The standard error is always smaller than the sample standard deviation. Propagation of Uncertainty Whenever we combine measurements via mathematical operations (addition, subtract, multiplication, and division) we need to properly handle the uncertainty. Doing so is called the propagation of uncertainty and it can get rather involved as the mathematical operations get more complicated. Instead of deriving the rules (using algebra or calculus), we will make use of the end result. Addition The uncertainty of the sum of measurements and is the sum the uncertainty of and the L J L uncertainty of J. Subtraction The uncertainty of the subtraction of measurements and is the sum the uncertainty of and L J L the uncertainty of J. Multiplication The uncertainty of the product of measurements and is slightly more complicated than for L J simple addition and subtraction. It is given by adding the relative uncertainties for and and L J multiplying the resulting sum by the product of the two measurements. You can obtain the percent uncertainty of the product by adding the percent uncertainties for L and J. Division The uncertainty of the quotient of the measurements and is similar to the rule above for L J multiplication. Simply replace the product with the quotient (LJ) (L/J). You can obtain the percent uncertainty of the quotient by adding the percent uncertainties for L and J. Multiplication by a Constant The uncertainty for multiplying a measurement and its uncertainty by a constant is given by
The standard error is always smaller than the sample standard deviation. Propagation of Uncertainty Whenever we combine measurements via mathematical operations (addition, subtract, multiplication, and division) we need to properly handle the uncertainty. Doing so is called the propagation of uncertainty and it can get rather involved as the mathematical operations get more complicated. Instead of deriving the rules (using algebra or calculus), we will make use of the end result. Addition The uncertainty of the sum of measurements L and J is the sum the uncertainty of L and the uncertainty of J. Subtraction The uncertainty of the subtraction of measurements L and J is the sum the uncertainty of L and the uncertainty of J. Multiplication The uncertainty of the product of measurements L and J is slightly more complicated than for simple addition and subtraction. It is given by adding the relative uncertainties for L and J and multiplying the resulting sum by the product of the two measurements. You can obtain the percent uncertainty of the product by adding the percent uncertainties for L and J. Division The uncertainty of the quotient of the measurements L and J is similar to the rule above for multiplication. Simply replace the product (LJ) with the quotient (L/J). You can obtain the percent uncertainty of the quotient by adding the percent uncertainties for L and J. Multiplication by a Constant The uncertainty for multiplying a measurement and its uncertainty by a constant is given by