Question

How are significant figures determined and rounded?

Summary

Determination and rounding of significant figures.

Answer

Each analytical measurement has accuracy limitations due to the chemical nature of the procedure, instrumentation and/or methodology. The number of significant figures reported indicates the accuracy to which the measurement was made. The objective is to report as many digits as were accurately measured while avoiding reporting digits not known. When this is accomplished, meaningful information is not lost, and the data does not indicate greater accuracy than is warranted. Significant figures represent all digits known to be correct plus one that is uncertain. Reporting the number 14.72 indicates the measurement is accurate to the tenths place (14.7) while the hundredths place (14.72) is uncertain. The number 14.72 has four significant figures.

Mathematicians call any positive or negative whole number or zero an integer. All digits that are nonzero integers are significant. The number 4 has one significant figure, the number 16 has two significant figures, and the number 18.3 has three significant figures. Zeros surrounded by nonzero integers are significant. The number 2001 has four significant figures, and the number 607,402 has six significant figures. Zeros following a decimal point preceded by a nonzero integer are significant. The number 4.30 has three significant figures, and the number 6.100 has four significant figures. Zeros following a decimal point that are not preceded by a nonzero integer are not significant. For example, the number 0.004 has only one significant figure. In this case the zeros only mark the position of the decimal place. The number 0.0040 has two significant figures. The first three zeros mark the decimal place and are not significant. The last zero indicates accurate measurement to the third decimal place with some uncertainty at the fourth decimal place, so the 4 and last 0 are considered significant. Determining significant figures for zeros following a nonzero integer is more difficult than our previous examples; the digits may or may not be significant. For example, the number 100 may have from one to three significant figures depending upon the accuracy of the measurement. If the measurement is accurate to the tens place (100) with some uncertainty for the ones place (100), there are three significant figures. But if the measurement is accurate only to the hundreds place (100) with some uncertainty for the tens place (100), there are two significant figures. If the number is reported as 100.0, there are four significant figures because the measurement is accurate to the ones place (100.0) with some uncertainty for the tenths place (100.0).

Once the number of significant figures for a measurement is determined, it may be necessary to round the measurement off to the appropriate number of significant figures. If a series of measurements have different numbers of significant figures, it is necessary to round off before the numbers are manipulated. The kind of math operation used to manipulate the data determines when the measurements are rounded off. If the math operation is addition, multiplication or division, round off after the operation is completed. If the operation is subtraction, round off before subtracting. If a series of math operations is used, round off after completing the calculations. The kind of math operation used to manipulate the data determines when the measurements are rounded off. If the math operation is addition, multiplication or division, round off after the operation is completed. If the operation is subtraction, round off before subtracting. If a series of math operations is used, round off after completing the calculations. If the number following the digit to round off is greater than 5, raise the digit to the next number. The number 4.352 rounded to two significant figures is 4.4 because 52 is greater than 50. If the number following the digit to round off is less than 5, leave the digit unchanged. The number 4.342 rounded to two significant figures is 4.3 because 42 is less than 50. If the number following the digit to round off is equal to 5, the convention is to raise odd digits to the next number while leaving even digits unchanged. The number 6.275 rounded to three significant figures is 6.28 because 7 is an odd number. The number 6.265 rounded to three significant figures is 6.26 because 6 is an even number. The reason for this convention is to eliminate biasing the measurement; always rounding up when the digit equals 5 produces an artificially high measurement. Rounding odd digits up and leaving even digits unchanged also eliminates this biasing problem when used consistently.

Mathematicians call any positive or negative whole number or zero an integer. All digits that are nonzero integers are significant. The number 4 has one significant figure, the number 16 has two significant figures, and the number 18.3 has three significant figures. Zeros surrounded by nonzero integers are significant. The number 2001 has four significant figures, and the number 607,402 has six significant figures. Zeros following a decimal point preceded by a nonzero integer are significant. The number 4.30 has three significant figures, and the number 6.100 has four significant figures. Zeros following a decimal point that are not preceded by a nonzero integer are not significant. For example, the number 0.004 has only one significant figure. In this case the zeros only mark the position of the decimal place. The number 0.0040 has two significant figures. The first three zeros mark the decimal place and are not significant. The last zero indicates accurate measurement to the third decimal place with some uncertainty at the fourth decimal place, so the 4 and last 0 are considered significant. Determining significant figures for zeros following a nonzero integer is more difficult than our previous examples; the digits may or may not be significant. For example, the number 100 may have from one to three significant figures depending upon the accuracy of the measurement. If the measurement is accurate to the tens place (100) with some uncertainty for the ones place (100), there are three significant figures. But if the measurement is accurate only to the hundreds place (100) with some uncertainty for the tens place (100), there are two significant figures. If the number is reported as 100.0, there are four significant figures because the measurement is accurate to the ones place (100.0) with some uncertainty for the tenths place (100.0).

Once the number of significant figures for a measurement is determined, it may be necessary to round the measurement off to the appropriate number of significant figures. If a series of measurements have different numbers of significant figures, it is necessary to round off before the numbers are manipulated. The kind of math operation used to manipulate the data determines when the measurements are rounded off. If the math operation is addition, multiplication or division, round off after the operation is completed. If the operation is subtraction, round off before subtracting. If a series of math operations is used, round off after completing the calculations. The kind of math operation used to manipulate the data determines when the measurements are rounded off. If the math operation is addition, multiplication or division, round off after the operation is completed. If the operation is subtraction, round off before subtracting. If a series of math operations is used, round off after completing the calculations. If the number following the digit to round off is greater than 5, raise the digit to the next number. The number 4.352 rounded to two significant figures is 4.4 because 52 is greater than 50. If the number following the digit to round off is less than 5, leave the digit unchanged. The number 4.342 rounded to two significant figures is 4.3 because 42 is less than 50. If the number following the digit to round off is equal to 5, the convention is to raise odd digits to the next number while leaving even digits unchanged. The number 6.275 rounded to three significant figures is 6.28 because 7 is an odd number. The number 6.265 rounded to three significant figures is 6.26 because 6 is an even number. The reason for this convention is to eliminate biasing the measurement; always rounding up when the digit equals 5 produces an artificially high measurement. Rounding odd digits up and leaving even digits unchanged also eliminates this biasing problem when used consistently.