Band Calculator returns Half-Width of the Confidence Band of 4.29407. Change any input above and every figure here is recalculated in your browser.

Inputs used

Predicted value at this x
150
Residual standard errorminimum 0
12
Sample sizeminimum 3
80
Distance of x from its mean
5
Sum of squared deviations of xminimum 0
1,200

Results

Half-Width of the Confidence Band
4.29407
Lower Confidence Band
145.7059
Upper Confidence Band
154.2941
Standard Error of the Fitted Value
2.19089
Band Half-Width at the Mean of X
2.62957

Step by step

  1. 1Values used
  2. 2Confidence Band
  3. 3Half-Width of the Confidence Band= 4.29407
  4. 4Lower Confidence Band= 145.7059
  5. 5Upper Confidence Band= 154.2941
  6. 6Standard Error of the Fitted Value= 2.19089
  7. 7Band Half-Width at the Mean of X= 2.62957

Assumptions and limits

  • The result is derived only from the 5 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How is Confidence Band calculated?

SE(fit) = s √(1/n + (x − x̄)² ÷ Σ(x − x̄)²) The confidence band is narrowest at the mean of x and widens toward the extremes, producing the characteristic bow-tie shape. It describes uncertainty in the fitted line, not in individual observations.

Why does Confidence Band matter?

Confusing a confidence band with a prediction interval understates the spread of actual observations substantially.

What values do I need to enter?

This calculator takes 5 inputs: Predicted value at this x, Residual standard error, Sample size, Distance of x from its mean, Sum of squared deviations of x. The pre-filled defaults are a realistic starting point — replace them with figures from your own environment for a result you can act on.

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