Significant Digit Considerations for Weighing Applications
Every force measurement device (load cells, strain gauges, and scales) comes with a data sheet listing its performance metrics. These include repeatability, hysteresis error, full scale output, input resistance, zero balance. Through the testing process, these metrics identify the device’s acceptable specification tolerances. It would be impossible, and not terribly meaningful, to include every digit provided by test equipment displays. Instead, these specification values are rounded with a method that considers the level of the device’s required accuracy to some number of significant digits. This is one of several significant digit considerations for weighing applications discussed in this document.
Load cell specifications and their tolerances show the device’s ability to provide acceptable data for an application. Understanding the number of significant digits and rounding method behind these specs avoids inconsistencies in measurement results and data gathering. That is, the process for recording weigh system data should use the same significant digits and rounding methodology as used by the internal load cell’s Original Equipment Manufacturer (OEM) in the published data sheet. This process should be formal and included in Management Quality System documentation; moreover, it should compare these specifications with the customer’s measuring intent to ensure that the data output meets customer expectations.
Definitions
- Significant Digit – Any of the figures 0 through 9, excepting leading zeros and some trailing zeros, whose place denotes a numerical quantity to some desired approximation.
- Absolute Method – A method of applying data where all observed or calculated digits are considered significant in determining conformance to specifications.
- Rounding Method – A procedure applied to an observed or calculated data result where numbers are adjusted and truncated to the nearest value having a designated number of digits.
Significant Digits
In today’s electronic world, spreadsheets, calculators, digital readouts can generate many digits for just one measurement. However, the measurement’s precision may be very different than the actual precision of its corresponding specification and tolerance criteria. The use of significant digits ensures that a data result from a weighing application contains no more digits than the least precise datum used to derive it.
Some basic rules for significant digits are:
- Non-zero digits are considered significant.
- The most significant digit in a data result is the left-most non-zero digit. For example, in the result: 865,241.5307; the 8 is the most significant digit.
- The least significant digit of a data result containing a decimal point is the right-most digit. It can be a zero or non-zero. For example, in the result: 865,241.5307; the 7 is the least significant digit.
- If there is no decimal point in the data result, then the right-most non-zero digit is the least significant digit. For example, in the result: 865,240; the 4 is the least significant digit.
- The number of significant digits is the quantity of digits between and including the most and least significant digit. For example, in the result: 865,241.5307; there are 10 significant digits.
Some additional examples are in the table below, with reference to the above rules.
Determining Significant Digits
| Data Result | Significant Digits |
|---|---|
| 0.0024 | 2 |
| 0.00240 | 3 |
| 3.14159 | 6 |
| 870 | 2 |
| 2000 | 1 |
| 2000.0 | 5 |
| 4007 | 4 |
We can apply this knowledge to data sheets for load cells, strain gauges, and scales. Let’s assume the following specifications are given: Non-linearity <±0.030%; Repeatability <±0.017%; Input Resistance 800Ω±20, Full Scale Output 1.3 ±0.1%. According to the above rules, the Repeatability and Input Resistance have three significant digits; Non-linearity and Full-Scale Output have two significant digits.
Data Handling Methods
It is important to consider all issues that may affect weighing application data results. In this section we consider the issue of significant digits with specific examples. Let’s assume a weighing system operator records data displayed on a 6-place electronic digital readout from a load cell. The operator takes repeated measurements with a calibration weight of 15 tons, with a tolerance range of \(\pm\) 0.25 tons. The employee records the tonnage as follows:
Example Data Collection Table
| Reading Number | Weight (t) |
|---|---|
| 1 | 15.0396 |
| 2 | 14.9775 |
| 3 | 15.2571 |
| (…and so forth) | … |
In this example, the customer may have problems with their quality management system’s registration auditor. The third recording can be argued to be out-of-tolerance. In other words, maximum tolerance is 15.25 tons and the recorded weight is 15.2571 tons.
This example illustrates the importance of employing standard data handling methods in weighing applications, taking into account significant digits. These methods clarify the intended meaning of the specification and tolerance when determining a load cell’s conformance to them using observed or calculated data. Also, by using a standardized methodology, a consumer can compare like specifications across different manufacturers’ products to make a valid product purchase decision.
Two methods for applying data results based on significant digits are the Absolute Method and the Rounding Method. Both are explained below. The one chosen by a manufacturer is typically arbitrary and may be determined by industry convention. Therefore corporate documentation should specify the method.
Absolute Method
This method considers all digits in an observed or calculated data result as significant to determine conformance to the specification within its tolerance. Where applicable, the absolute method compares an observed or a calculated data result, unrounded, to the specification criteria. Conformance or non-conformance is based on this comparison. It is a good practice to clearly document the absolute method’s usage in any weighing application’s test data report; likewise collection process documentation should clearly note its use.
Rounding Method
The rounding method is the second way to express output data, given the number of significant digits. In this case, a limited number of digits in the observed or calculated data result are treated as significant to determine conformance to a specification within its tolerance. The numeric data are either “rounded up” or “rounded down” per the rules below, to the nearest value with the same placeholder digit as the specification.
To do this, one must know the “placeholder” of the specification. For example, if the specification is “100 tons”, the placeholder is the hundreds place; if “10 tons”, the placeholder is the tens place. Likewise if “2 tons”, the placeholder is the unit; if “0.2 tons”, the tenths unit; if “0.25 tons”, the hundredth unit, and so forth. Conformance or non-conformance to the specification is based on the rounded data result. Once again, use of the rounding method should be clearly documented.
Rounding Rules
The basic rules for rounding a data result are as follows:
- If the next digit beyond the placeholder digit is less than 5, then just truncate the data point. For example, the specification 17 \(\pm\)1.0 tons has two significant digits, and the placeholder digit is the units place. If the data result is 17.4 tons, then the data result is rounded to 17 tons.
- If the next digit beyond the placeholder digit is greater than 5, then increase the last significant digit by 1, and then truncate the remaining digits. For example, if the specification 17 \(\pm\)1.0 tons, a data result of 17.6 tons is rounded to 18 tons.
- If the next digit beyond the place holder digit is a 5:
- (a) If the last placeholder digit is an even number, then just truncate the data point. For example, the specification 1,602 \(\pm\)1.0 tons has four significant digits. Given a data result of 1,602.5 tons, the data result is rounded to 1,602 tons.
- (b) If the last placeholder digit is an odd number, then increase the last significant digit by 1 and truncate the remaining digits. For example, given the same specification 1,602 \(\pm\)1.0 tons, a data result of 1,601.5 tons is rounded to 1,602 tons.
- (a) If the last placeholder digit is an even number, then just truncate the data point. For example, the specification 1,602 \(\pm\)1.0 tons has four significant digits. Given a data result of 1,602.5 tons, the data result is rounded to 1,602 tons.
Rounding Examples
Some additional examples of rounding per the above rules are in the table below, assuming 3 significant digits in the specification. Red digits in the table indicate the decision digits needed for rounding.
Rounding When There Are 3 Significant Digits in the Comparative Specification
| Initial Data Result | Significant Digits +1 Digit | Rounded Data Result | Reason |
|---|---|---|---|
| 15.0536 | 15.05 | 15.0 | 3(a) |
| 15.0078 | 15.00 | 15.0 | 1 |
| 14.9687 | 14.96 | 15.0 | 2 |
| 14.9421 | 14.94 | 14.9 | 1 |
| 14.9555 | 14.95 | 15.0 | 3(b) |
Note that if the specification is a range (i.e., 6 – 12 Lbs.), then a data result of 7, 8.5, 9 or 11 is an acceptable data result. If the data result is 12.01 and you record 12.01, then the door is open for interpretation by others.
Retaining Significant Digits in Calculations and Data Results
Any significant digits consideration involves some loss of information. The degree of rounding should avoid a misleading impression of precision yet prevent information loss from coarse resolution. To accomplish this, consider these guidelines:
- Initially record all measurement digits displayed on a digital readout disregarding significant digits. If using analog readouts, record all digits known with certainty plus one estimated digit. For example if a needle indicator on an analog scale reads approximately a third of the way between two graduations, estimate a last digit that would reflect this.
- Avoid rounding a calculation from a data result. Perform calculations with all data digits and round only the final result. If this result is used to determine conformance to a specification within its tolerances, that final result is rounded to the specification tolerance’s significant digits.
- When adding or subtracting data, the result should contain no significant digits beyond the place of the last significant digit of any datum. Therefore, if one figure has two significant digits right of a decimal and the other has three, the result should have only two significant digits right of the decimal.
- When multiplying or dividing data, the result should contain no more significant digits than the factor, dividend or divisor with the smaller number of significant digits.
- When using logarithms and exponents, the digits of ln(x) or log10(x) are significant through the nth place after the decimal when x has n significant digits. The number of significant digits of ex or 10x is equal to the place of the last significant digit in x after the decimal.
- A mathematical constant is treated as having an infinite number of significant digits.
- A rounded standard deviation is most often rounded to two significant digits.
- A rounded average is most often rounded to the last place of significant digits required by the specific weighing application.
Conclusion
The recorded data from any precision weighing device are only as good as the user’s adherence to standard data handling practices. Without the consistency of applying significant digits and rounding methods, data results may falsely indicate conformance or non-conformance to specifications. This is especially true when calibrating a measuring system. Practicing acceptable data handling methods achieves consistency in measurement results. This enables valid decision-making with the measured data.
References
- ASTM Designation E29-93a, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, 1999.
- Food and Drug Administration, Office of Regulatory Affairs, ORA Laboratory Manual Volume III Section 4, Basic Statistics and Data Presentation, Rev. 2, 08/13/2019.
