Enter the desired confidence level into the calculator to determine the coverage factor (k). If you also enter a standard deviation (σ), the calculator will compute the corresponding expanded uncertainty (U) using U = k·σ. (This calculator assumes a normal (Gaussian) distribution and a two-sided (central) confidence level.)

Expanded Uncertainty & Coverage Factor Calculator

Enter any 2 values to calculate the missing variable (σ, confidence level, or U). The coverage factor (k) is computed from the confidence level (or from U/σ if confidence is missing).

Coverage Factor (k): —

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Coverage Factor Formula

For a normal (Gaussian) distribution and a two-sided (central) confidence level, the coverage factor equals the corresponding critical value (Z-score).

k = Z

The expanded uncertainty (interval half-width) is then:

U = k \sigma = Z\sigma

Variables:

  • k is the coverage factor (dimensionless)
  • Z is the Z-score corresponding to the desired two-sided (central) confidence level
  • σ is the standard deviation (standard uncertainty) of the measurement process (has units)
  • U is the expanded uncertainty (has the same units as σ and the measurement)

To compute expanded uncertainty, multiply the coverage factor by the standard deviation. To compute the coverage factor itself (for a normal distribution), determine the Z-score for the chosen two-sided confidence level.

What is a Coverage Factor?

The coverage factor, denoted as k, is a dimensionless multiplier used in measurement uncertainty analysis. It is applied to a standard uncertainty (often a standard deviation, such as σ or a combined standard uncertainty) to obtain an expanded uncertainty, commonly written as U = k·σ. The resulting interval (e.g., measurement ± U) is intended to include a stated fraction of values that could reasonably be attributed to the measurand. For a normal distribution, k is the same as the Z-score for the selected two-sided (central) confidence level (for example, k ≈ 1.96 for 95%).

How to Calculate Coverage Factor?

The following steps outline how to calculate the Coverage Factor.

  1. Determine the standard deviation (σ) (or standard uncertainty) of the measurement process, if you plan to compute expanded uncertainty.
  2. Choose a desired confidence level for a two-sided (central) interval (often called a coverage probability in measurement uncertainty contexts).
  3. Find the Z-score (Z) that corresponds to that confidence level for a normal distribution; this Z-score is the coverage factor k.
  4. If needed, compute expanded uncertainty using U = k·σ, and report the interval as measurement ± U.
  5. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Standard Deviation (σ) = 1.5

Confidence Level (%) = 95

Note: For a normal distribution with a two-sided (central) 95% confidence level, the Z-score (and thus the coverage factor) is approximately k ≈ 1.96.

Expanded uncertainty: U = k·σ = 1.96 × 1.5 = 2.94 (in the same units as the measurement).