Data set S consists of the four integers 4, 8, 12, and 16. A fifth integer k, which is gre… — GMAT

Data set $S$ consists of the four integers $4, 8, 12,$ and $16$ . A fifth integer $k$ , which is greater than the mean of data set $S$ , is added to the set to form data set $T = \{4, 8, 12, 16, k\}$ . If the standard deviation of data set $T$ is equal to the standard deviation of data set $S$ , what is the value of $k$ ?

Answer: 15

Answer

The value of

k

is 15.

The variance of the original set

S = \{4, 8, 12, 16\}

is 20 with a mean of 10. Adding

k

creates a 5-element set

T

with mean

\frac{40+k}{5}

. Setting the variance of set

T

equal to 20 yields the quadratic equation

k^2 - 20k + 75 = 0

, which factors as

(k-5)(k-15) = 0

. Since

k

must be greater than 10, the correct answer is 15.

Step-by-Step Solution

Calculate the mean and variance of data set

S

Mean

\mu_S = 10

and variance

\sigma_S^2 = 20

Determining the baseline mean and variance for the initial dataset of 4 elements.

Express the mean and variance of data set

T = \{4, 8, 12, 16, k\}

algebraically in terms of

k

Mean

\mu_T = \frac{40+k}{5}

and variance

\sigma_T^2 = \frac{480 + k^2 - 5\left(\frac{40+k}{5}\right)^2}{5}

Formulating the standard deviation equation for the expanded dataset.

Equate

\sigma_T^2

\sigma_S^2 = 20

and solve for

k

The resulting quadratic equation

k^2 - 20k + 75 = 0

yields solutions

k = 5

and

k = 15

Finding all potential values of

k

that leave the dataset's standard deviation unchanged.

Apply the condition that

k

must be strictly greater than the mean of data set

S

(

\mu_S = 10

k = 15

Rejecting

k = 5

because it is less than 10.

Key Concept

Standard deviation measures the average distance of data points from the mean. Adding a value to a dataset shifts the mean and alters variance; finding when the standard deviation remains unchanged requires solving the quadratic relationship generated by the addition of the new element.

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