[Notes] - Percent Error (Percent Deviation, Relative Error)

Accuracy

When scientists need to compare the results of two different measurements, the absolute 
difference between the values is of very little use.  The magnitude of error of being off by 
10 cm depends on whether you are measuring the length of a piece of paper or the distance
from New Orleans to Houston.  To express the magnitude of the error (or deviation) between
two measurements scientists invariably use percent error .

If you are comparing your value to an accepted value, you first subtract the two values so 
that the difference you get is a positive number.  This is called taking the absolute value 
of the difference.  Then you divide this result (the difference) by the accepted value to 
get a fraction, and then multiply by 100% to get the percent error.

So,     % error =   | your result - accepted value |    x  100 % 
                                          accepted value

Several points should be noted when using this equation to obtain a percent error.

1) When you subtract note how many significant figures remain after the subtraction, and 
express your final answer to no more than that number of digits.

2)  If neither of the two values being compared is an "accepted value", then use either 
number in the denominator to get the fraction.  If one value is more reliable than the
other, choose it for the denominator.

3) Treat the % symbol as a unit. The fraction is dimensionless because units
in the values will cancel.

4) Notice that the error is a positive number if the experimental value is too high, and is a
 negative number if the experimental value is too low.

Example:  A student measures the volume of a 2.50 liter container to be 2.38 liters. 
What is the percent error in the student's measurement?

Ans.      % error = (2.50 liters - 2.38 liters)  x   100% 
                                  2.50 liters

                       =  (.12 liters)     x    100% 
                            2.50 liters

                        =  .048     x   100%

                        =  4.8% error 
  
(Note only two sig figs left in the answer after the subtraction) 
  

Precision

Frequently in science, an accepted or true value is not known. The accuracy of a measurement cannot be reported if an accepted value is unavailable.  Precision is a measure of how reproducible experimental measurements are.  Precision is reported as Deviation or Difference of values.

The Absolute Deviation, or Absolute Difference, of each measurement is the difference of each measure­ment from the mean or average:

                Absolute deviation = Measured value - Mean

The Average Deviation, or Average Difference, is the average of all of the absolute deviations.  The Percent Deviation, or Percent Difference, is the average deviation reported as a percentage:

Percent Deviation =    Average Deviation   x 100%
                                                Mean