Range and Standard Deviation Alıştırma Soruları — GMAT

Soru 1Soru →

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 $x$ ?

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Cevap: 9

Cevap

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

or

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.

Adım Adım Çözüm

1

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.

2

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.

3

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.

4

Solve the quadratic equation for

x

.

x = 4

or

x = 9

.

Factoring

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

gives two potential solutions.

5

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.

Anahtar Kavram

Variance calculation and dataset order statistics

Soru 2Soru →

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 $k$ ?

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Cevap: 15

Cevap

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.

Adım Adım Çözüm

1

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.

2

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.

3

Equate

\sigma_T^2

to

\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.

4

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.

Anahtar Kavram

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.

Soru 3Soru →

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 $S^2$ ?

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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

1

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.

2

Define Set B relative to Set A.

The largest element of Set

A

is

x+4

. The smallest element of Set

B

is

(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.

3

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.

4

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

Soru 4Soru →

During a quality control inspection, five manufactured items were measured and found to have lengths of $44\text{ mm}$ , $48\text{ mm}$ , $50\text{ mm}$ , $52\text{ mm}$ , and $56\text{ mm}$ . What is the standard deviation, in millimeters, of the lengths of these five items?

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Cevap: 4

Cevap

The standard deviation of the lengths of the five items is 4 millimeters.

The arithmetic mean of the five measurements is 50 mm. The sum of the squared deviations from 50 is 36 + 4 + 0 + 4 + 36 = 80. Dividing 80 by 5 yields a variance of 16. Taking the principal square root of 16 gives a standard deviation of 4 mm.

Adım Adım Çözüm

1

Calculate the arithmetic mean of the dataset

Mean = 50

The standard deviation measures dispersion relative to the mean.

2

Find the squared difference of each data point from the mean

Squared deviations are 36, 4, 0, 4, and 36

Squaring ensures all deviations are non-negative and penalizes larger deviations.

3

Compute the mean of the squared deviations (variance)

Variance = 80 / 5 = 16

Variance is the average squared distance from the mean.

4

Take the non-negative square root of the variance

Standard deviation = sqrt(16) = 4

Standard deviation converts variance back to the original unit of measurement.

Anahtar Kavram

Standard Deviation Calculation for a Data Set

Soru 5Soru →

A dataset consists of five identical integers: $12, 12, 12, 12,$ and $12$ . What is the standard deviation of this dataset?

0

12

-12

144

2\sqrt{3}

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Cevap: $0$

Cevap

0

The correct value is

0

. Standard deviation measures how spread out the numbers in a dataset are around the mean. Since every value in the set is

12

, the mean is

12

and every value lies exactly on the mean (distance of

0

). Thus, the standard deviation is

0

.

Adım Adım Çözüm

1

Calculate the arithmetic mean of the dataset

Mean

\mu = \frac{12 + 12 + 12 + 12 + 12}{5} = 12

The mean is needed to measure individual deviations.

2

Calculate the deviation of each data point from the mean

Deviations are

12 - 12 = 0

for all five elements

Standard deviation measures the average distance of data points from the mean.

3

Compute the standard deviation

Standard Deviation

SD = \sqrt{\frac{0^2 + 0^2 + 0^2 + 0^2 + 0^2}{5}} = 0

When all data points are identical, the spread around the mean is zero.

Anahtar Kavram

Standard deviation measures the dispersion or spread of a set of numbers around its mean. If all elements in a set are identical, there is no variation, so both the range and the standard deviation are equal to 0.

Soru 6Soru →

Set $S$ consists of five distinct integers: $14, 22, 18, 9,$ and $31$ . What is the range of the numbers in Set $S$ ?

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Cevap: 22

Cevap

The range of the numbers in Set

S

is

22

.

To find the range of a set of numbers, subtract the minimum value from the maximum value in the set. For Set

S = \{14, 22, 18, 9, 31\}

, the maximum value is

31

and the minimum value is

9

. Subtracting the minimum from the maximum gives

31 - 9 = 22

.

Adım Adım Çözüm

1

Identify the maximum and minimum elements in the given set.

The maximum element is

31

and the minimum element is

9

.

The range of a dataset is defined as the difference between its greatest and least values.

2

Subtract the minimum value from the maximum value.

31 - 9 = 22

.

Applying the formula

\text{Range} = \text{Maximum} - \text{Minimum}

yields the range of the set.

Anahtar Kavram

Range of a Numerical Data Set

Soru 7Soru →

Data Set $A$ consists of $n$ numbers with mean $\mu$ and standard deviation $\sigma$ , where $\sigma > 0$ . Data Set $B$ is created by transforming each element $x$ in Data Set $A$ into $y = 5 - 2x$ . If the variance of Data Set $B$ is equal to $k$ , which of the following expressions represents the standard deviation of Data Set $A$ in terms of $k$ ?

\frac{\sqrt{k}}{2}

-\frac{\sqrt{k}}{2}

\frac{k}{4}

\frac{\sqrt{k - 5}}{2}

\sqrt{\frac{k}{2}}

Cevabı ve açıklamayı göster

Cevap: $\frac{\sqrt{k}}{2}$

Cevap

\frac{\sqrt{k}}{2}

Under a linear transformation

y = ax + b

, constant shifts do not change data dispersion, while multiplying elements by

a

scales the standard deviation by

|a|

and the variance by

a^2

. Here,

y = -2x + 5

, so

a = -2

. The variance of Data Set

B

is

\text{Var}(B) = (-2)^2 \cdot \text{Var}(A) = 4\sigma^2

. Given

\text{Var}(B) = k

, we get

4\sigma^2 = k

, which leads to

\sigma^2 = \frac{k}{4}

. Taking the positive square root yields

\sigma = \frac{\sqrt{k}}{2}

.

Adım Adım Çözüm

1

Analyze the linear transformation rule for standard deviation and variance.

For

y = ax + b

,

\text{SD}(y) = |a| \cdot \text{SD}(x)

and

\text{Var}(y) = a^2 \cdot \text{Var}(x)

.

Adding a constant

b

shifts all data points equally without altering dispersion, while multiplying by

a

scales distance from the mean by

|a|

.

2

Apply the transformation

y = -2x + 5

to find the variance of Data Set

B

in terms of

\sigma

.

\text{Var}(B) = (-2)^2 \cdot \text{Var}(A) = 4\sigma^2.

The coefficient of

x

is

a = -2

, so variance scales by

a^2 = (-2)^2 = 4

.

3

Set the variance equal to

k

and solve for the standard deviation

\sigma

of Data Set

A

.

4\sigma^2 = k \implies \sigma^2 = \frac{k}{4} \implies \sigma = \frac{\sqrt{k}}{2}.

Standard deviation is strictly non-negative (

\sigma > 0

), requiring the positive square root.

Anahtar Kavram

Linear Transformation of Dispersion Measures

Tahmini Süre:2m 0s

Soru 8Soru →

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 $\sigma^2$ ?

6.4

7.2

7.6

8.0

8.4

Cevabı ve açıklamayı göster

Cevap: $7.6$

Cevap

The maximum possible value of the variance

\sigma^2

is

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

.

Adım Adım Çözüm

1

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.

2

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.

3

Evaluate valid dataset configurations for

a = 7

that satisfy the unique mode condition.

If

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.

4

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.

Anahtar Kavram

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

Soru 9Soru →

Data Set $P$ consists of five distinct integers with a range of $20$ and a standard deviation of $d_1$ , where $d_1 > 0$ . A new Data Set $Q$ is formed by multiplying each integer in Data Set $P$ by $-2$ and then adding $7$ to the product. What are the range and standard deviation of Data Set $Q$ in terms of $d_1$ ?

Range is

40

; standard deviation is

2d_1

Range is

40

; standard deviation is

-2d_1

Range is

-40

; standard deviation is

-2d_1

Range is

47

; standard deviation is

2d_1 + 7

Range is

20

; standard deviation is

2d_1

Cevabı ve açıklamayı göster

Cevap: Range is $40$ ; standard deviation is $2d_1$

Cevap

Range is

40

; standard deviation is

2d_1

The correct answer accounts for both linear transformation rules for measures of dispersion: multiplying every value in a dataset by a scalar

c

multiplies both the range and standard deviation by

|c|

, and adding a constant

k

to every value does not alter either measure. Here, multiplying by

-2

scales the range from

20

to

40

and the standard deviation from

d_1

to

2d_1

. Adding

7

has no effect on either measure.

Adım Adım Çözüm

1

Analyze the effect of multiplying each element by a constant

c = -2

.

The range scales by

|c| = |-2| = 2

, giving a new range of

20 \times 2 = 40

. The standard deviation scales by

|c| = |-2| = 2

, giving a new standard deviation of

2d_1

.

Measures of dispersion (range and standard deviation) are non-negative distance metrics, so they scale by the absolute value of any multiplicative constant.

2

Analyze the effect of adding a constant

k = 7

to each element.

The range remains

40

, and the standard deviation remains

2d_1

.

Adding a constant shifts every data point by the exact same distance, leaving all pairwise distances between data points unchanged.

Anahtar Kavram

Linear Transformations on Range and Standard Deviation

Soru 10Soru →

Data set $A$ consists of five consecutive even integers. What is the ratio of the standard deviation of data set $A$ to the range of data set $A$ ?

\frac{\sqrt{2}}{4}

-\frac{\sqrt{2}}{4}

\frac{\sqrt{10}}{8}

\frac{\sqrt{2}}{2}

1

Cevabı ve açıklamayı göster

Cevap: $\frac{\sqrt{2}}{4}$

Cevap

\frac{\sqrt{2}}{4}

For any set of five consecutive even integers, the deviations from the mean are always

-4, -2, 0, 2, 4

, yielding a variance of 8 and a standard deviation of

2\sqrt{2}

. The range of any five consecutive even integers is always

8

. Taking the ratio of the standard deviation to the range gives

\frac{2\sqrt{2}}{8}

, which simplifies to

\frac{\sqrt{2}}{4}

.

Adım Adım Çözüm

1

Represent the dataset algebraic terms

Let the five consecutive even integers be

x, x+2, x+4, x+6, x+8

.

Choosing symmetric terms simplifies calculating the mean and deviations.

2

Calculate the mean of the dataset

Mean

\mu = \frac{x + (x+2) + (x+4) + (x+6) + (x+8)}{5} = x+4

.

The mean of an evenly spaced set of consecutive numbers is equal to the middle element.

3

Calculate the deviations from the mean and sum of squared deviations

Deviations:

-4, -2, 0, 2, 4

.
Squared deviations:

16, 4, 0, 4, 16

.
Sum of squared deviations

= 40

.

Standard deviation measures dispersion from the mean.

4

Calculate the variance and standard deviation

Variance

\sigma^2 = \frac{40}{5} = 8

.
Standard deviation

\sigma = \sqrt{8} = 2\sqrt{2}

.

Variance is the average of squared deviations, and standard deviation is its non-negative square root.

5

Calculate the range and the ratio of standard deviation to range

Range

= (x+8) - x = 8

.
Ratio

= \frac{\sigma}{\text{Range}} = \frac{2\sqrt{2}}{8} = \frac{\sqrt{2}}{4}

.

Range is the difference between the maximum and minimum values.

Anahtar Kavram

Properties of Standard Deviation and Range for Consecutively Spaced Datasets

Range and Standard Deviation

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