STP 303-1 Standard Test Procedures Manual Section: GUIDELINES FOR THE USE OF SIGNIFICANT FIGURES 1. SCOPE 1.1. Description of Procedure This Procedure provides a process for the rounding of significant figures used in reporting observed or calculated values and comparing them to the specification limits. 1.2 Application of Test 2. TERMINOLOGY The following applies to all specified limits in the specification. For the purposes of determining conformance with the specification limits, an observed or calculated value shall be rounded to the nearest unit in the last right hand digit used in expressing the specification limit. The observed or calculated value shall be rounded in accordance with the method described in Section 3 of this Standard Test Procedure. 2.1. Significant Digit n A significant digit is any of the figures 0 through 9, except leading zeros and some trailing zeros, which is used with its place to denote a numerical quantity to some desired approximation. 2.2 Use of Zeros 0 The digit zero may either indicate a specific value or indicate place only in accordance with the following guidelines. 2.2.1 Zeros leading the first non-zero digit of a number indicate order of magnitude only and are not significant digits. For example, the number 0.0054 has two significant digits. 2.2.2 Zeros trailing the last non-zero digit for numbers represented with a decimal point are significant digits. For example, 1270. and 32.00 both have four significant digits. 2.2.3 The significance of trailing zeros for numbers represented without the use of a decimal point can only be identified from knowledge of the source of the value. For example, a modulus strength of 140,000 Pa, may have as many as 6 and as Date: 1996 04 09 Page 1 of 6
few as 2 significant digits. To eliminate ambiguity, exponential notation may be used. Thus 1.40 x 10 5 Pa indicates that the modulus is reported to the nearest 0.01 x 10 5 Pa or 1,000 Pa. The use of SI prefixes is recommended to reduce the need for trailing zeros of uncertain significance. Thus, 140 kpa and 0.140 Mpa each indicate that the modulus is reported to the nearest 1 kpa or 1,000 Pa while 140 kpa may have two or three significant digits. 3. PROCEDURE 3.1. Rounding Procedure The actual rounding procedure shall be as follows: 3.1.1 When the next digit beyond the last figure to be retained is less than five, do not change the digit in the last place to be retained. For example, if a test result was found to be 5.14% and it was to be reported to the nearest 0.1%, the 4 is less than 5, so the test result would be reported as 5.1%. 3.1.2 When the next digit beyond the last figure to be retained is greater than five, increase by one the digit in the last place to be retained. For example, if a test result was found to be 5.16% and it was to be reported to the nearest 0.1%, the 6 is greater than 5, so the test result would be reported as 5.2%. 3.1.3 When the next digit beyond the last figure to be retained is five, and there are no digits beyond this five, or only zeros, increase by one the digit in the last place to be retained if it is odd, if it is even leave the digit unchanged. For example, if a test result was found to be 5.150% and it was to be reported to the nearest 0.1%, the 1 is an odd number, so the test result would be reported as 5.2%. If the test result was found to be 5.250% and it was to be reported to the nearest 0.1%, the 2 is an even number so the test result would be reported as 5.2%. 3.1.4 When the next digit beyond the last figure to be retained is five, and there are digits beyond this five, increase by one the digit in the last place to be retained. For example, if a test result was found to be 5.151% and it was to be reported to the nearest 0.1%, there are non-zero digits beyond the five so the test result would be reported as 5.2% 3.1.5 The rounded value should be obtained in one step by direct rounding of the most precise value available and not in two or more successive roundings. For example, 89,490 Pa rounded to the nearest 1,000 Pa is 89,000 Pa. It is incorrect to first round to the nearest 100 Pa giving 89,500 Pa then to round to the nearest 1,000 Pa giving 90,000 Pa. 3.1.6 If the standard indicates that values are to be rounded to the nearest 50, 5, 0.5, 0.005, etc. the recorded or observed value should be doubled, then rounded off to the nearest 100, 10, 1, 0.1, etc. in accordance with the procedures listed in Section Date: 1996 04 09 Page: 2 of 6
3.1.1 to 3.1.5. The rounded number would then be divided by two and reported. For example, in rounding 6,025 to the nearest 50, 6,025 is doubled giving 12,050. This number becomes 12,000 when rounded to the nearest 100 (using Section 3.1.3). When 12,000 is divided by 2, the resulting number, 6,000 is the rounded value of 6,025. In rounding 6,075 to the nearest 50, 6,075 is doubled giving 12,150 (using Section 3.1.3) which becomes 12,200 when rounded to the nearest 100. When 12,200 is divided by 2, the resulting number, 6,100, is the rounded value of 6,075. 3.2 Recording Test Data Unless otherwise specified in a Standard Test Procedure, the following shall apply when recording direct measurements on a graduated cylinder, dial, or ruler. All digits known exactly from the measurement device plus one digit, which is uncertain due to estimation, should be recorded. For example, if a graduated cylinder is graduated in increments of 0.1 ml, then an observation would be recorded as 9.76 ml where it is observed between the 9.7 and 9.8 marks on the cylinder and estimated to be about six tenths of the way between those marks. When the measuring device has a vernier scale, the last digit recorded shall be the one from the vernier. 3.3 Calculations of Test Results from Test Data When calculating a test result from test data, avoid rounding of intermediate quantities. As far as is practical with the calculating device or form used, carry out calculations with the test data exactly and round only the final result. For example: Given: 1. Calculated Asphalt Content (ASPH) = 5.142 % 2. Specification Limit for Asphalt Content = 5.3% ± 0.3% 3. Bulk Specific Gravity of Mix (BSGM) = 2.186 4. Bulk Specific Gravity of Aggregate (BSG) = 2.6359 5. Specification Limit for Voids in Mineral Aggregate = 14.5 % Minimum (VMA) The asphalt content has been calculated to be exactly 5.142 %. In order to compare this value against the specification limit which is significant to the nearest 0.1 % (i.e. 5.3 %), the calculated asphalt content would be rounded to the nearest 0.1 % and be reported as 5.1 % (Section 1.2 and Section 3.1.1). Date: 1996 04 09 Page: 3 of 6
BSGM 2.186 VMA = 100x 1- = 100x 1 = 21.12397906% ASPH 5.142 1 + x BSG 1 + x2.6359 100 100 The VMA has been exactly calculated to be 21.12397906%. In order to compare this value against the specification limit which is significant to the nearest 0.1% (i.e. 14.5%), the calculated VMA would be rounded to the nearest 0.1% and reported as 21.1% (Section 1.2 and Section 3.1.1). The exact asphalt content value was used in the calculation of VMA not the rounded value. This is because the calculation of Asphalt Content is an intermediate calculation in the computation of the VMA. In accordance with Section 3.3, the intermediate values should not be rounded, the final result is rounded and compared to the specification limits. 3.4 Rounding and Significant Figures (No Indication of Significant Figures) The following rules will be used to derive the number and place of significant digits to be reported if the number and place of significant digits are not defined. 3.4.1 When adding or subtracting test data the result shall contain no significant digits beyond the place of the last significant digit of any datum. 1. 11.24 + 9.3 + 6.32 = 26.9, because the number 9.3 is significant to only one decimal place. The value 26.9 is obtained by rounding the exact sum of 26.86 to one decimal place (Section 3.1.2). 2. 926 923.4 = 3, because the last significant digit of 926 is the last digit or the nearest 1. The value 3 is obtained by rounding the exact difference of 2.6 to the nearest 1 (Section 3.1.2). 3. 140,000 + 91,460 = 231,000, because the number 140,000 is known to have been recorded to the nearest thousand. The value 231,000 is obtained by rounding the exact value of 231,460 to the nearest thousand (Section 3.1.1). 3.4.2 When multiplying or dividing the result shall contain no more significant figures than the value with the least number of significant digits. Date: 1996 04 09 Page: 4 of 6
1. 11.38 x 4.3 = 49, since the factor 4.3 has two significant digits the exact product, 48.934, is rounded and reported to two significant digits (Section 3.1.2). 2. ( 926 923.4) = 0.6, in this case only one figure is significant because 4.3 the numerator difference has only one significant figure (Section 3.4.1 Example 2). As a result, the exact quotient, 0.6046511628 is rounded and reported to one significant digit (Section 3.1.1). 3.4.3 When using logarithms the digits of ln(x) or log 10 (x) are significant through the n th place after the decimal when x has n significant digits. When using exponents the number of significant digits of e x or 10 x is equal to the place of the last significant digit in x after the decimal. 1. ln(3.46) = 1.241, since 3.46 has 3 significant digits, the exact value, 1.241268589, is rounded and reported to 3 places after the decimal (Section 3.1.1). 2. 10 3.46 = 2,900, since 3.46 is significant to two places after the decimal, the exact value, 2,884.031503, is rounded and reported to two significant digits (Section 3.1.2). 3.4.4 The rule for numbers representing exact counts or mathematical constants is that they are to be treated as having an infinite number of significant digits. 0.23 1. 1 x = 1 = 0. 88, in this case 1 and 2 are exact constants 2 2 with two significant digits and 0.23 is a measured quantity with two significant digits. There will be two significant digits in the quotient (Section 3.4.2 Example 2) and in the difference (Section 3.4.1 Example 2). As a result, the exact value, 0.885, is rounded and reported to two significant digits (Section 3.1.3). 2. A count of 50 pieces times a measured thickness of 0.124 mm is 50 x 0.124 = 6.20 mm, since the factor of 0.124 has three significant digits Date: 1996 04 09 Page: 5 of 6
4. REPEATABILITY 4.1 Sources of Error and the exact count of 50 has an infinite number of significant digits, the exact product, 6.2, is rounded and reported to three significant digits (Section 3.4.2 Example 1). 3. A measurement of 1.634 inches is to be converted to millimetres. The result is 1.634 x 25.4 = 41.50 mm, since the factor of 1.634 has four significant digits and the constant value 25.4 has an infinite number of significant digits, the exact product, 41.5036, is rounded and reported to four significant digits (Section 3.1.1 and Section 3.4.2 Example 1). Rounding test results avoids a misleading impression of precision while preventing loss of information due to coarse resolution. Any approach to retaining only the necessary significant figures involves some information loss. As a result, the level of rounding is carefully selected by considering the planned and potential uses for the data. The number of significant digits must be adequate for comparison against specification limits. For some purposes, such as where calculations involve differences of measurements that are close in magnitude and in some statistical calculations, such as Students t-tests and autocorrelations, data should be reported to at least two more significant figures than required by the specifications limits. 5. ADDED INFORMATION 5.1 References ASTM E29-93a Standard Practice for Using Significant Digits in Test Data to Determine Acceptance with Specifications. Date: 1996 04 09 Page: 6 of 6