Standard Uncertainty Calculator (Type A and Type B)

Historical Background

The concept of standard uncertainty is fundamental in measurement science. It helps quantify the confidence in a given result by accounting for variability (Type A uncertainty, based on repeated measurements) and instrument precision (Type B uncertainty, based on instrument specifications or other available data).

Calculation Formula

1. Standard Deviation (S_d):
   The standard deviation, a measure of data spread, is calculated as:

   \[
   Sd = \sqrt{\frac{1}{n-1} \sum{i=1}^{n} (x_i - \bar{x})^2}
   \]

   Where:

   • \( n \) is the number of measurements.
   • \( x_i \) is the ith measurement.
   • \( \bar{x} \) is the mean of the measurements.

2. Type A Uncertainty (u_A):
   The uncertainty from repeated measurements is calculated as:

   \[
   u_A = \frac{S_d}{\sqrt{n}}
   \]

3. Type B Uncertainty (u_B):
   The uncertainty based on instrument precision is:

   \[
   uB = \frac{\Delta{ins}}{\text{divisor}}
   \]

   Where Δ_ins is the instrument uncertainty and the divisor is typically √3.

4. Combined Uncertainty (u_C):

   \[
   u_C = \sqrt{u_A^2 + u_B^2}
   \]

Example Calculation

Given:

• Measurements: 1.001, 1.002, 1.003, 1.001, 1.000 mm
• Δ_ins = 0.004 mm
• Divisor = √3

1. Standard Deviation S_d:

\[ Sd = \sqrt{\frac{1}{5-1} \sum{i=1}^{5} (x_i - \bar{x})^2} \approx 0.00112 \text{ mm}
\]

2. Type A Uncertainty u_A:

\[ u_A = \frac{0.00112}{\sqrt{5}} \approx 0.00050 \text{ mm}
\]

3. Type B Uncertainty u_B:

\[ u_B = \frac{0.004}{\sqrt{3}} \approx 0.00231 \text{ mm}
\]

4. Combined Uncertainty u_C:

\[ u_C = \sqrt{(0.00050)^2 + (0.00231)^2} \approx 0.00236 \text{ mm}
\]

Importance and Usage Scenarios

Understanding standard uncertainty is critical in scientific research, engineering, and quality control, where high-precision measurements are required. Quantifying and combining uncertainties allows professionals to assess the reliability and accuracy of their measurements.

Common FAQs

1. What is Standard Deviation (S_d)?

   • Standard deviation measures the spread or variability of data points in a set of measurements.

2. Why is Type A uncertainty important?

   • Type A uncertainty captures the random variability in measurements, crucial for assessing repeatability.

3. Can I change the divisor for Type B uncertainty?