Ultra-condensed description of propagation of errors

Suppose that you measure two quantities, x +/- dx cm and y +/- dy cm. What happens to their uncertainties if you need to combine these values?

If you add the quantities, their uncertainties just add.



                  total length  =  (x + y)    cm

   uncertainty in total length  =  (dx + dy)  cm

If you subtract the quantities, again, their uncertainties ADD (not subtract).



         difference in  length  =  (x - y)    cm

   uncertainty in difference    =  (dx + dy)  cm

If you multiply the quantities, you must add their fractional (or percentage) uncertainties to find the fractional (or percentage) uncertainty in the product.



           area of rectangle    =  (x * y)                 square cm


   uncertainty in area              dx   dy
   -------------------          = ( -- + -- )              (pure fraction)
         area                        x    y

                                    dx   dy
   uncertainty in area          = ( -- + -- ) * ( area )   square cm
                                     x    y

If you divide the quantities, you must again add their fractional (or percentage) uncertainties to find the fractional (or percentage) uncertainty in the ratio.


                                      x 
      ratio of length to width  =    ---                   (pure fraction)
                                      y


   uncertainty in ratio             dx   dy
   -------------------          = ( -- + -- )              (pure fraction)
         ratio                       x    y

                                    dx   dy
   uncertainty in ratio         = ( -- + -- ) * ( ratio )  (pure fraction)
                                     x    y

If you raise a value to a power N, you multiply its fractional (or percentage) uncertainty by N to find the fractional (or percentage) uncertainty in the result.


                                     3                              3
         length cubed           =   x                             cm


  uncertainty in length cubed        ( dx )
  ----------------------------  =    (----) * 3            (pure fraction)
         length cubed                ( x  )

                                       dx                            3
  uncertainty in length cubed   =    (----) * 3 * (length cubed)   cm
                                       x