## 2019

MEASURE FOR MEASURE

# Best of Both Worlds

## What to do when dealing with two different units of measurement

by Dilip Shah

In a previous column, I wrote about the importance of considering instrument resolution in addition to accuracy and precision when estimating measurement uncertainty.1 Let’s take that same consideration further.

The United States is one country where imperial units and the international system of units (SI) are used widely. In most other countries, only the use of SI—or metric units—is permitted for commerce and legal metrology.

Some measuring instruments allow you to display either metric or imperial units at the touch of a button. The digital micrometer is one such instrument. Most digital micrometers display the imperial units in 0.00005-inch resolution. When switched to SI, it displays a resolution of 0.001 mm.

This raises the question of whether two measurement uncertainty budgets are required for the different units of measurement. One response I hear is, "We do not need to have two measurement uncertainty budgets for the two units of measurement. We can always convert the measurement uncertainty from one set of units to another."

True or false?

Let’s examine the micrometer example with the two units and their corresponding display resolution using a 0-1 inch (0-25 mm) digital micrometer.

First, look at all the possible measurement uncertainty contributors:

1. Gage block calibration uncertainty (for measurement traceability).
2. Micrometer resolution (for inch and millimeter scale).
3. Micrometer repeatability (measured in inches and millimeters).
4. Coefficient of thermal expansion (in µin/in/oF and µm/m/oC).

For now, assume these are all of the significant contributors and proceed to develop the measurement uncertainty budget for both units of measurement.

Perform a repeatability study, which involves measuring the micrometer with a calibrated, traceable gage block for a minimum of 10 times (30 times is ideal). Normally, this repeatability study is performed across the entire range of the micrometer. For example, it would be characterized at 25%, 50% and 75% of the usable range to perform the micrometer study.

For the sake of brevity, we’ll perform the study at 50% of the range (0.5 inches or 12.7 mm):

• If you’re using a steel gage block, the coefficient of thermal expansion of steel is 13 µm/m/oC or 7.3 µin/in/oF.
• The maximum temperature differential while performing the repeatability study is assumed to be 0.5oC ( 0.9oF).
• The uncertainty of measurement for the gage block calibration is (1.7 + 1.2L) µin or (66.9 + 47.2L) nm, in which L is the length of the gage block. The measurement uncertainty is quoted at k = 2 for 95% confidence interval.

The repeatability study is performed using a 0.5-inch gage block with both units of measurement (see Table 1). Please note that you cannot perform the study using one unit and convert the results to another unit. Two separate repeatability studies need to be performed.

The gage block calibration measurement uncertainty from the calibration supplier was (1.7 + 1.2L) µin or (66.9 + 47.2L) nm. The gage block size used was 0.5 inches (12.7 mm), and the overall uncertainty needs to be calculated. Because the uncertainty was supplied at 95% confidence interval (k = 2), it needs to be divided by two to report everything at one standard uncertainty (one standard deviation).

The coefficient of thermal expansion (CTE) also needs to be multiplied by the length (L) of the gage block and the resultant temperature change (ΔT) while performing the measurement in the following way:

CTE x L x ΔT

Next, the resolution of the device at the unit setting is considered and entered into the measurement uncertainty budget. The two completed measurement uncertainty budgets are shown in Tables 2 and 3.

A comparison of the results for the two measurement uncertainty budgets reveals that resolution and repeatability of two different units on the same instrument measuring the same part but reporting results in a different unit may have a significant effect on the measurement uncertainty consideration.

It is not sufficient to make good measurements in one unit and convert its associated measurement uncertainty to another unit. It is always a good idea to check by performing measurement uncertainty estimates for all the units the equipment is capable of measuring.

### Reference

1. Dilip Shah, "Keep Your Resolution," Quality Progress, March 2011, pp. 56–58.

### Bibliography

International Organization for Standardization, ISO/IEC Guide 99:2007—International vocabulary of metrology—Basic and general concepts and associated terms.

Stein, Philip, "All You Ever Wanted to Know about Resolution," Quality Progress, July 2001.

Dilip Shah is president of E = mc3 Solutions in Medina, OH. He is a past chair of ASQ’s Measurement Quality Division and Akron–Canton Section 0810, and is co–author of The Metrology Handbook (ASQ Quality Press, 2004). Shah is an ASQ–certified quality engineer and calibration technician, and a senior member of ASQ.

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