Set A consists of 5 consecutive even integers, and Set B consists of 5 consecutive even in… — GMAT

Set $A$ consists of 5 consecutive even integers, and Set $B$ consists of 5 consecutive even integers. The smallest element of Set $B$ is 6 greater than the largest element of Set $A$ . If $S$ represents the standard deviation of the combined set of 10 integers formed by merging Set $A$ and Set $B$ , what is the value of $S^2$ ?

Cevap: 57

Cevap

The value of

S^2

is 57.

Each subset of 5 consecutive even integers has an internal variance of 8. Because the mean of Set A is

x

and the mean of Set B is

x+14

, the combined mean is

x+7

. Each group mean is 7 units away from the overall mean. By the parallel axis theorem for variance, the combined variance is the within-group variance plus the squared distance from the combined mean:

8 + 7^2 = 57

Adım Adım Çözüm

Define Set A algebraically and calculate its mean and variance.

Set

A = \{x-4, x-2, x, x+2, x+4\}

. Mean

\mu_A = x

, and variance

\sigma_A^2 = \frac{(-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2}{5} = \frac{40}{5} = 8

Symmetrically defining 5 consecutive even integers simplifies variance calculation.

Define Set B relative to Set A.

The largest element of Set

A

x+4

. The smallest element of Set

B

(x+4) + 6 = x+10

. Thus, Set

B = \{x+10, x+12, x+14, x+16, x+18\}

with mean

\mu_B = x+14

and variance

\sigma_B^2 = 8

Translating the word problem into exact algebraic representations for Set B.

Calculate the mean of the combined 10-element set.

The combined mean

\mu = \frac{5(x) + 5(x+14)}{10} = x+7

The combined mean is essential to evaluate overall variation.

Calculate the variance

S^2

of the combined set.

Using the combined variance formula

S^2 = \frac{1}{2}[\sigma_A^2 + (\mu_A - \mu)^2] + \frac{1}{2}[\sigma_B^2 + (\mu_B - \mu)^2] = \frac{1}{2}[8 + (-7)^2] + \frac{1}{2}[8 + 7^2] = 8 + 49 = 57

Combines internal set variances with the variance between group means.

Anahtar Kavram

Combined Variance and Standard Deviation of Sets

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