Accuracy is about whether a measurement agrees with the true value. If a measurement is accurate then it is correct. This can be difficult to determine unless it is possible to look up a result in a trusted source.
Precision is about whether several measurements of something agree with each other. Precision can be measured using the range of values. One half of this range is the amount by which the true value may vary above and below the average value (this is called the plusorminus amount—the ± amount). In lab activities we make several measurements in order to be able to calculate just how precise you are. Chances are good, but not certain, that if your measurements have a small range then your results are accurate, too.
The precision of a result is reflected in the number of digits reported. If you have a bathroom scale and measure the mass of a book with it you might find it weighs 7 pounds. A more precise scale might reveal that the weight of the book is actually 7.15 pounds. This second number is more precise. When a measurement is expressed with more digits it is generally more precise than a measurement of the same thing showing fewer digits. The significant figures of a measurement are all the digits known for sure plus one estimated digit.
Errors in the lab come in three types. First, human error. Human error is when you make a boneheaded mistake. For example, say you intend to measure out 1.0 g of a chemical but instead measure out 10.0 g. This type of mistake can be avoided and will be noticed while you are working if you are paying attention. Second, there is systematic error. Systematic error leads to measurements being always high or always low. The change is in the same direction for each measurement and is always the same size. This type of error occurs, for example, if you use a ruler to make measurements by pushing the end of the ruler up against something. Many rulers do not have their zero mark right at the end. This results in all your measurements being too long by the extra bit of ruler. Third, there is random error. Random error can be seen in a set of data that represent measurements of the same quantity. The quantity is a constant and has some true value but the measurements of it are usually a bit too high or a bit too low. For each individual measurement the amount that the measurement is too high or low is random. It can be safely assumed that random error will affect every set of measurements that anyone makes. It occurs because we estimate the last digit of a measurement and different people estimate that last digit differently. Random error may also occur because the value displayed by an electronic instrument may fluctuate and it is up to the observer to make a judgment about the value. Random error can be measured by using the range of measurements to calculate a plusorminus amount. The plusorminus amount is onehalf of the range. It can also be measured using the percent random error. This is calculated by dividing the plusofminus amount by the average and multiplying by 100%.
Unless otherwise noted the final significant figure (the last digit) in a measurement is assumed to be ±1. So 9.45 kg is 9.45 ± 0.01 kg—that is, the true value is assumed to be between 9.44 kg and 9.46 kg. Random error’s influence over results can be reduced by taking multiple measurements of the same thing and averaging the reults. This evens out the toohigh measurements with the toolow measurements.
The maximum possible precision is defined as one tenth (1/10 or 0.1 times) the smallest division on the measuring instrument. On the first ruler shown below there are marks only every centimeter. So the best precision comes by estimating the last digit which is 0.1 × 1 cm = 0.1 cm. The maximum possible precision can be expressed as a plusorminus amount and is ±0.1 cm. For the second ruler the plusorminus amount is 0.1 × 0.1 cm = 0.01 cm. This ruler is ten times more precise than the one above it at ±0.01 cm.
Look at the rulers below and use each one in turn to measure the gray line. Remember, precision is limited to one tenth of the smallest division. For each ruler estimate the last digit. That is, for the first ruler, estimate to the tenths place. For the second ruler estimate to the hundredths place.
To do the estimating you must imagine that the space between the markings on the ruler can be divided into ten smaller spaces. Use these imaginary spaces to measure the last digit.
Here is a key point: When you estimate the last digit you are actually making your measurement more precise. First, you have added a significant figure: more significant figures means better precision. Second, you are making the digit before the last digit more secure due to the fact that you are observing the last digit so closely.
There are three rules for counting the number of significant figures in a measurement.
It is always easiest to tell the number of significant figures if you use scientific notation. In scientific notation you never have trailing zeros or leading zeros. Those zeros only tell you how far the measurement is from the decimal point. In scientific notation this job is taken care of by the exponent.
When you do a calculation using measurements you must be sure that the result is never expressed with more significant figures than you started with. If you do then you are pretending you know more than you really know. The number of significant figures tells you how precise your data are. If you use too many figures you are exaggerating your precision. Here are two simple rules:
Important: Calculations often end up producing answers that have a lot more digits (in the calculator) than the measurements you started with. These digits are not all significant! It is impossible to increase the precision of measurements by doing calculations with them. Just because the calculator says there are more digits doesn't mean they are all meaningful.
For this lab you must turn in your lab handout with all checkins, the completed data tables and answer the lab questions at the end of this handout. Answer the questions on a separate piece of paper. All numbers must be expressed with the correct number of significant figures. If you must type your tables to make them legible then please use Excel or another spreadsheet program. Be sure to use borders on your tables.
You will be graded on the quality of your writing, the professionalism of your work’s appearance, and the quality of the answers to the questions.
For the following measurements and calculations pay close attention to the number of significant figures and use the rules for doing calculations with significant figures.
Use the following data table to collect your data. If you find that the boxes are too small then make the data table in your lab notebook. Remember, every individual student must have their own copy of these data so everyone must write it down. The numbers 1  4 in this data table represent different individuals making measurements of exactly the same object.
Here are some sample data I collected for the paperclip in cm: 3.30, 3.28, 3.29, 3.30. Average: 3.29 cm, Range: 0.02 cm, ± Amount: 0.01 cm, Result: 3.29 ± 0.01 cm, Percent Error: 0.30%.
Paperclip  Book  Lab Bench  
Length (cm)  Length (cm)  Width (cm)  Height (cm)  Volume (cm^{3})  Length (cm)  Width (cm)  Area (A_{1}) (cm^{2})  Sink Length (cm) 
Sink Width (cm) 
Area (A_{2}) (cm^{2})  Useable Area A_{1} – A_{2} (cm^{2}) 

1  
2  
3  
4  
Average  
Range  
± Amount  
Result  
Percent Error 
When you have completed your work in this section check in with your teacher. Initials will be given for a complete set of data and calculations.
Your teacher has provided several objects for you to weigh.
There are several graduated cylinders at the back of the room. These instruments are used to measure the volume of liquids. Each one has a limit to its precision. See if you can find out whether there is a relationship between the size of a graduated cylinder and its precision.
For this lab you must turn in your lab handout with all checkins, the completed data tables and answer the lab questions at the end of this handout. Answer the questions on a separate piece of paper. All numbers must be expressed with the correct number of significant figures. If you must type your tables to make them legible then please use Excel or another spreadsheet program. Be sure to use borders on your tables.
You will be graded on the quality of your writing, the professionalism of your work’s appearance, and the quality of the answers to the following questions.