Significant Figures in Calculations Uncertainty in Calculations

Measured quantities are often used in calculations. The precision of the calculation is limited by the precision of the measurements on which it is based.

Addition and Subtraction

When measured quantities are used in addition or subtraction, the uncertainty is determined by the absolute uncertainty in the least precise measurement (not by the number of significant figures). Sometimes this is considered to be the number of digits after the decimal point.

31.49 30.9471 = 0.5439 The position of greatest uncertainty in the example is in the 10-3 position of the first value and should be reflected in the 10-3 position in the answer; 0.544 g.

32.383 31.857 = 0.526 In this example the greatest uncertainty is in the 10-4 position in both values. Since the calculated answer doesnt contain enough digits, digit(s) [zeros] need to be added to represent the uncertainty. In this case one zero is added to the 10-4 position; 0.5260 g.

80 + 0.874 = 890.874 The greatest source of uncertainty in this example is in the 10s position; 890 g.

1.06 997.2 = When presented with different units the first step is to convert the values to the same unit. 100 997.2 = 62.8 The greatest source of uncertainty is in the 10s position of the first value; 60 mL.

3.6 1015 + 1.72 1013 6.875 1014 = There are two methods for dealing with scientific notation. The first method involves lining up the decimal points vertically. First the values need to be converted so that they all have the same exponent. 3.6 1015 + 172 1015 68.75 1015 = 3.6 1015 + 17 1015

68.75 1015 Line up the values vertically aligning the decimal points. Perform the operation(s) [addition and or subtraction]. Starting from the left side look for the first uncertain figure which in the case of the example above is the ones position in the second value. This is the position with the greatest uncertainty. This position will be the uncertain figure for the answer.

10. 849 1015 10 1015 .

The second method involves the use of a calculator while identifying the position of greatest uncertainty for each value. For example in the first value: 3.6 1015 The position of uncertainty is in the 10-16 position; the decimal moves one position to the right resulting in a change to the exponent from 10-15 to 10-16. 1.72 1013 For the second value the uncertainty is in the 10-15; the decimal moves two positions to the right resulting in a change to the exponent from 10-13 to 10-15. 6.8751 1014 Finally in the third value the uncertainty is in the 10-18 position; the decimal moves four positions to the right resulting in a change to the exponent from 10-14 to 10-18. The largest of these numbers is the 10-15 and so the answer will be uncertain in the 15 position. Inputting the original values into a calculator results in the answer: 1.0849 1013 After rounding to the 15 position the final answer is:

.

Multiplication and Division

When experimental quantities are multiplied or divided, the number of significant figures in the result is the same as that in the quantity with the smallest number of significant figures. If, for example, a density calculation is made in which 25.624 grams is divided by 25 mL, the density should be reported as 1.0 g/mL, not as 1.0000 g/mL or 1.000 g/mL. The key here is largest relative uncertainty and the rule is the number of significant figures in the answer will match the number of significant figures in the value with the fewest significant figures.

1.08 1 150.11 = 7.194724 103 Cancel out the units (grams in this case) and you are left with the unit for the answer (mol in this case). 1.08 1 150.11 = 7.194724 103 When determining the significant figures in the example above, 1.08g has 3 SFs and 150.11g has 5 SFs. The 1 mol value is not a measurement but rather an exact figure denoting the relationship between grams and moles; exact values have no significant figures as they are exact and not uncertainties. Therefore the 1 mol is not used to determine SFs in the answer. The first value 1.08g with 3 SFs has the least number of SFs and therefore determines the number of SFs in the answer: 7.19 x 10-3 mol.

$4.129 13.6 = $56.1544 13.6 $4.1291 = $56.1544 In this example 13.6 gallons has 3 SFs and $4.129 has 4 SFs; again we ignore the 1 gal as it is an exact value. The answer should contain 3 SFs after rounding. $56.1544 rounds to $56.2.

15.671 1 1000 = 0.015671 15.671 1 1000 = 0.015671 In this example both the 1 L and the 1000 mL are exact values with both denoting the relationship between milliliters and Liters. Therefore the only value used to determine the SFs is 15.671 mL with 5 SFs and so the answer will have 5 SFs. The original calculated answer, 0.015671 L has 5 SFs and is therefore the final answer.

In summary, with multiplication and division the answer will have the same number of significant figures as the value going into the equation having the fewest significant figures. Watch out for exact values as these are not used to determine significant figures.

Compound Operations When performing compound operations, which is operations that contain both addition/subtraction and multiplication/division the significant figures are determined using the rules for the individual operations in the order that the operations are performed (order of operations).

1.07 0.88260.762 11 In this example the first operation performed will be the subtraction in the numerator. 1.07 0.88260.762 11 = 0.1740.762 11 In the subtraction the first value, 1.07 has 2 decimal places (or the uncertainty is in the 10-2 position) and the second value, 0.8826 has decimal places (or the uncertainty is in the 10-4 position); our answer will have 2 decimal places or the uncertainty will be in the 10-2 as indicated in bold red type. Note that in the answer the extraneous digits remain for the purpose of completing the calculation however the location of the uncertainty is noted for use in determining the significant figures in the final answer. Now the division can be completed. 1.07 0.88260.762 11 = 0.1740.762 11 = 0.2459 11 In this case the numerator, 0.1874 has 2 SFs (remember we determined this in first step) and the denominator, 0.762 has 3 SFs so the answer will have 2 SFs; in rounding the 0.2459 to 2 SFs the 4 becomes a 5 as it is followed by a 5 for a final answer of 0.25 M-1cm-1.

0.91 + 1.2 + 8.43.700 In this example the addition is performed first. 0.91 + 1.2 + 8.43.700 = 10.13.700 Using the rules for addition/subtraction the position of uncertainty is in the 10-1 place or 1 decimal place as indicated by the bold red type. 0.91 + 1.2 + 8.43.700 = 10.13.700 = 2.8405 Next the division operation is performed. Using the rules for multiplication/division we next determine the number of significant figures for the answer. The numerator has 3 SFs as determined in the first step and the denominator has 4 SFs so the answer will have 3 SFs; 2.84.

Miscellaneous

Losing Significant Figures Sometimes significant figures are 'lost' while performing calculations. For example, if you find the mass of a beaker to be 53.110 g, add water to the beaker and find the mass of the beaker plus water to be 53.987 g, the mass of the water is 53.987-53.110 g = 0.877 g The final value only has three significant figures, even though each mass measurement contained 5 significant figures. Rounding and Truncating Numbers There are different methods which may be used to round numbers. The usual method is to round numbers with digits less than '5' down and numbers with digits greater than '5' up (some people round exactly '5' up and some round it down).

Example: If you are subtracting 7.799 g - 6.25 g your calculation would yield 1.549 g. This number would be rounded to 1.55 g, because the digit '9' is greater than '5'. In some instances numbers are truncated, or cut short, rather than rounded to obtain appropriate significant figures. In the example above, 1.549 g could have been truncated to 1.54 g.

Exact Numbers Sometimes numbers used in a calculation are exact rather than approximate. This is true when using defined quantities, including many conversion factors, and when using pure numbers. Pure or defined numbers do not affect the accuracy of a calculation. You may think of them as having an infinite number of significant figures. Pure numbers are easy to spot, because they have no units. Defined values or conversion factors, like measured values, may have units. Practice identifying them!

Example: You want to calculate the average height of three plants and measure the following heights: 30.1 cm, 25.2 cm, 31.3 cm; with an average height of (30.1 + 25.2 + 31.3)/3 = 86.6/3 = 28.87 = 28.9 cm. There are three significant figures in the heights; even though you are dividing the sum by a single digit, the three significant figures should be retained in the calculation.

Additional information and videos reinforcing the above concepts is available at the following website:

http://www.kentchemistry.com/links/Measurements/calcswithsigfigs.htm Works Cited: The above information was copied and or derived from: "Calculations Using Significant Figures (Sig Figs)." Mr. Kent's Chemistry Regents Help and AP Chemistry Exam Review Pages. Web. 18 Feb. 2012. .