Question

Range Rule of Thumb for Estimating s The 20 brain volumes (cm3 ) from Data Set 8 “IQ and Brain Size” in Appendix B vary from a low of 963 cm3 to a high of 1439 cm3. Use the range rule of thumb to estimate the standard deviation s and compare the result to the exact standard deviation of 124.9 cm3.

Short answer

The estimated value of the standard deviation is equal to 119.0 ${\text{cm}}^3$.

The estimated value of is approximately equal to the exact value of ${\text{124.9cm}}^3$.

Step-by-step solution

Given information

The data given shows the total brain volume $\left({\text{cm}}^3\right)$of 20 identical twins.

The highest and the lowest volumes are equal to 1439${\text{cm}}^3$ and 963${\text{cm}}^3,$ respectively.

Estimation of standard deviation

Standard deviation(s) can be estimated by the range rule of thumb, which says that one by four of the range of a set of values gives the estimated value of the standard deviation. Mathematically,

$s\approx \frac{range}{4}$

Here, the range can be calculated by subtracting the minimum (or lowest) value from the maximum (or highest) value.

Substituting the values in the above formula,

.$$\begin{gathered} s\approx \frac{1439-963}{4} \\ =\frac{476}{4} \\ =119.0 \end{gathered}$$

Thus, the estimated value of the standard deviation is 119${\text{cm}}^3$

Comparison with the exact value

It can be seen that the estimated value of the standard deviation equal to 119${\text{cm}}^3$ is quite close to the exact value of 124.9 ${\text{cm}}^3$. Thus, it is a good estimate.