Dataset P consists of the five numbers 10, 20, 30, 40, and 50. Dataset Q is created by rep… — GRE General Test

Dataset $P$ consists of the five numbers $10, 20, 30, 40,$ and $50$ . Dataset $Q$ is created by replacing the minimum value in Dataset $P$ with $18$ and the maximum value with $42$ , leaving the remaining three numbers unchanged. Which of the following statements correctly compares the mean and standard deviation of Dataset $Q$ to those of Dataset $P$ ?

The mean of Dataset $Q$ is equal to the mean of Dataset $P$ , and the standard deviation of Dataset $Q$ is less than the standard deviation of Dataset $P$ .Answer
B
The mean of Dataset $Q$ is equal to the mean of Dataset $P$ , and the standard deviation of Dataset $Q$ is greater than the standard deviation of Dataset $P$ .
C
The mean of Dataset $Q$ is equal to the mean of Dataset $P$ , and the standard deviation of Dataset $Q$ is equal to the standard deviation of Dataset $P$ .
D
The mean of Dataset $Q$ is greater than the mean of Dataset $P$ , and the standard deviation of Dataset $Q$ is less than the standard deviation of Dataset $P$ .
E
The mean of Dataset $Q$ is less than the mean of Dataset $P$ , and the standard deviation of Dataset $Q$ is greater than the standard deviation of Dataset $P$ .

Answer

The mean of Dataset

Q

is equal to the mean of Dataset

P

, and the standard deviation of Dataset

Q

is less than the standard deviation of Dataset

P

The mean of both datasets is 30 because the decrease of 8 from 50 balances the increase of 8 to 10. Standard deviation quantifies how far data points deviate from the mean. Because 18 and 42 are closer to the mean of 30 than 10 and 50 are, the spread of Dataset Q around the mean is strictly smaller, making its standard deviation smaller.

Step-by-Step Solution

Calculate the mean of Dataset

P

\text{Mean of } P = \frac{10 + 20 + 30 + 40 + 50}{5} = \frac{150}{5} = 30

To find the baseline central tendency before the dataset values are modified.

Calculate the mean of Dataset

Q

\text{Mean of } Q = \frac{18 + 20 + 30 + 40 + 42}{5} = \frac{150}{5} = 30

Replacing

10

with

18

(+8) and

50

with

42

(-8) results in a net change of zero to the sum, so the mean remains unchanged.

Compare the dispersion of Dataset

Q

relative to Dataset

P

In Dataset

P

, the squared deviations of the modified points from the mean are

(10 - 30)^2 = 400

and

(50 - 30)^2 = 400

. In Dataset

Q

, the squared deviations of these points are

(18 - 30)^2 = 144

and

(42 - 30)^2 = 144

Standard deviation measures the average distance of data points from the mean. Since the outer values in Dataset

Q

are closer to the mean than in Dataset

P

, the overall dispersion and standard deviation decrease.

Key Concept

Standard deviation measures the spread of data points around their mean; bringing extreme values closer to the mean reduces the standard deviation.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic