Calculate expanded uncertainty, coverage factor k, standard deviation, or confidence level from any two of the three values for a normal distribution.

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): —

Related Calculators

Expanded Uncertainty & Coverage Factor Formula

The calculator assumes a normal distribution and a two-sided, central confidence level. Expanded uncertainty is found by multiplying the standard deviation by the coverage factor.

U = kσ
σ = U / k
k = U / σ
k = ⁻¹((1 + CL / 100) / 2)
CL = (2(k) - 1) * 100
  • U = expanded uncertainty
  • σ = standard deviation, also called standard uncertainty in many measurement contexts
  • k = coverage factor
  • CL = two-sided confidence level, entered as a percent
  • Φ = standard normal cumulative distribution function
  • Φ-1 = inverse standard normal cumulative distribution function

If you enter standard deviation and confidence level, the calculator first converts the confidence level to a coverage factor, then calculates expanded uncertainty with U = kσ.

If you enter expanded uncertainty and confidence level, it calculates the same coverage factor from the confidence level, then solves for standard deviation with σ = U/k.

If you enter standard deviation and expanded uncertainty, it calculates the coverage factor with k = U/σ, then converts that coverage factor into the matching two-sided confidence level.

Common Coverage Factors for Two-Sided Confidence Levels

These values are for a standard normal distribution. They are useful for checking whether a result is in the expected range.

Two-Sided Confidence Level Coverage Factor k Meaning
68.27% 1.000 About one standard deviation
90% 1.645 Common lower confidence setting
95% 1.960 Common reporting level
95.45% 2.000 Often approximated as k = 2
99% 2.576 Wider interval

Example Calculations

Example 1: Find expanded uncertainty

Suppose the standard deviation is 0.20 and the two-sided confidence level is 95%.

k = ⁻¹((1 + 0.95) / 2) = 1.959964
U = kσ = 1.959964 * 0.20 = 0.391993

The expanded uncertainty is approximately 0.391993.

Example 2: Find confidence level from U and σ

Suppose the expanded uncertainty is 1.0 and the standard deviation is 0.5.

k = U / σ = 1.0 / 0.5 = 2
CL = (2(2) - 1) * 100 = 95.45%

The confidence level is approximately 95.45%.

FAQ

What is the difference between standard deviation and expanded uncertainty?

Standard deviation, or standard uncertainty, describes the one-standard-deviation spread of the measurement result. Expanded uncertainty multiplies that value by a coverage factor. The expanded value gives a wider interval tied to a chosen confidence level.

Why is the coverage factor for 95% confidence not exactly 2?

For a normal distribution, a two-sided 95% confidence level gives k = 1.959964, usually rounded to 1.96. A coverage factor of 2 corresponds to about 95.45% confidence, not exactly 95%.

Can this be used for small sample uncertainty calculations?

Use this result when the normal distribution assumption is appropriate. If you have a small sample and need to account for degrees of freedom, a Student’s t coverage factor may be more appropriate than the normal coverage factor used here.