Question

Verify the claim made in a section 4.4 that if all results of a repeated experiment are equal, the standard deviation, equation (4.13) Will be0.

Short answer

The standard deviation is 0.

Step-by-step solution

Step 1:Explanation of solution.

For a given set of results, the standard deviation is the same and equal to 0.

Standard Deviation Definitions

We can solve for both quantities by directly applying the mean and standard deviation definitions given by equations (1) and (2) respectively. Here, refers to the outcome of each measurement, and refers to the number of these measurements.

$\overline{\mathbf{Q}}\mathbf{=}\frac{{\displaystyle \mathbf{\sum }_{\mathbf{i}}}{\mathbf{Q}}_{\mathbf{i}}{\mathbf{n}}_{\mathbf{i}}}{{\displaystyle \mathbf{\sum }_{\mathbf{i}}}{\mathbf{n}}_{\mathbf{i}}}$………………(1)

role="math" localid="1658322728674" $\mathbf{∆}\mathit{Q}\mathbf{=}\sqrt{\frac{{\displaystyle \mathbf{\sum }_{\mathbf{i}}{\mathbf{(}{\mathbf{Q}}_{\mathbf{i}}\mathbf{-}{\overline{\mathbf{Q}}}_{\mathbf{i}}\mathbf{)}}^{\mathbf{2}}{\mathbf{n}}_{\mathbf{i}}}}{{\displaystyle \mathbf{\sum }_{\mathbf{i}}{\mathbf{n}}_{\mathbf{i}}}}}$……………………(2)

Calculation.

When Q is a constant and can be removed from the equation, the expression is calculated as,

$\begin{array}{rcl}∆Q& =& {\left[\frac{{\displaystyle \sum _i}{(Q_i-\overline{Q_i})}^2{n}_i}{{\displaystyle \sum _i}{n}_i}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ & =& 0\end{array}$

As a result, the standard deviation for a particular set of outcomes is equal 0.