Significant Figures and Measurements

In making measurements, scientists need to be accurate. Almost any measurement has limits to its precision (exactness). The digits in a measurement that are known to be precised (exact), along with ta final digit about which there is some uncertainty, are know as significant digits (figures). How many significant digits a measurement contains depends on the nature of the measuring device used.

1b.2 sf.gif The top ruler measures accurately to the tenth, the measurement of uncertainty is to the hundredths. The bottom ruler measures accurately to the ones, thus the measurement of uncertainty is to the tenths. The same principles will apply to the images below.

measur37 sf.jpg

Basic Significant Rules:

All non-zero numbers are significant

(135 cows has 3 significant figures)

Zero's between non-zero numbers are significant

(103 grams has 3 significant figures)

Leading zero's are NOT significant

(0.00102 meters has 3 significant figures and can be written 1.02x10^-3m)

Trailing zeros are NOT significant unless a decimal point is present

(1020 m has 3 significant figures, can be written as 1.02x10³ m)
0.0000100 cm has 3 significant figures, can be written as 1.00x10^-5 cm)

More examples of the Basic Significant Figure Rules

1. Nonzero digits are always significant.

2.4	two significant digits
23.5	three significant digits
12.54	four significant digits

2. Leading zeros: Zeroes that appear at the start of a number are never significant because they act only to fix a position of the decimal point in a number less than 1.

0.456	three significant digits
0.001275	four significant digits
0.05	one significant digits

3. Confined Zeros: Zeroes that appear between nonzero numbers are always significant.

101	three significant digits
10.04	four significant digits
5040	three significant digits

4. Trailing zeros: Zeroes at the end of a number are significant only if the number (a) contains a decimal points or (b) contains an overbar.

13.00	four significant digits
0.00400	three significant digits
12.50	four significant digits

Rounding

The use of significant digits creates interesting problems. For example, suppose we want toe xpress to three significant digits the percent water in a sample and our calculator give 63.764? Scientists have a certain simple rules that are used.

Rules for rounding:

1. If the first non-significant digits is less than 5, drop it. The last significant digits remains the same.

Example: 47.21 to three significant figures is 47.2
Example: 12.654 to four significant figures is 12.65

2. If the first non-significant digit is more than 5 or is 5 followed by numbers other than zero, drop the non-significant digit(s) and increase the last significant digit by 1.

Example: 47.26 to three significant figures is 47.3
Example: 56.355 to three significant figures is 56.4

3. If the first non-significant digit is 5 and is followed by zeros, drop the 5 and

increase the last significant digit by one if it is odd or
leave the last significant digit the same if it is even.

Example: 47.250 to three significant digits is 47.2

Example: 78.350 to three significant digits is 78.4

4. Non-significant digits to the left of a decimal point are not discarded but are replaced by zeros.

Example: 1781 to three significant digits is 1780
Example: 3,327 to two significant digits is 3,300

Simple Math and Significant figures

1. In addition and subtraction, the answer must not contain a smaller place (that is, decimal, units, tens, and so on) than the number with the smallest place.

25.1 + 22.11 = 47.21

since 25.1 is the larger place holder (tenths) the answer should be expressed as = 47.2

4.732 - 3.62 = 1.112

Since 3.62 is the larger place holder (hundredths) the answer should be expressed as = 1.11

Addition and Subtraction using SF Video

2. In multiplication and division, the answer must not contain any more significant digits than the least number of significant digits in the numbers used in the multiplication or division.

17.21 x 11.1 = 191.031

The answer must be express to only three significant digits because 11.1 has 3 S.F. and 17.21 has 4 SF. The answer is 191.

26.32 / 2.2 = 11.9636

The answer must be express to only two significant digits because 2.2 has 2 S.F. and 26.32 has 4 SF. The answer is 12

Multiplication and Division using SF Video
		Significant Figures Calculations Worksheet.docx

A Special Rule: Exact numbers

Occasionally, a calculation involves an exact number. Exact numbers are precisely known and can have as many significant digits as a calculation requires, and so they are not used to determine the number of significant digits for an answer.

For example: 12 inches in 1 foot. 18 g in 1 mole. 22.4 L in one mole.

Accuracy and Precision:

Accuracy is defined if the results are very close to the true or accepted value.

Precision is defined if the results (different trials) yield very similar results, when repeated in the same manner

Archer's Bow Analogy.png

Answer the following questions using the below Archers results:

Which Archer has the best precision?
Which Archer has the best accuracy

accuracy and precision archer example.jpg

Given the data below:

Trial 1	Trial 2	Trial 3	Trial 4
4.52 g/mL	5.75 g/mL	5.34 g/mL	5.57 g/mL

The Actual Value was to be 5.58 g/mL

The student's results are???

precise and accurate
precise but not accurate
accurate but not precise
neither accurate nor precise