A dataset of 25 student test scores has a mean of 70, a standard deviation of 8, and an in… — GRE General Test

A dataset of 25 student test scores has a mean of $70$ , a standard deviation of $8$ , and an interquartile range ( $\text{IQR}$ ) of $12$ . A instructor creates a modified dataset by multiplying each original test score by $1.5$ and then adding $10$ to the result. What is the standard deviation of the modified dataset?

A
$8$
$12$ Cevap
C
$18$
D
$22$
E
$28$

Cevap

For any dataset, if each value

x

is transformed into

a \cdot x + b

, where

a

and

b

are constants, the new standard deviation is given by

|a| \cdot \sigma_{\text{old}}

. In this problem,

a = 1.5

and

b = 10

. Therefore, the new standard deviation is

1.5 \times 8 = 12

. The addition of

10

affects measures of center (such as the mean and median) but has no effect on measures of dispersion (such as standard deviation, IQR, and range).

Adım Adım Çözüm

Analyze the effect of multiplying each data point by a constant on standard deviation.

Multiplying every value in a dataset by a constant

c = 1.5

multiplies the standard deviation

\sigma

|c|

. The new intermediate standard deviation becomes

1.5 \times 8 = 12

Standard deviation measures dispersion around the mean, so scaling all data points by a factor scales the distance between each point and the mean by that same factor.

Analyze the effect of adding a constant to each data point on standard deviation.

Adding a constant

k = 10

to every value shifts the entire distribution without changing the distances between data points or the relative spread around the mean. Thus, standard deviation remains

12

Adding a constant shifts both the individual values and the mean by the same amount, leaving

(x_i - \bar{x})

unchanged.

Anahtar Kavram

Linear Transformations on Measures of Dispersion

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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