Significant figures

Significant Figures is a method of determining propogation of error in a scientific experiment that is often taught in high school science classes. It is often reffered to as "sig figs" or as "sig digs" (significant digits).

Table of contents

1 Measuring with Significant Figures 2 Counting Significant Figures 3 Multiplying and Dividing using Significant Figures 4 Adding and Subtracting using Significant Figures 5 Even-Odd Rule

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 insturment. For example, if a graduated cylinder was marked off at every mililiter, the observer should measure the amount of volume contained in the cylinder to the nearest tenth of a mililiter. In order to express the degree of percision 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 12 mililiters, he would write the value as 12.0, which would indicate that the tenths place was an estimation. If the cylinder was marked off to every tenth of a milimeter, he would write the value as 12.00. Note that exact numbers obtained by counting should not be subject to the rules of 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. 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:

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

Adding and Subtracting using Significant Figures

When adding and subtracting numbers together, the sum or differences 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. The number of significant figures in each number is irrelevant. For instance, using significant figures rules:

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

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:

• 6m x 0.75m must be rounded to one significant figure
  • 4.5m2 should be rounded to 4, since 4 is even
• 14m / 4m 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.