Discussion on the Verification Results of Spring Tube Precision Pressure Gauge and Vacuum Gauge

According to the requirements of the verification regulations of JJG49-1999 "Spring Tube Precision Pressure Gauge and Vacuum Gauge", the reading value of the verification result should be read to 1/10 of the minimum scale value, and the arithmetic mean of the forward and reverse stroke readings should be rounded to 1/10 of the minimum scale value. So, how to round off the data reasonably? In

the error theory, there are the following rules for rounding numbers:

(1) If the value of the discarded part is greater than half a unit of the last digit of the retained part (that is, the first digit in the discarded digital segment is greater than 5), then add 1 to the last digit.
(2) If the value of the discarded part is less than half a unit of the last digit of the retained part (that is, the first digit in the discarded digital segment is less than 5), then the last digit remains unchanged.
(3) If the value of the discarded part is equal to half a unit of the last digit of the retained part (that is, the first digit in the discarded digital segment is equal to 5 and the last digit is not 0), then the last digit is rounded to an even number, that is, when the last digit is an even number, the last digit remains unchanged; when the last digit is an odd number, add 1 to the last digit.

We know that the error caused by rounding is called rounding error. According to the above rules, the rounding error does not exceed half a unit of the last digit of the retained number. It can be seen from Article (3) of the rule that the rounded number is not rounded up as soon as it is 5, which makes the rounding error a random error. In connection with the relevant knowledge of probability theory, this rounding method ensures that the probability of "rounding up" and "rounding up" is equal (both are 50%), so that the average value of the rounding error tends to zero.

The following is a discussion on the processing of the calibration results of spring tube pressure gauges and vacuum gauges in combination with the above data rounding rules:

(1) The "half unit" mentioned in the above rules refers to the number "5". For spring tube pressure gauges and vacuum gauges, because their 1/10 grid has 0.0005, 0.0002, 0.0001, 0.005..., their significant figures are either "5", "2", or "1". In connection with the above rules, the "half unit" here is not the "5" mentioned in the rules. For example, for a spring tube precision pressure gauge with a calibration level of 0.4 and an upper limit of 1.0 MPa, its 1/10 grid is 0.0005 MPa. When the standard value is 0.2 MPa, if the positive stroke reading is 0.1990 MPa and the reverse stroke reading is 0.1995 MPa, the average value is 0.19925 MPa, which should be retained to the fourth decimal place as required. According to the above rules, the rounded value is 0.1992 MPa, which is obviously not rounded to 1/10 grid.

The average value 0.19925 can be analyzed as follows: 0.19925-0.1990+0.00025. Since the data is required to be rounded to 1/10 grid, the number to be selected here should be 0.00025 instead of 0.00005. The so-called "half unit" should be understood as half of 0.0005MPa of 1/10 grid, that is, 0.00025MPa. In other words, the number to be selected should be compared with 0.00025. Similar conclusions can be drawn by analyzing the case where 1/10 grid is other values. For example, when 1/10 grid is 0.0002MPa, the "half unit" compared with the number to be selected should be 0.0001MPa; when 1/10 grid is 0.0001MPa, the "half unit" is 0.00005MPa.

In conclusion, for spring tube pressure gauges and vacuum gauges, combined with the provisions on data rounding in error theory, when rounding off the calibration results, "half a unit" should be understood as "half of a 1/10 grid".

(2) The above rules are fundamentally intended to ensure that the probabilities of "rounding off" and "inclusion" are equal. However, directly rounding off the calibration results according to the above rules cannot ensure that the probabilities of "rounding off" and "inclusion" are equal in some cases.

As shown in Table 1, for a spring tube precision pressure gauge with a calibration level of 0.25 and an upper limit of 0.4MPa, its 1/10 grid is 0.0002MPa, where "half a unit" is "0.0001MPa". According to the provisions of Article (3) of the above rules, when the number to be rounded is "half a unit", no matter how the forward and reverse stroke readings change, the half unit value of 0.0001MPa will be rounded off, and the probability of "inclusion" is zero.

For example, at the point of 0.04MPa, the average value of the forward and reverse stroke readings is (0.0402+0.0404)/2=0.0403=0.0402+0.0001, and the last digit is an even number "2", so 0.0001 is discarded.

This is obviously unreasonable (because the probabilities of "rounding down" and "entering in" are not equal). So, what kind of rounding method can be adopted to make the probabilities of "rounding down" and "entering in" equal for the part to be rounded? Let n be the number of digits after the decimal point of the 1/10 grid of the inspected table. The analysis shows that: in order to ensure that the probabilities of "rounding down" and "entering in" are equal, the following analysis is made for the case where the part to be rounded down is half of the 1/10 grid of the inspected table:

(1) When the effective digit of the 1/10 grid of the inspected table is "2" (such as 0.0002 for 1/10 grid), the choice should be determined by the parity of the corresponding n-1th digit of the 1/10 grid.

(2) When the effective digit of the 1/10 grid of the meter being tested is "5" (such as 0.005 for 1/10 grid) or "1" (such as 0.0001 for 1/10 grid), the parity of the nth or n-1th digit can be used to determine the choice (for the sake of uniformity, it is recommended to determine the choice based on the parity of the n-1th digit).

Discussion on the article "Discussion on the Verification Results of Spring Tube Precision Pressure Gauges and Vacuum Gauges"

This article mainly discusses Wu Jianquan's article "Discussion on the Verification Results of Spring Tube Precision Pressure Gauges and Vacuum Gauges" published in "China Metrology" No. 11, 2008 (hereinafter referred to as "Wu's article").

Spring tube precision pressure gauges and vacuum gauges are analog instruments. The national metrology verification regulations and their publicity materials have never clarified the rounding of their verification results, especially how to round off the "2" and "5" interval readings.

GB8170-1987 "Numerical Rounding Rules" or GB3101-1993 "General Principles for Quantities, Units and Symbols" have clear rounding rules for "1", "2" and "5", but how to apply these rounding rules in actual work? According to the requirements of the verification regulations of JJG49-1999 "Spring Tube Precision Pressure Gauges and Vacuum Gauges", the reading of its verification results should be read to 1/10 of the minimum scale value. According to the national standard of precision pressure gauges, its scale value is specified as "1", "2", "5" intervals. If it is estimated as 1/10, the result is always "1", "2", "5" intervals. Because the rounding result is 1/10 of the scale value, it must reach this 1/10, otherwise the significant digits of the indicated value will be incorrect, reflecting that the value was not taken as required during the measurement. The following examples illustrate the rounding results of the intervals of "1", "2", and "5":

1. "1" interval rounding

is generally carried out according to the "rounding rule", and the last digit of the rounding result is from 0 to 9.

If the calibration result data can be divided to the estimated reading, there is no rounding problem. For example, if the calibration data is 1.003 and 1.005 (unit omitted, the same below), the average value of the two data is 1.004, and there is no rounding problem. The rounding result can be estimated from the dial of the precision meter.

If the calibration result data cannot be divided to the estimated reading, there is a rounding problem. For example, if the calibration data are 1.003 and 1.004, the average value of the data is 1.0035, and 1.0035 cannot be estimated on the dial of the precision meter. According to the "1" interval rounding, the result is 1.004, and the rounding digit is 1.003 odd, so it is carried and becomes an even number, that is, 1.004. [page]

2. "2" interval rounding

First divide by 2 and then round according to the "1" interval, the rounding result must be a multiple of 2, that is, 2, 4, 6, 8, 0 and other data.

If the calibration result data can be divided to the same estimated reading digit and is a multiple of 2, there is no rounding problem. For example, if the calibration data are 1.002 and 1.006 respectively, the average value of the data is 1.004. 4 is a multiple of 2, and the result 1.004 can also be estimated on the dial of the precision meter.

If the calibration data are 1.002 and 1.004, and the average value of the data is 1.003, then 1.003 cannot be estimated on the dial of the precision meter. First, divide 1.003 by 2 to get 0.5015, and round it off with a "1" interval, the result is 0.502, and then 0.502×2=1.004, and the result 1.004 can be estimated on the dial of the precision meter. However, it is impossible to perform such troublesome rounding in daily work. Generally, the rounding result is taken as the principle of whether the last 2 digits of the rounding result are divisible by 4. If 1.003 is between 1.002 and 1.004, the rounding result is either 1.002 or 1.004. 02 cannot be divided by 4 (0.5), and 04 is divisible by 4 (1), so 1.004 is taken.

3. Rounding with "5" intervals

First divide by 5 and then round according to the "1" interval. The mantissa of the rounded result must be a multiple of 5, that is, the data with a mantissa of 5 and 0.

If the calibration result data can be divided to the same estimated reading position and is a multiple of 5, there is no rounding problem. If the calibration data are 1.005 and 1.015 respectively, the average of the two data is 1.010. The number with a mantissa of 0 is a multiple of 5, and the result 1.010 can also be estimated on the dial of the precision meter.

If the calibration data are 1.005 and 1.010, the average of the data is 1.0075, and 1.0075 cannot be estimated on the dial of the precision meter. First, divide 1.0075 by 5 to get 0.2015. Round it off to 0.202. Then, 0.202×5=1.010. The result 1.010 can be estimated on the dial of the precision meter. In work, for the "5" interval between two numbers, the rounding digit is "0" as the rounding result. For example, if 1.0075 is between 1.005 and 1.010, the rounding result is either 1.005 or 1.010. Just take 1.010 as the rounding result.

The rounding result in Wu's article (see Table 1) is rounded off according to the rules of this article.

Table 1 Verification results and rounding results in "Wu Wen" Unit: MPa
Note: Accuracy level: 0.25 level, upper limit: 0.4MPa; 1/10 grid: 0.0002MPa

The rounding results in Wu's Table 1: the last 2 digits of the data such as 0.0402, 0.0806, 0.2398, 0.2794, 0.3198 cannot be divided by 4, and they are not rounded off at intervals of "2". Therefore, Wu's Table 1 is rounded off according to the method described in this article, and the results are shown in Table 2.

Table 2 Verification results and rounding results Unit: MPa

The last two digits of the rounded result in 2 are all divisible by 4, that is, the rounding is performed according to the rounding interval.