Error Analysis

Every lab report must include step 1. A straightforward error analysis includes steps 1 and 2. A sophisticated error analysis includes steps 1-4.

• Report uncertainties with each measured value
  
• Based on estimates of measuring device uncertainty . For single measurements, an estimate of uncertainty can be assumed from the precision of the instrumental output. For example, if an absorbance reading measures 0.47, then the uncertainty can be assumed to be +/- 0.01 if no further information is available. For some measuring devices (pipettes, graduated cylinders) the uncertainty is known and standardized.
  
• Based on repetitive measurements . A much more accurate method of determining uncertainty is to find the standard deviation of a set of measurements on the same sample. For example, the absorbance of a solution was measured five times and the values were 0.47, 0.51, 0.48, 0.46, 0.47. The average and uncertainty in this case would be 0.48 +/- 0.2.
  
• Report uncertainties with calculated values
  
• Using standard formulas .
• When adding numbers, add their uncertainties.
  For example, (1.2 +/- 0.3) + (-0.2 +/- 0.1) = 1.0 +/- 0.4
• When multiplying or dividing numbers, add their percent uncertainties.
  For example, (1.2 +/- 0.3)(-0.2 +/- 0.1) = (1.2 +/- 25%)(-0.2 +/- 50%)
  = (0.24 +/- 75%) or (0.24 +/- 0.18)
  
• Using calculus to derive the formulas . In less-simple calculations, it is necessary to use calculus to determine uncertainties. The differential dP represents the uncertainty in the value for P.
  
  For example, if you are given
  P = rsin q , with r = (7 +/- 1) and q = ( p /4 +/- 0.1)
  This means that r = 7, dr = 1, q = p/4, , and d q = 0.1
  
  Take the total differential of both sides to yield
  dP = sin q dr + rcos q d q
  
  So calculate P = rsin q = (7)(sin p /4) = 4.95
  and calculate dP = sin q dr + rcos q d q = (sin p /4)(1) + (7)(cos p /4)(0.1) =1.2
  and report that P = 5.0 +/- 1.2
  
• Report uncertainties from graphical data analysis (have an Excel template for this.) Based on slope and intercept uncertainties, what are the uncertainties in calculated values?
  
• Compare your measured value to the true, accepted, or expected value. Decide whether your measured value agrees with the expected value within the bounds of your error. Critically evaluate the accuracy and legitimacy of your results.