Scientific Methods: Significant figures

There are two kinds of numbers in the world:

• exact:
• example: There are exactly 12 eggs in a dozen.
• inexact numbers:
• example: any measurement.
If I quickly measure the width of a piece of notebook paper, I might get 220 mm (2 significant figures). If I am more precise, I might get 216 mm (3 significant figures). An even more precise measurement would be 215.6 mm (4 significant figures).

PRECISION VERSUS ACCURACY

Accuracy refers to how closely a measured value agrees with the correct value.
Precision refers to how closely individual measurements agree with each other.

 accurate(the average is accurate) not precise precisenot accurate accurateandprecise

In any measurement, the number of significant figures is critical. The number of significant figures is the number of digits believed to be correct by the person doing the measuring. It includes one estimated digit.

A rule of thumb: read a measurement to 1/10 or 0.1 of the smallest division. This means that the error in reading (called the reading error) is 1/10 or 0.1 of the smallest division on the ruler or other instrument. If you are less sure of yourself, you can read to 1/5 or 0.2 of the smallest division.

Rules for Significant Figures:

1. Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant only if the decimal point is specified.
Hint: Change the number to scientific notation. It is easier to see.

The last digit retained is set by the first doubtful digit.

3. Multiplication or Division:
The answer contains no more significant figures than the least accurately known number.

EXAMPLES:

 Example Number ofSignificant Figures Scientific Notation 0.00682 3 6.82 x 10 -3 Leading zeros are not significant. 1.072 4 1.072 (x 10 0) Imbedded zeros are always significant. 300 1 3 x 102 Trailing zeroes are significant only if the decimal point is specified. 300. 3 3.00 x 102 300.0 4 3.000 x 102

EXAMPLES
 Addition Even though your calculator gives you the answer 8.0372, you must round off to 8.04. Your answer must only contain 1 doubtful number. Note that the doubtful digits are underlined. Subtraction Subtraction is interesting when concerned with significant figures. Even though both numbers involved in the subtraction have 5 significant figures, the answer only has 3 significant figures when rounded correctly. Remember, the answer must only have 1 doubtful digit. Multiplication The answer must be rounded off to 2 significant figures, since 1.6 only has 2 significant figures. Division The answer must be rounded off to 3 significant figures, since 45.2 has only 3 significant figures.

Notes on Rounding

• When rounding off numbers to a certain number of significant figures, do so to the nearest value.
• example: Round to 3 significant figures: 2.3467 x 104 (Answer: 2.35 x 104)
• example: Round to 2 significant figures: 1.612 x 103 (Answer: 1.6 x 103)

• What happens if there is a 5? There is an arbitrary rule:
• If the number before the 5 is odd, round up.
• If the number before the 5 is even, let it be.
The justification for this is that in the course of a series of many calculations, any rounding errors will be averaged out.

• example: Round to 2 significant figures: 2.35 x 102 (Answer: 2.4 x 102)
• example: Round to 2 significant figures: 2.45 x 102 (Answer: 2.4 x 102)

• Of course, if we round to 2 significant figures: 2.451 x 102, the answer is definitely 2.5 x 102 since 2.451 x 102 is closer to 2.5 x 102 than 2.4 x 102.

QUIZ:
 Question 1 Give the correct number of significant figures for 4500, 4500., 0.0032, 0.04050 Question 2 Give the answer to the correct number of significant figures: 4503 + 34.90 + 550 = ? Question 3 Give the answer to the correct number of significant figures: 1.367 - 1.34 = ? Question 4 Give the answer to the correct number of significant figures: (1.3 x 103)(5.724 x 104) = ? Question 5 Give the answer to the correct number of significant figures: (6305)/(0.010) = ?

Answers: (1) 2, 4, 2, 4 (2) 5090 (3 significant figures - round to the tens place - set by 550) (3) 0.03 (1 significant figure - round to hundredths place) (4) 7.4 x 107 (2 significant figures - set by 1.3 x 103) (5) 6.3 x 105 (2 significant figures - set by 0.010)