Question

Question: Given a data set with a largest value of 760 and a smallest value of 135, what would you estimate the standard deviation to be? Explain the logic behind the procedure you used to estimate the standard deviation. Suppose the standard deviation is reported to be 25. Is this feasible? Explain

Short answer

s = 156.25

Step-by-step solution

Estimating the standard deviation 

Range = largest value - Smallest value

=760 – 135

= 625

$$\begin{gathered} \mathbf{StandardDeviation}\mathbf{=}\frac{\mathbf{Range}}{\mathbf{4}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{625}}{\mathbf{4}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{156}\mathbf{.}\mathbf{25} \end{gathered}$$

Therefore, the estimated s is 156.25.

Explaining the logic behind the method 

Based on the Chebyshev and Empirical rule, we know that most measurements will fall in the 2 and 3 standard deviation range from the mean. Therefore, the standard deviation should lie within that range.

We also know that the standard deviation cannot be more than ¼th of the range. Therefore, to estimate the standard deviation, we use$\mathrm{s}=\frac{\mathrm{Range}}{4}$formula.

Checking the feasibility

The standard deviation as 25 can't be feasible because the standard deviation would lie in the 104.16 to 156.25 range.

We found the lower range by applying the same procedure in step 1 except for the denominator. We will change it to 6 because most observations will lie in the 3 standard deviation range.

$$\begin{gathered} \mathbf{Lower}\mathbf{}\mathbf{range}\mathbf{=}\frac{\mathbf{Range}}{\mathbf{6}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{625}}{\mathbf{6}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{104}\mathbf{.}\mathbf{16} \end{gathered}$$