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