Precision and Accuracy, Significant Figures, Rounding Rules

Accuracy vs. Precision

Hi there again. You have probably heard these two words "Accuracy" and "Precision" at least once in your life time. And you have possibly encountered at least 10 people who have been using these two words interchangeably; possibly you're even one of them yourself, which makes it at least 11 people so far. And maybe up until this very moment, you still have completely no idea what I'm trying to point out here. Okay, let me keep this real simple: Accuracy ≠ Precision. So you've been using BAD English all along. (Haha, we're not trying to critize our enthusiastic supporters here, just keep in mind that, we still love you very much, even if you use bad English ,dear!) Now, let us take a deeper look at these two words by CORRECTLY defining them.

Definitions:

Accuracy is how close the measurement is to true value of the measured quantity.

Precision is how close the measurement is to other ones.

As can be seen in the chart below showing the dart boards of drunk/sober men with good/bad aim, accuracy is being sober, and precision is having a steady hand.

Significant Figures

• more precise digits means there are going to be more significant figures
• the last digit of a value/measurement is usually estimated or uncertain (eg. 2938.234, 4 would be the uncertain number)
• to calculate you must include all of the certain digits and only ONE uncertain digits
• leading zeros are not counted as significant figures(eg. 0.0000001, the number of significant figures is 1)
• trailing zeros without a decimal point do not count. (eg. 10, 1000, 10,000, there is only one significant figure)
• trailing zeros AFTER a decimal point are significant digits. (eg, 32.000 has 5 significant figures) 

Rounding Rules

Calculations With Significant Figures

Addition and Subtraction
Problem: Kofia Dicted adds 1.3 grams more sugar in her morning coffee than the average person, which is 12.544 grams. How much sugar does she add in her coffee?

      12.544 g

+      1.3     g

=     13.844 g

As Kofia Dicted is aware of significant figures, she knows that her 1.3 added grams are imprecise, and that the total amount of sugar she adds cannot be more precise than 1.3grams. Thus she rounds off the sum to the first uncertain digit.

13.844g →13.8g
Answer: Kofia Dicted puts approximately 13.8grams of sugar in her coffee.

Multiplication and Division (Programming techniques mentioned are all real)
Problem: Terri Buljo Oaks is making a computer program that multiples with significant figures in mind to avoid actually thinking while doing chemistry homework. The first question is 12.544g x 1.3g

Below is her program's source written in pseudocode.

String.amountOfSigFigs(factor1)=sigfig1;
String.amountOfSigFigs(factor2)=sigfig2; //Finds the number of sig figs each number has
Variable Short Sigfigs=Math.Min(sigfig1,sigfig2); //Finds the lesser number of sig figs
Variable Double Float: product=factor1*factor2; //Finds the product of the two numbers
Print(Math.Round(product,sigfigs)); //Rounds to the number of lesser sig figs and displays the result

Variables used:
factor1: 12.544
factor2: 1.3
sigfig1: 5
sigfig2: 2
SigFigs: 2
product: 16.3072
Printed result: 16
     12.544 g

x      1.3     g 
        3.7632g²
+   12.544  g²

=   16.3072g²

16.3072g² →16g²

Unfortunately Terri's program does not account for non-decimal trailing zeros or units, so she got all of those wrong and naturally failed the course for life.