Data set A consists of five consecutive even integers. What is the ratio of the standard d… — GMAT

Data set $A$ consists of five consecutive even integers. What is the ratio of the standard deviation of data set $A$ to the range of data set $A$ ?

\frac{\sqrt{2}}{4}

For any set of five consecutive even integers, the deviations from the mean are always

-4, -2, 0, 2, 4

, yielding a variance of 8 and a standard deviation of

2\sqrt{2}

. The range of any five consecutive even integers is always

8

. Taking the ratio of the standard deviation to the range gives

\frac{2\sqrt{2}}{8}

, which simplifies to

\frac{\sqrt{2}}{4}

Represent the dataset algebraic terms

Let the five consecutive even integers be

x, x+2, x+4, x+6, x+8

Choosing symmetric terms simplifies calculating the mean and deviations.

Calculate the mean of the dataset

Mean

\mu = \frac{x + (x+2) + (x+4) + (x+6) + (x+8)}{5} = x+4

The mean of an evenly spaced set of consecutive numbers is equal to the middle element.

Calculate the deviations from the mean and sum of squared deviations

Deviations:

-4, -2, 0, 2, 4

.
Squared deviations:

16, 4, 0, 4, 16

.
Sum of squared deviations

= 40

Standard deviation measures dispersion from the mean.

Calculate the variance and standard deviation

Variance

\sigma^2 = \frac{40}{5} = 8

.
Standard deviation

\sigma = \sqrt{8} = 2\sqrt{2}

Variance is the average of squared deviations, and standard deviation is its non-negative square root.

Calculate the range and the ratio of standard deviation to range

Range

= (x+8) - x = 8

.
Ratio

= \frac{\sigma}{\text{Range}} = \frac{2\sqrt{2}}{8} = \frac{\sqrt{2}}{4}

Range is the difference between the maximum and minimum values.

Properties of Standard Deviation and Range for Consecutively Spaced Datasets

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