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# Example Determination of Combined Uncertainty for Simple Subtraction Calculation

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Consider the need to determination of the remaining wall thickness between the bottom of a drilled hole and the surface. This determination would require the depth of the hole to be measured and subtracted from the measured value of the total thickness of the block. Five readings for each measurement were taken and summarized in the table below.

 Reading # Specimen Thickness, mm Hole Depth, mm 1 21.06 16.60 2 21.02 16.68 3 21.10 16.58 4 21.04 16.56 5 21.08 16.64 Mean 21.06 16.61

For this example, two possible source of uncertainty in the measurement will be considered: the resolution of the dial gage and the repeatability of the measurement. Since there are multiple sources of uncertainty in this calculation, the evaluation process will be broken down into the following steps:

1. Calculate the standard uncertainty due to the resolution of the dial gage.
2. Calculate the standard uncertainty due to the repeatability of each measurement.
3. Calculate the combined standard uncertainty for each of the two measurements.
4. Calculated the combined standard uncertainty for the calculated remaining wall thickness.
5. State the uncertainty in terms of an uncertainty interval, coverage factor and level of confidence.

Uncertainty of Individual Measurements Due to Resolution of Dial Gage
First, consider the uncertainty of each of the two measurements separately. For this example, two possible source of uncertainty in the measurement will be considered: the resolution of the dial gage and the repeatability of the measurement. The dial has a resolution of 0.02mm and since this is an analog device, a triangular pdf will be used to determine the standard uncertainty due to the device resolution. Since the dial is being read to the nearest division, a reading could be off by ± 0.01mm. Since 0.01mm is half of the interval of possible values that would be rounded up or down to get to a division marking, the standard uncertainty due to the resolution of the caliper will by 0.01/√6 or ±0.00408mm. This will be the same for both the specimen thickness and the hole depth measurements and since it is an intermediate result, it will be left unrounded.

Uncertainty of Individual Measurements Due to Measurement Repeatability
The uncertainty due to the measurement repeatability is the standard deviation of the mean of the repeat readings. The standard deviation of the specimen thickness measurement is ±0.031623mm. This value is divided by the square root of the number of measurements to produce a standard uncertainty of ±0.014142mm. The standard deviation of the hole depth measurement is ±0.048166mm and the standard uncertainty is ±0.021541mm. Again, since these standard uncertainties are intermediate results, they were left unrounded.

Combined Uncertainty of Individual Measurements
The combined standard uncertainty for the specimen thickness measurement is then the root sum of the squares of 0.00408mm and 0.014142mm and this equals ±0.01472mm. The combined standard uncertainty for the hole depth measurement is then the root sum of the squares of 0.00408mm and 0.021541mm, which is ±0.02192mm.

Combined Uncertainty of Calculated Remaining Wall Thickness
The remaining wall thickness is the specimen thickness minus the hole depth, which is 21.06mm minus 16.61mm, equals 4.45mm. The combined standard uncertainty for this value is then root sum of the squares of 0.01472mm and 0.02192mm or ±0.0264mm. To increase the confidence level to 95%, 0.0264 would be multiplied by a coverage factor of two to get 0.0528. Since this is the final combined uncertainty, this value will be rounded to 0.053mm.

State the Uncertainty in Terms of an Uncertainty Interval and Level of Confidence.
The final value for the remaining wall thickness would then be reported as 4.45mm ± 0.053mm with a 95% confidence level.

Last updated on November 24, 2010