Measurement, Calculation, and Error

In reports of scientific work, each quantity has two parts: a numerical value and a unit of measure. In a laboratory context, the word error refers not to a mistake, but to your degree of confidence in a reported value. The precision of measured values (and calculations based on those measured values) is limited by the nature of the system measured and the tools used to make the measurements.

 

Reading digital instruments: Record all digits. For example, if a balance display reads 27.100 g, write 27.100 g, not 27.1 g.

 

Reading linear instruments: Read and record to 0.1 of the smallest division. When using a gradu­ated cylinder whose smallest division is 1 mL, record to 0.1 mL.

           

                       

 

Retain extra digits measured or shown on your calculator until all calculations of a value are complete, then round to the correct number of significant figures (# s.f.). See the discussion of significant figures in your lecture text  (Chang, Chapter 1).

 

Example: The mass of a 10.00 mL sample of liquid is 10.176 g.

 

            Density = 1.0176 g/mL, rounds to 1.018 g/mL (4 s.f.)

 

Absolute Error is the absolute value of the difference between the true value (or expected value) and your measured or calculated value.

 

            Absolute Error = |True Value – Found Value|

 

Relative Error is the absolute error expressed as a fraction of the true value.

 

            Relative Error = |True Value – Found Value| / True Value

 

Percent Error is the relative error expressed as a percentage of the true value.

 

            Percent Error = Relative Error x 100% = x 100%

 

The Range is the difference between the largest and smallest measured or calcu­lated value of a quantity.

 

Examples:

 

1.  Three measurements of the density of a solution give values of 1.017, 1.016, and 1.020 g/mL. The average of these values, taken as the found value, is 1.018 g/mL. The true value of the density is known to be 1.020 g/mL. Calculate the absolute error, the relative error, the percent error and the range.

 

      Absolute Error = |1.020 – 1.018| = 0.002 g/mL

 

      Relative Error = = 0.0019607 = 0.002

 

      Percent Error = Relative Error x 100% = 0.19607% = 0.2%

 

      Range = 1.020 – 1.016 = 0.004 g/mL

 

2.  The percentage of SrCl2.6H2O in a mixture is found to be 61.2%. The true value is 60.1%. Calculate the absolute error, the relative error and the percent error.

 

      Absolute Error = |60.1 – 61.2| = 1.1%

 

      Relative Error =   = 0.0183028 = 0.018

 

      Percent Error =   x 100%= 1.83028% = 1.8%