Significant figures

Significant Figures (also called sig figs, significant digits, or sig digs) is a method used to determine propagation of error in a scientific experiment or in statistics when complete precision is not attainable or not required. Scientific notation is often used in significant figures.

Counting Significant Figures

Before calculations can be done according to the rules of significant figures, one must know how many significant digits are in each number being used in the calculations. Note that because of the rounding, a number to n significant figures is not necessarily the same as the first n digits of that number. Rules for determining the significance of digits are as follows:

• Each non-zero number is a significant figure
• All zeroes between two non-zero numbers are significant figures
• All zeroes at the end of a number after a decimal point are significant figures
• The number of significant figures is then totaled

In order to correctly show which digits are significant, figures such as "2000" should be expressed in scientific notation to the correct number of significant figures. If two digits are significant the number is 2.0x10³, if three are significant then it's 2.00x10³.

Multiplying and Dividing using Significant Figures

When multiplying and dividing numbers together, the product or quotient is rounded to the number of significant figures of that of the factor with the least. For instance, using significant figures rules:

• 8 x 8 = 60
• 8 x 8.0 = 60
• 8.0 x 8.0 = 64
• 8.02 x 8.02 = 64.3

When squaring or taking the square root of a value, the number of significant figures decreases by one using some systems of significant digits.

Another system of significant figures states that what multiplying a number having n significant figures with a number having m significant figures. For example:

Adding and Subtracting using Significant Figures

When adding or subtracting it is not the number but the position of the significant figures that determines the significant figures of the result. To write the sum or difference of two numbers to the correct position, the sum or difference is rounded to the place farthest to the right of the decimal point of the number with the least amount of digits after the decimal point. For instance, using significant figures rules:

• 1 + 1.1 = 2
• 1.0 + 1.1 = 2.1
• 100 m + 110 = 100
• 1.0 x 10² + 111 = 210

Even-Odd Rule

As with all rounding procedures, if the number directly to the right of the digit to be rounded to is less than five, the digit stays the same; if more than five, the digit is rounded up. However, to always round up or down if the digit is equal to exactly five would skew data in one direction or the other. Thus, when using the significant figures system and rounding in such situation, the even-odd rule is used: round in whichever direction would make the last digit of the final product even. For example:

• 4.5² should be rounded to 4, since 4 is even
• 14/4 must be rounded to one significant figure
• 3.5 should be rounded to 4, since 4 is even

In this way, the even-odd rule avoids skewing data either upwards or downwards.

Measuring with Significant Figures

The significant figures method teaches that when measuring using a non-electronic instrument, the observer should estimate within the nearest tenth of a division marked on the instrument. For example, if a graduated cylinder was marked off at every milliliter (ml), the observer should measure the amount of volume contained in the cylinder to the nearest tenth of a milliliter. In order to express the degree of precision to which a value was measured, decimals are used. When using significant figures rules, it should be assumed that the last significant digit of every value was estimated. Using the previous example, if the observer read the amount of liquid in the cylinder to be exactly at the 12 ml mark, the observer would write the value as 12.0 ml, which would indicate that the tenths place was the precision obtained. If the cylinder was marked off to every tenth of a ml, the observer would write the value as 12.00 ml. Note that exact numbers obtained by counting should not be subject to the rules of significant figures.

External link

• Worksheets for Practicing Rounding