A dataset X consists of five positive integers with a mean of 12, a median of 12, a unique… — GMAT

A dataset $X$ consists of five positive integers with a mean of 12, a median of 12, a unique mode of 12, and a range of 8. If $\sigma^2$ represents the variance (the square of the standard deviation) of dataset $X$ , what is the maximum possible value of $\sigma^2$ ?

A
$6.4$
B
$7.2$
$7.6$ Answer
D
$8.0$
E
$8.4$

Answer

The maximum possible value of the variance

\sigma^2

7.6

To maximize variance

\sigma^2 = \frac{1}{5}\sum (x_i - 12)^2

, data points must be placed as far from the mean of 12 as possible. For five ordered positive integers

a \le b \le c \le d \le e

, the mean of 12 gives

a+b+c+d+e=60

, the median of 12 gives

c=12

, and the range of 8 gives

e=a+8

. Substituting yields

2a+b+d=40

. Since

b \le 12

and

d \le a+8

b+d \le a+20

, leading to

3a \ge 20

, so

a \ge 7

. Testing

a=7

gives

e=15

and

b+d=26

. To keep 12 as the unique mode,

b=12

and

d=14

, forming dataset

\{7, 12, 12, 14, 15\}

. The squared deviations from 12 are

(-5)^2 + 0^2 + 0^2 + 2^2 + 3^2 = 25 + 0 + 0 + 4 + 9 = 38

. Dividing by 5 yields the maximum variance of

7.6

Step-by-Step Solution

Express the five integers in ascending order and set up equations from given statistics.

Let the integers be

a \le b \le c \le d \le e

. Mean = 12 implies

a + b + c + d + e = 60

. Median = 12 implies

c = 12

. Range = 8 implies

e = a + 8

Translating statistical measures into algebraic constraints provides the boundary conditions for the dataset.

Substitute known values into the sum equation and determine constraints on

a

a + b + 12 + d + (a + 8) = 60 \implies 2a + b + d = 40

. Since

b \le 12

and

d \le e = a + 8

, we have

b + d \le a + 20

. Thus

2a + (a + 20) \ge 40 \implies 3a \ge 20 \implies a \ge 7

Determining the minimum value of

a

is critical because smaller values of

a

allow larger deviations from the mean of 12.

Evaluate valid dataset configurations for

a = 7

that satisfy the unique mode condition.

a = 7

, then

e = 15

, and

b + d = 26

. With

b \le 12

and

d \le 15

, and requiring 12 to be the unique mode, the only valid integer set is

\{7, 12, 12, 14, 15\}

The set must maintain 12 as the unique mode while maximizing the dispersion of values.

Calculate the variance

\sigma^2

for the set

\{7, 12, 12, 14, 15\}

Deviations from 12 are

-5, 0, 0, 2, 3

. Squared deviations are

25, 0, 0, 4, 9

. Sum of squared deviations = 38. Variance

\sigma^2 = \frac{38}{5} = 7.6

Variance is defined as the mean of the squared deviations from the dataset mean.

Key Concept

Properties of Variance and Standard Deviation under Range and Central Tendency Constraints

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