Experimental Skills - Significant Figures

A measurement is the result some process of observation or experiment. The aim of any measurement is to estimate the “true” value of some physical quantity. However, we can never know the “true” value and so there is always some uncertainty  associated with the measurement (except for some simple counting processes).

 

A rough method of indicating the degree of uncertainty is through quoting the correct number of significant figures. The usual convention is to quote no more than one uncertain figure. So when you write down a number, the last figure in that number should be the one that is in doubt.

 

Rules for assigning significance to a digit:

• Digits other than zero are always significant.
• Final zeros after a decimal point are always significant.
• Zeros between two other significant digits are always significant.
• Zeros at the end of a number maybe ambiguous in counting the number of significant figures.

 

Example 1

In a lab activity, four students calculated the mass of a brass block. The measurements they recorded were

            M₁ = 2000.2041578 g

            M₂ = 2002 g   

M₃ = 2000 g

            M₄ = 2000.2 g

Measurement 1 is given to 11 significant figures, the recording of such a measurement is ridiculous, the mass could not be calculated to this number of significant figures.

Measurement 2 has 4 significant figures.

Measurement 4 has 5 significant figures.

But, what about measurement 3 – it is ambiguous. The best way to clearly indicate the correct number of significant figures is to write the number in scientific notation with one digit to the left of the decimal place, so for measurement 3, we could write

            M₃ = 2 ´10³ g (1 significant figure)

            M₃ = 2.0 ´10³ g (2 significant figures)

            M₃ = 2.00 ´10³ g (3 significant figures)

M₃ = 2.000 ´10³ g (4 significant figures)

 

Example 2

Remember, the last digit is usually the one in doubt. For example, you probably know your height to a few centre metres and you could write it as

            h = 1.73 m

The 3 is uncertain.

For a tall friend of yours, you can only guess their height and so you would record

            h = 1.7 m

In this case the 7 is the doubtful number.

 

Example 3

0.00341 (3 significant figures since 0.00341 = 3.41´10^-3).

2.0040 ´10⁴ (5 significant figures).

 

Operations with Significant Digits

· Any number that represents a numerical count or an exact definition has an unlimited number of significant figures(sf).

· When superfluous digit(s) is/are less than 5, the preceding figure is retained without change. When the digit(s) to be dropped is/are greater than 5, the last place retained is increased by 1. When the digit(s) to be dropped is/are 5 exactly (5, 50, 500, etc), the last retained column is rounded off to be even.

· When adding or subtracting numbers, arrange the numbers in columns. Retain no column that is to the right of a column containing a doubtful digit. For example, 160.45 + 6.732. The answer could be expressed to the nearest hundredths position (i.e.: two positions to the right of the decimal) since 160.45 is the least precise, we have

            160.45 + 6.73 = 167.21

· In multiplication and division, the result should have no more sf. than the factor having the fewest number of sf. For example, 0.00172 ´ 120.46

0.000172 has only 3 significant digits, and 120.46 has 5. So according to the rule the product answer could only be expressed with 3 significant digits.

0.00172 ´ 120.46 = 0.207

· The root or power of a number should have as many sf as the number itself.