Let r be the range and S^2=(1)/(n-1) ∑i=1^n(xi-x̅)^2 be the S.D. of a set of observations x1, x2

step 1 In problem 10 we have r is the range and s squared is the standard deviation of a set of observa

Let r be the range and S^2=(1)/(n-1) ∑i=1^n(xi-x̅)^2 be the...

Let $r$ be the range and $S^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}$ be the
S.D. of a set of observations $x_{1}, x_{2}, \ldots, x_{n}$, then
(A) $S \leq r \sqrt{\frac{n}{n-1}}$
(B) $S=r \sqrt{\frac{n}{n-1}}$
(C) $S \geq r \sqrt{\frac{n}{n-1}}$
(D) None of these

Key Concepts

Maximizing Variance Under a Fixed Range

When the range is held fixed, the maximum possible variance is attained by pushing the data points to the extremes. In an idealized scenario, this would involve partitioning the observations into two groups, one at the minimum and one at the maximum. This idea underlies the derivation of an upper bound on the standard deviation in terms of the range.

Inequalities Relating Dispersion Measures

In comparing measures of dispersion, it is often useful to establish bounds on one measure in terms of another. In this context, one can show by considering extreme configurations (data taking only on the extreme values) that an inequality of the form S ? r?(n/(n–1)) must hold. This highlights how different summaries of spread are related, even if the bound is not the best possible.

Sample Variance and Standard Deviation

These are measures of dispersion that quantify how far the data points are spread out from their mean. The sample variance is defined as the average of the squared deviations from the sample mean (dividing by n – 1) and its square root is the sample standard deviation. They are fundamental in statistics for describing variability and are used in numerous inferential procedures.

Range as a Measure of Dispersion

The range is the simplest measure of dispersion and is defined as the difference between the maximum and minimum values in a data set. While it gives a quick idea of the spread of the data, it is very sensitive to outliers and does not take into account how the data are distributed between the extremes.

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