Data Set P consists of five distinct integers with a range of 20 and a standard deviation … — GMAT

Data Set $P$ consists of five distinct integers with a range of $20$ and a standard deviation of $d_1$ , where $d_1 > 0$ . A new Data Set $Q$ is formed by multiplying each integer in Data Set $P$ by $-2$ and then adding $7$ to the product. What are the range and standard deviation of Data Set $Q$ in terms of $d_1$ ?

Range is $40$ ; standard deviation is $2d_1$ Answer
B
Range is $40$ ; standard deviation is $-2d_1$
C
Range is $-40$ ; standard deviation is $-2d_1$
D
Range is $47$ ; standard deviation is $2d_1 + 7$
E
Range is $20$ ; standard deviation is $2d_1$

Answer

Range is

40

; standard deviation is

2d_1

The correct answer accounts for both linear transformation rules for measures of dispersion: multiplying every value in a dataset by a scalar

c

multiplies both the range and standard deviation by

|c|

, and adding a constant

k

to every value does not alter either measure. Here, multiplying by

-2

scales the range from

20

40

and the standard deviation from

d_1

2d_1

. Adding

7

has no effect on either measure.

Step-by-Step Solution

Analyze the effect of multiplying each element by a constant

c = -2

The range scales by

|c| = |-2| = 2

, giving a new range of

20 \times 2 = 40

. The standard deviation scales by

|c| = |-2| = 2

, giving a new standard deviation of

2d_1

Measures of dispersion (range and standard deviation) are non-negative distance metrics, so they scale by the absolute value of any multiplicative constant.

Analyze the effect of adding a constant

k = 7

to each element.

The range remains

40

, and the standard deviation remains

2d_1

Adding a constant shifts every data point by the exact same distance, leaving all pairwise distances between data points unchanged.

Key Concept

Linear Transformations on Range and Standard Deviation

Question

Answer

Step-by-Step Solution

Key Concept

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