A dataset consists of five integers: 2, 5, 8, 11, and x. The variance of the dataset is 10… — GMAT

A dataset consists of five integers: $2, 5, 8, 11,$ and $x$ . The variance of the dataset is $10$ . If the median of the dataset is $8$ , what is the value of $x$ ?

Answer: 9

Answer

The value of

x

is 9.

By setting up the equation for variance in terms of

x

, we obtain the quadratic equation

x^2 - 13x + 36 = 0

, which yields

x = 4

x = 9

. Arranging the set in ascending order for

x = 9

gives

\{2, 5, 8, 9, 11\}

, where the middle number (median) is 8, satisfying all conditions.

Step-by-Step Solution

Calculate the mean of the dataset in terms of

x

The mean is

\mu = \frac{26 + x}{5}

The mean of a dataset is the sum of all elements divided by the number of elements.

Set up the variance equation using the definition of population variance.

Variance

\sigma^2 = \frac{(2-\mu)^2 + (5-\mu)^2 + (8-\mu)^2 + (11-\mu)^2 + (x-\mu)^2}{5} = 10

Variance measures the average squared deviation from the mean.

Substitute

\mu = \frac{26+x}{5}

, clear denominators, expand the algebraic expression, and set up the quadratic equation.

x^2 - 13x + 36 = 0

Multiplying through by 25 and simplifying the quadratic terms yields a standard quadratic form.

Solve the quadratic equation for

x

x = 4

x = 9

Factoring

(x-4)(x-9) = 0

gives two potential solutions.

Evaluate the median condition for both potential values of

x

For

x = 9

, the ordered dataset is

\{2, 5, 8, 9, 11\}

, which has a median of 8.

For

x = 4

, the ordered set is

\{2, 4, 5, 8, 11\}

with median 5, which fails the given condition.

Key Concept

Variance calculation and dataset order statistics

Question

Answer

Step-by-Step Solution

Key Concept

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