Measures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles) Alıştırma Soruları — GRE General Test

Soru 1Soru →

A quality control engineer records the thickness, in millimeters, of 9 sample components: $11, 13, 15, 17, 19, 21, 23, 25,$ and $27$ . Each thickness measurement $x$ is then converted to a scaled rating $y$ using the linear formula $y = 1.5x + 4.8$ . What is the interquartile range (IQR) of the transformed dataset of $y$ -values?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The interquartile range of the transformed dataset is 15.

The interquartile range (IQR) measures the spread of the middle 50% of the data (

Q_3 - Q_1

). For the original dataset

11, 13, 15, 17, 19, 21, 23, 25, 27

, the median is

19

.

Q_1

is the median of the lower half

\{11, 13, 15, 17\}

, which is

\frac{13+15}{2} = 14

.

Q_3

is the median of the upper half

\{21, 23, 25, 27\}

, which is

\frac{23+25}{2} = 24

. Thus, the original

\text{IQR} = 24 - 14 = 10

. Under a linear transformation

y = ax + b

, measures of position shift by

a x + b

, so

Q_1(y) = 1.5(14) + 4.8 = 25.8

and

Q_3(y) = 1.5(24) + 4.8 = 40.8

. Subtracting these yields

\text{IQR}(y) = 40.8 - 25.8 = 15

. Notice that this is simply

1.5 \times 10

, as adding a constant shifts the location of the distribution but leaves measures of spread unchanged.

Adım Adım Çözüm

1

Find the quartiles of the original 9-element dataset.

Q_1 = 14

and

Q_3 = 24

The median of the dataset is 19 (the 5th value). The lower half of the data consists of

11, 13, 15, 17

, so

Q_1 = \frac{13 + 15}{2} = 14

. The upper half consists of

21, 23, 25, 27

, so

Q_3 = \frac{23 + 25}{2} = 24

.

2

Calculate the interquartile range of the original dataset.

\text{IQR}_x = 10

\text{IQR}_x = Q_3 - Q_1 = 24 - 14 = 10

.

3

Apply the linear transformation rules to find the transformed interquartile range.

\text{IQR}_y = 15

For a linear transformation

y = ax + b

, the interquartile range scales by

|a|

, so

\text{IQR}_y = |a| \cdot \text{IQR}_x = 1.5 \times 10 = 15

. The constant addition of

4.8

shifts all values equally and does not affect the spread/IQR.

Anahtar Kavram

Effect of linear transformations on measures of dispersion (IQR, standard deviation, range)

Alternatif Yöntem

Transform the individual quartiles directly:

Q_1(y) = 1.5(14) + 4.8 = 25.8

and

Q_3(y) = 1.5(24) + 4.8 = 40.8

. Then calculate the new IQR directly as

40.8 - 25.8 = 15

.

Tahmini Süre:1m 30s

Soru 2Soru →

A dataset of employee salaries has a standard deviation of $6$ . If every salary in the dataset is multiplied by $3$ and then increased by $10$ , what is the standard deviation of the resulting dataset?

6

18

28

16

54

Cevabı ve açıklamayı göster

Cevap: $18$

Cevap

The standard deviation of the resulting dataset is

18

.

Multiplying each data point by

3

increases the spread around the mean by a factor of

3

, changing the standard deviation from

6

to

18

. Adding

10

to each value shifts all values by the same amount, which leaves the distance between data points and the mean unchanged. Thus, the standard deviation remains

18

.

Adım Adım Çözüm

1

Evaluate the effect of multiplying each value in the dataset by a positive constant

c = 3

.

The standard deviation is scaled by

3

, giving a new standard deviation of

6 \times 3 = 18

.

Scaling every data value by a constant factor

c

multiplies the measures of dispersion (such as standard deviation, range, and IQR) by

|c|

.

2

Evaluate the effect of adding a constant

k = 10

to each value in the dataset.

The standard deviation remains

18

.

Adding a constant to every data value shifts the entire distribution along the number line without changing the relative distances between values or the overall spread.

Anahtar Kavram

Linear Transformations on Standard Deviation

Tahmini Süre:45s

Soru 3Soru →

Dataset $D$ consists of $40$ distinct real numbers arranged in strictly increasing order: $x_1 < x_2 < \dots < x_{40}$ . Let $R_D$ , $Q_D$ , and $\sigma_D$ denote the range, interquartile range, and standard deviation of Dataset $D$ , respectively.

A new dataset, Dataset $E$ , is created by replacing each of the $10$ largest numbers ( $x_{31}, x_{32}, \dots, x_{40}$ ) with the value of $x_{30}$ , which represents the $75\text{th}$ percentile of Dataset $D$ . Let $R_E$ , $Q_E$ , and $\sigma_E$ denote the range, interquartile range, and standard deviation of Dataset $E$ , respectively.

Which of the following statements MUST be true? Select all such statements.

Geçerli olan tümünü seçin

The range of Dataset

E

is strictly less than the range of Dataset

D

(

R_E < R_D

).

The interquartile range of Dataset

E

is equal to the interquartile range of Dataset

D

(

Q_E = Q_D

).

The standard deviation of Dataset

E

is strictly less than the standard deviation of Dataset

D

(

\sigma_E < \sigma_D

).

The median of Dataset

E

is strictly less than the median of Dataset

D

.

The interquartile range of Dataset

E

is strictly less than the interquartile range of Dataset

D

(

Q_E < Q_D

).

Cevabı ve açıklamayı göster

Cevap: The range of Dataset $E$ is strictly less than the range of Dataset $D$ ( $R_E < R_D$ ).; The interquartile range of Dataset $E$ is equal to the interquartile range of Dataset $D$ ( $Q_E = Q_D$ ).; The standard deviation of Dataset $E$ is strictly less than the standard deviation of Dataset $D$ ( $\sigma_E < \sigma_D$ ).

Cevap

The statements asserting that the range of Dataset E is strictly less than the range of Dataset D (

R_E < R_D

), the interquartile range of Dataset E is equal to the interquartile range of Dataset D (

Q_E = Q_D

), and the standard deviation of Dataset E is strictly less than the standard deviation of Dataset D (

\sigma_E < \sigma_D

) are all correct.

The range of Dataset E decreases because its maximum value is reduced from

x_{40}

to

x_{30}

while its minimum value

x_1

remains the same. The interquartile range remains unchanged because the boundary values defining the

25\text{th}

percentile (

x_{10}

) and

75\text{th}

percentile (

x_{30}

) are not altered. The standard deviation strictly decreases because reducing the

10

largest values brings them closer to the rest of the distribution, decreasing total variation.

Adım Adım Çözüm

1

Analyze the change in the maximum and minimum values to evaluate the Range.

In Dataset

D

,

\text{Range } R_D = x_{40} - x_1

. In Dataset

E

, the maximum element becomes

x_{30}

while the minimum remains

x_1

. Since all numbers are distinct and ordered,

x_{30} < x_{40}

, so

R_E = x_{30} - x_1 < R_D

. Thus,

R_E < R_D

is true.

Range is defined as

\text{Maximum} - \text{Minimum}

.

2

Analyze the

25\text{th}

percentile (

Q_1

) and

75\text{th}

percentile (

Q_3

) to evaluate the Interquartile Range.

For

N=40

,

Q_1 = x_{10}

and

Q_3 = x_{30}

. Dataset

E

modifies only the values strictly above

x_{30}

(

x_{31}

through

x_{40}

). Thus

x_{10}

and

x_{30}

retain their exact values in Dataset

E

. Therefore,

Q_E = x_{30} - x_{10} = Q_D

.

Interquartile range depends solely on

Q_3 - Q_1

, which relies on position cutoffs that were unaffected.

3

Analyze the effect of pulling extreme upper values inward on the Standard Deviation.

Standard deviation measures overall spread around the mean and can be calculated via pairwise differences

\text{Var}(X) = \frac{1}{2N^2} \sum_{i,j} (x_i - x_j)^2

. Replacing

x_{31}, \dots, x_{40}

with

x_{30}

strictly reduces the distances between these upper points and all other points, ensuring

\sigma_E < \sigma_D

.

Reducing the distance of tail observations from the rest of the dataset strictly decreases overall variance and standard deviation.

Anahtar Kavram

Impact of Data Transformations on Measures of Dispersion and Position

Soru 4Soru →

A constant value of $7$ is added to each number in a dataset to form a new dataset. Which of the following statistical measures of the new dataset are equal to the corresponding measures of the original dataset? Select all that apply.

Geçerli olan tümünü seçin

The range

The standard deviation

The interquartile range

The mean

The median

Cevabı ve açıklamayı göster

Cevap: The range; The standard deviation; The interquartile range

Cevap

The range, the standard deviation, and the interquartile range of the new dataset are equal to those of the original dataset.

Adding a constant to every value in a dataset shifts the entire distribution uniformly without altering its overall shape or spread. Consequently, measures of dispersion—including the range, standard deviation, and interquartile range—remain completely unchanged.

Adım Adım Çözüm

1

Evaluate the effect of adding a constant on measures of central tendency

If each data point

x_i

is replaced by

x_i + 7

, the mean

\mu

becomes

\mu + 7

and the median

M

becomes

M + 7

.

Measures of central tendency shift directly by the added constant, so they are not equal to the original measures.

2

Evaluate the effect of adding a constant on measures of dispersion

The new range is

(x_{\text{max}} + 7) - (x_{\text{min}} + 7) = x_{\text{max}} - x_{\text{min}}

. The new interquartile range is

(Q_3 + 7) - (Q_1 + 7) = Q_3 - Q_1

. Each deviation from the mean is

(x_i + 7) - (\mu + 7) = x_i - \mu

, so the standard deviation is unchanged.

Measures of dispersion reflect the relative distances between data points or distance from the mean, which are unaffected by shifting the whole distribution.

Anahtar Kavram

Effect of adding a constant on measures of central tendency versus measures of dispersion

Soru 5Soru →

In a group of 200 student test scores, a score of $S$ is located at the 75th percentile of the distribution. Which of the following statements must be true?

Score

S

is equal to 75.

Approximately 75 students scored at or below

S

.

Approximately 150 students scored at or below

S

.

Score

S

is 75 percent greater than the average score of the distribution.

The standard deviation of the test scores is equal to 75.

Cevabı ve açıklamayı göster

Cevap: Approximately 150 students scored at or below $S$ .

Cevap

Approximately 150 students scored at or below

S

.

The 75th percentile of a distribution indicates that approximately

75\%

of the data points lie at or below that value. For a group of 200 students,

75\%

of 200 is calculated as

0.75 \times 200 = 150

. Therefore, approximately 150 students achieved a score at or below

S

.

Adım Adım Çözüm

1

Understand the definition of a percentile rank

The 75th percentile means that approximately

75\%

of all observations in the dataset are less than or equal to that score.

By definition, the

p

-th percentile represents the value below which

p

percent of the observations fall.

2

Calculate 75% of the total number of students

0.75 \times 200 = 150

students.

The dataset contains 200 total student scores, so

75\%

of 200 yields the number of students at or below score

S

.

Anahtar Kavram

Percentile Position

Tahmini Süre:45s

Soru 6Soru →

Dataset $X$ consists of seven numerical values: $\{12, 16, 20, 24, 28, 32, 36\}$ . Dataset $Y$ is formed by replacing the value $36$ in Dataset $X$ with $56$ , while keeping all other six values the same. Which of the following statements correctly compares the interquartile range (IQR) and standard deviation of Dataset $Y$ to those of Dataset $X$ ?

The IQR remains unchanged, while the standard deviation increases.

Both the IQR and the standard deviation increase.

The IQR increases, while the standard deviation remains unchanged.

Both the IQR and the standard deviation remain unchanged.

The IQR decreases, while the standard deviation increases.

Cevabı ve açıklamayı göster

Cevap: The IQR remains unchanged, while the standard deviation increases.

Cevap

The IQR remains unchanged, while the standard deviation increases.

The statement specifying that the IQR remains unchanged while the standard deviation increases is correct because the quartiles Q1 (16) and Q3 (32) are unaffected by changing the largest value from 36 to 56, maintaining an IQR of 16. In contrast, standard deviation measures spread relative to the mean, so pulling the maximum value further out increases overall deviation.

Adım Adım Çözüm

1

Calculate the IQR of Dataset X

For Dataset X = {12, 16, 20, 24, 28, 32, 36}, the median is 24. The lower half is {12, 16, 20} with Q1 = 16. The upper half is {28, 32, 36} with Q3 = 32. Thus, IQR_X = 32 - 16 = 16.

IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).

2

Calculate the IQR of Dataset Y

For Dataset Y = {12, 16, 20, 24, 28, 32, 56}, the median is still 24. The lower half remains {12, 16, 20} (Q1 = 16). The upper half becomes {28, 32, 56}, so Q3 is still 32. Thus, IQR_Y = 32 - 16 = 16.

The third quartile is the middle number of the upper half, which remains 32 regardless of replacing 36 with 56.

3

Compare the standard deviation of Dataset X and Dataset Y

Replacing 36 with a significantly higher value 56 increases the distance of the maximum data point from the mean, increasing overall variance and thus increasing the standard deviation.

Standard deviation measures the average distance of data points from the mean and is highly sensitive to extreme values/outliers.

Anahtar Kavram

Sensitivity of Measures of Dispersion to Outliers

Soru 7Soru →

The high temperatures, in degrees Fahrenheit, recorded in a city over a 7-day period were 64, 58, 75, 67, 61, 83, and 72. What is the interquartile range of these temperatures, in degrees Fahrenheit?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The interquartile range of the recorded temperatures is 14.

To calculate the interquartile range, first arrange the dataset in ascending order: 58, 61, 64, 67, 72, 75, 83. The median of the 7 values is 67. The first quartile (

Q_1

) is the median of the lower three values (58, 61, 64), which is 61. The third quartile (

Q_3

) is the median of the upper three values (72, 75, 83), which is 75. Subtracting

Q_1

from

Q_3

yields an interquartile range of

75 - 61 = 14

.

Adım Adım Çözüm

1

Order the dataset from least to greatest

58, 61, 64, 67, 72, 75, 83

Finding quartiles requires data to be arranged in ascending order.

2

Find the first quartile (

Q_1

) and third quartile (

Q_3

)

Q_1 = 61

and

Q_3 = 75

The median of the dataset (the 4th value) is 67. The lower half of the data consists of 58, 61, 64 (median 61), and the upper half consists of 72, 75, 83 (median 75).

3

Compute the difference between

Q_3

and

Q_1

75 - 61 = 14

The interquartile range is defined as

IQR = Q_3 - Q_1

.

Anahtar Kavram

Interquartile Range (IQR)

Soru 8Soru →

The table below shows the distribution of scores achieved by 20 students on a final exam:

Score	Frequency
60	3
70	5
80	6
90	4
100	2

What is the interquartile range (IQR) of the scores?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The interquartile range (IQR) of the scores is 20.

For a set of 20 ordered scores, the first quartile

Q_1

is the median of the first 10 scores (average of the 5th and 6th values), and the third quartile

Q_3

is the median of the last 10 scores (average of the 15th and 16th values). Using the cumulative frequencies, the 5th and 6th scores are both 70 (

Q_1 = 70

), and the 15th and 16th scores are both 90 (

Q_3 = 90

). Thus, the interquartile range is

IQR = Q_3 - Q_1 = 90 - 70 = 20

.

Adım Adım Çözüm

1

Determine the cumulative frequency to locate quartile positions.

Score 60 occupies positions 1 to 3; Score 70 occupies positions 4 to 8; Score 80 occupies positions 9 to 14; Score 90 occupies positions 15 to 18; Score 100 occupies positions 19 to 20.

Tracking data positions in a frequency distribution allows efficient determination of medians and quartiles without expanding the raw list.

2

Calculate the first quartile (

Q_1

).

Q_1 = \frac{70 + 70}{2} = 70

.

With 20 total values, the lower half consists of the first 10 values (positions 1 through 10). The median of these 10 values is the average of the 5th and 6th values, which are both 70.

3

Calculate the third quartile (

Q_3

).

Q_3 = \frac{90 + 90}{2} = 90

.

The upper half consists of the last 10 values (positions 11 through 20). The median of these 10 values is the average of the 15th and 16th values, which are both 90.

4

Subtract

Q_1

from

Q_3

to find the interquartile range.

IQR = Q_3 - Q_1 = 90 - 70 = 20

.

The interquartile range represents the spread of the middle 50% of the dataset.

Anahtar Kavram

Interquartile Range (IQR) from a Frequency Distribution

Tahmini Süre:1m 30s

Soru 9Soru →

A dataset consists of 15 numerical values with a mean of 50 and a standard deviation of 8. If a constant value of 5 is added to every number in the dataset, what is the standard deviation of the resulting dataset?

8

13

40

55

64

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The standard deviation of the resulting dataset is 8.

Adding a constant to every data point in a set shifts the central location (the mean and median) by that constant, but leaves the distances between points unchanged. Because standard deviation depends solely on the distances between each point and the mean, the standard deviation remains 8.

Adım Adım Çözüm

1

Recall the effect of adding a constant to data points on measures of dispersion.

Measures of dispersion (range, IQR, standard deviation) describe the spread or scatter of data relative to its center.

When a constant

k

is added to every value

x_i

, the new mean becomes

\text{mean} + k

.

2

Evaluate the distance of each transformed data point from the new mean.

(x_i + k) - (\text{mean} + k) = x_i - \text{mean}

.

The deviation of each point from the mean remains identical to its original deviation.

3

Determine the standard deviation of the new dataset.

The standard deviation is unchanged and remains 8.

Because all individual deviations from the mean are preserved, the average distance from the mean does not change.

Anahtar Kavram

Invariance of standard deviation under addition of a constant

Soru 10Soru →

A dataset of $20$ numerical measurements has a range of $30$ and a standard deviation of $6.4$ . If every measurement in the dataset is multiplied by $-2$ and then increased by $5$ to construct a new dataset, what are the range and standard deviation of the new dataset, respectively?

Range is

65

and standard deviation is

17.8

Range is

-60

and standard deviation is

-12.8

Range is

60

and standard deviation is

12.8

Range is

60

and standard deviation is

25.6

Range is

30

and standard deviation is

6.4

Cevabı ve açıklamayı göster

Cevap: Range is $60$ and standard deviation is $12.8$

Cevap

The range of the new dataset is

60

and the standard deviation is

12.8

.

Under a linear transformation

Y = aX + b

, all measures of dispersion (range, interquartile range, standard deviation) are multiplied by

|a|

and are unaffected by the additive constant

b

. Here,

a = -2

and

b = 5

. The absolute multiplier is

|-2| = 2

. Therefore, the new range is

2 \times 30 = 60

and the new standard deviation is

2 \times 6.4 = 12.8

.

Adım Adım Çözüm

1

Analyze the general transformation formula for measures of dispersion under linear transformations of the form

Y = aX + b

.

For any constant multiplier

a

and constant shift

b

, the new range is

|a| \times \text{Range}_X

and the new standard deviation is

|a| \times \sigma_X

.

Measures of dispersion reflect spread; multiplying each data value by

a

scales the spread by

|a|

, while adding a constant shift

b

translates all values equally without expanding or contracting their relative distance.

2

Calculate the new range using

a = -2

and

b = 5

.

\text{New Range} = |-2| \times 30 = 2 \times 30 = 60.

The range scales by the absolute value of the multiplier

|-2| = 2

, while the additive constant

5

has no impact on the spread.

3

Calculate the new standard deviation using

a = -2

and

b = 5

.

\text{New Standard Deviation} = |-2| \times 6.4 = 2 \times 6.4 = 12.8.

The standard deviation scales by

|-2| = 2

, and the shift of

+5

does not change the dispersion around the mean.

Anahtar Kavram

Effect of Linear Transformations on Measures of Dispersion

Tahmini Süre:1m 30s

Soru 11Soru →

An original dataset $X$ consists of $20$ distinct positive numerical values with standard deviation $\sigma$ , range $R$ , and interquartile range $IQR$ . A new dataset $Y$ is created by transforming each data value $x_i$ in $X$ according to the rule $y_i = -3x_i + 5$ . Which of the following statements about the dispersion metrics of dataset $Y$ compared to dataset $X$ must be true? Select all such statements.

Geçerli olan tümünü seçin

The standard deviation of dataset

Y

is

3\sigma

.

The interquartile range of dataset

Y

is

3 \times IQR

.

The standard deviation of dataset

Y

is

3\sigma + 5

.

The range of dataset

Y

is

-3R + 5

.

The interquartile range of dataset

Y

is equal to

-3 \times IQR

.

Cevabı ve açıklamayı göster

Cevap: The standard deviation of dataset $Y$ is $3\sigma$ .; The interquartile range of dataset $Y$ is $3 \times IQR$ .

Cevap

The statements asserting that the standard deviation of dataset Y is 3 times the standard deviation of dataset X, and that the interquartile range of dataset Y is 3 times the interquartile range of dataset X, are both correct.

For any data set undergoing a linear transformation

y = ax + b

, all measures of dispersion (including standard deviation, range, and interquartile range) are scaled by the absolute value of the multiplier

|a|

and remain completely unaffected by the additive constant

b

. Since

a = -3

,

|a| = |-3| = 3

. Therefore, both the standard deviation and the interquartile range scale by a factor of 3.

Adım Adım Çözüm

1

Recall the general rule for linear transformation of measures of dispersion.

For any linear transformation

y = ax + b

, measures of dispersion (Range, IQR, Standard Deviation) scale by

|a|

and are unaffected by

b

.

Measures of dispersion evaluate the spread/distance between points, which shifts unchanged when a constant is added, and scales non-negatively when multiplied by a constant.

2

Identify the values of the multiplicative constant

a

and additive constant

b

.

a = -3

and

b = 5

, giving

|a| = |-3| = 3

.

The transformation equation is

y_i = -3x_i + 5

.

3

Apply the scaling factor

|a| = 3

to each measure of dispersion.

Standard deviation of

Y

is

3\sigma

, IQR of

Y

is

3 \times IQR

, and Range of

Y

is

3R

.

All measures of dispersion scale by a factor of 3 regardless of the negative sign of the multiplier or the addition of 5.

Anahtar Kavram

Linear Transformations on Measures of Dispersion

Soru 12Soru →

Dataset $A$ consists of $n$ numerical values with a standard deviation of $\sigma_A$ and an interquartile range of $\text{IQR}_A$ . Dataset $B$ is constructed by multiplying each value in Dataset $A$ by $4$ and then subtracting $10$ from each resulting value. Which of the following correctly expresses the standard deviation $\sigma_B$ and the interquartile range $\text{IQR}_B$ of Dataset $B$ in terms of $\sigma_A$ and $\text{IQR}_A$ ?

\sigma_B = 4\sigma_A - 10

and

\text{IQR}_B = 4\text{IQR}_A - 10

\sigma_B = 4\sigma_A

and

\text{IQR}_B = 4\text{IQR}_A

\sigma_B = 16\sigma_A

and

\text{IQR}_B = 4\text{IQR}_A

\sigma_B = 4\sigma_A

and

\text{IQR}_B = \text{IQR}_A

\sigma_B = 4\sigma_A - 10

and

\text{IQR}_B = 4\text{IQR}_A

Cevabı ve açıklamayı göster

Cevap: $\sigma_B = 4\sigma_A$ and $\text{IQR}_B = 4\text{IQR}_A$

Cevap

\sigma_B = 4\sigma_A

and

\text{IQR}_B = 4\text{IQR}_A

When a dataset undergoes a linear transformation of the form

Y = cX + k

, any measure of dispersion

D

(such as standard deviation or interquartile range) transforms according to

D_Y = |c| \cdot D_X

. The constant shift

k

shifts all points equally and does not affect spread. Therefore, multiplying by

4

multiplies both

\sigma_A

and

\text{IQR}_A

by

4

, and subtracting

10

has no effect on either measure.

Adım Adım Çözüm

1

Analyze the effect of multiplying each value in a dataset by a positive constant

c = 4

.

Multiplying all data values by

c

scales all measures of spread (including standard deviation, range, and interquartile range) by

|c| = 4

. Thus, initial scaled values give

4\sigma_A

and

4\text{IQR}_A

.

Measures of dispersion reflect distances between data points, which scale linearly when all values are multiplied by a constant.

2

Analyze the effect of subtracting a constant

k = 10

from each data point.

Subtracting a constant shifts the entire distribution along the number line without altering the relative distances between any data points.

Because measures of dispersion measure relative spread rather than absolute location, adding or subtracting a constant has zero effect on standard deviation or interquartile range.

3

Combine the transformation effects to state

\sigma_B

and

\text{IQR}_B

.

\sigma_B = 4\sigma_A

and

\text{IQR}_B = 4\text{IQR}_A

.

The multiplicative factor scales both metrics by 4, and the additive shift of

-10

leaves both metrics unchanged.

Anahtar Kavram

Linear Transformations on Measures of Dispersion

Tahmini Süre:1m 15s

Soru 13Soru →

A research team measured the height, in centimeters, of 15 plants in a greenhouse. The dataset has a range of $24\text{ cm}$ , an interquartile range (IQR) of $10\text{ cm}$ , and a standard deviation of $5.2\text{ cm}$ . If the height of every plant is increased by exactly $8\text{ cm}$ , which of the following statements regarding the modified dataset must be true? Select all such statements.

Geçerli olan tümünü seçin

The range of the modified dataset is

32\text{ cm}

.

The range of the modified dataset is

24\text{ cm}

.

The interquartile range of the modified dataset is

18\text{ cm}

.

The standard deviation of the modified dataset is

5.2\text{ cm}

.

The standard deviation of the modified dataset is

13.2\text{ cm}

.

Cevabı ve açıklamayı göster

Cevap: The range of the modified dataset is $24\text{ cm}$ .; The standard deviation of the modified dataset is $5.2\text{ cm}$ .

Cevap

The statement that the range of the modified dataset is

24\text{ cm}

and the statement that the standard deviation of the modified dataset is

5.2\text{ cm}

are both true.

When a constant is added to every value in a dataset, the entire distribution shifts by that amount. Consequently, the relative positions and distances between data points do not change, leaving measures of dispersion (range, IQR, standard deviation) unchanged. Therefore, the range remains

24\text{ cm}

and the standard deviation remains

5.2\text{ cm}

.

Adım Adım Çözüm

1

Analyze the effect of adding a constant to a dataset on measures of dispersion.

Adding a constant

k

to every data value shifts the center (mean, median) by

k

, but all measures of spread (range, IQR, standard deviation) remain completely unchanged.

Measures of dispersion quantify the spread or distance between data values, which is invariant under a uniform translation.

2

Evaluate each statement against the unchanged measures of dispersion.

The range remains

24\text{ cm}

, the IQR remains

10\text{ cm}

, and the standard deviation remains

5.2\text{ cm}

.

Comparing the calculated original measures to the statements shows that the range of

24\text{ cm}

and standard deviation of

5.2\text{ cm}

are correct.

Anahtar Kavram

Effect of Linear Transformations (Addition of a Constant) on Measures of Dispersion

Soru 14Soru →

Dataset $S$ consists of 10 distinct numerical values. Dataset $T$ is created by replacing the maximum value of Dataset $S$ with a number that is strictly greater than that maximum value, while all other 9 values remain unchanged. Which of the following statements comparing Dataset $S$ and Dataset $T$ must be true? Select all that apply.

Geçerli olan tümünü seçin

The range of Dataset

T

is greater than the range of Dataset

S

.

The interquartile range (IQR) of Dataset

T

is equal to the interquartile range of Dataset

S

.

The standard deviation of Dataset

T

is equal to the standard deviation of Dataset

S

.

The median of Dataset

T

is strictly greater than the median of Dataset

S

.

The 75th percentile of Dataset

T

is strictly greater than the 75th percentile of Dataset

S

.

Cevabı ve açıklamayı göster

Cevap: The range of Dataset $T$ is greater than the range of Dataset $S$ .; The interquartile range (IQR) of Dataset $T$ is equal to the interquartile range of Dataset $S$ .

Cevap

The statements asserting that the range of Dataset

T

is greater than the range of Dataset

S

and that the interquartile range (IQR) of Dataset

T

is equal to the interquartile range of Dataset

S

are correct.

The statement regarding the range is correct because increasing the maximum value increases the difference between the maximum and minimum values. The statement regarding the interquartile range is correct because the 25th percentile (

Q_1

) and 75th percentile (

Q_3

) depend only on the first 8 ordered elements of a 10-element dataset, so changing the 10th element leaves

Q_1

and

Q_3

identical.

Adım Adım Çözüm

1

Analyze the impact on Range

\text{Range}(T) > \text{Range}(S)

Range is defined as

\text{Maximum} - \text{Minimum}

. Since the minimum is unchanged and the maximum increases, the range must increase.

2

Analyze the impact on Interquartile Range (IQR)

\text{IQR}(T) = \text{IQR}(S)

IQR is

Q_3 - Q_1

. For 10 ordered elements,

Q_1

and

Q_3

depend on the positions of the bottom 75% of the data. Modifying only the 10th (largest) element leaves

Q_1

and

Q_3

unchanged.

3

Analyze the impact on Median and Standard Deviation

Median is unchanged; Standard deviation increases.

The median depends on the 5th and 6th values, which are unaffected. Moving an extreme value further out increases distance from the mean, strictly increasing standard deviation.

Anahtar Kavram

Effect of outlier modification on measures of position (

Q_1, Q_3

, Median) versus measures of dispersion (Range, Standard Deviation, IQR).

Soru 15Soru →

Dataset $S$ consists of $80$ distinct numerical scores arranged in ascending order, with an interquartile range denoted by $\text{IQR}_S$ . A new dataset, $T$ , is created by adding four scores to Dataset $S$ : two scores that are strictly smaller than the minimum value of Dataset $S$ , and two scores that are strictly greater than the maximum value of Dataset $S$ . If $\text{IQR}_T$ represents the interquartile range of Dataset $T$ , which of the following statements must be true?

\text{IQR}_T > \text{IQR}_S

\text{IQR}_T = \text{IQR}_S

\text{IQR}_T < \text{IQR}_S

\text{IQR}_T = \text{IQR}_S + 4

The relationship between

\text{IQR}_T

and

\text{IQR}_S

cannot be determined from the given information.

Cevabı ve açıklamayı göster

Cevap: $\text{IQR}_T > \text{IQR}_S$

Cevap

The interquartile range of Dataset

T

is strictly greater than the interquartile range of Dataset

S

(

\text{IQR}_T > \text{IQR}_S

).

The statement that

\text{IQR}_T > \text{IQR}_S

is correct. Inserting two items below the minimum value shifts the rank position of the first quartile to lower-indexed elements of the original dataset, reducing

Q_1

. Inserting two items above the maximum value shifts the rank position of the third quartile to higher-indexed elements of the original dataset, increasing

Q_3

. Since the original numbers are strictly increasing,

Q_3

increases while

Q_1

decreases, resulting in a larger interquartile range for Dataset

T

.

Adım Adım Çözüm

1

Determine the quartile positions for the original dataset

S

.

Dataset

S

has

n = 80

distinct elements

x_1 < x_2 < \dots < x_{80}

. Its first quartile

Q_{1,S}

is the average of

x_{20}

and

x_{21}

, and its third quartile

Q_{3,S}

is the average of

x_{60}

and

x_{61}

. Thus,

\text{IQR}_S = Q_{3,S} - Q_{1,S} = \frac{x_{60} + x_{61}}{2} - \frac{x_{20} + x_{21}}{2}

.

Quartiles partition sorted data into four equal parts based on element indices.

2

Determine the new element positions for dataset

T

.

Dataset

T

has

n = 84

elements ordered as

a, b, x_1, x_2, \dots, x_{80}, c, d

, where

a, b < x_1

and

c, d > x_{80}

. The elements of

S

are now shifted right by 2 index positions in

T

.

Inserting two values at the bottom of the ordered list increases the rank index of all original elements by 2.

3

Calculate the first and third quartiles for dataset

T

.

In Dataset

T

, the first quartile

Q_{1,T}

is the average of the 21st and 22nd elements of

T

, which correspond to

x_{19}

and

x_{20}

of

S

. The third quartile

Q_{3,T}

is the average of the 63rd and 64th elements of

T

, which correspond to

x_{61}

and

x_{62}

of

S

.

The 25th and 75th percentile positions in an 84-element dataset correspond to indices 21/22 and 63/64.

4

Compare

\text{IQR}_T

with

\text{IQR}_S

.

Since

S

consists of distinct values in strictly ascending order (

x_{19} < x_{20} < x_{21}

and

x_{60} < x_{61} < x_{62}

),

Q_{1,T} = \frac{x_{19} + x_{20}}{2} < \frac{x_{20} + x_{21}}{2} = Q_{1,S}

and

Q_{3,T} = \frac{x_{61} + x_{62}}{2} > \frac{x_{60} + x_{61}}{2} = Q_{3,S}

. Therefore,

\text{IQR}_T = Q_{3,T} - Q_{1,T} > Q_{3,S} - Q_{1,S} = \text{IQR}_S

.

Subtracting a smaller number from a larger number increases the overall difference.

Anahtar Kavram

Interquartile Range and Position Indices under Data Insertion

Soru 16Soru →

Dataset $X$ consists of 2 numbers, each equal to 10. Dataset $Y$ consists of 8 numbers, each equal to 25. If Dataset $X$ and Dataset $Y$ are combined to form a single dataset of 10 numbers, what is the standard deviation of the combined dataset?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The standard deviation of the combined dataset is 6.

To find the standard deviation of the combined dataset, first calculate the combined mean:

\frac{2(10) + 8(25)}{10} = 22

. Next, compute the variance by finding the average of the squared deviations from 22:

\frac{2(10 - 22)^2 + 8(25 - 22)^2}{10} = \frac{2(144) + 8(9)}{10} = \frac{360}{10} = 36

. Taking the square root of the variance yields a standard deviation of

\sqrt{36} = 6

.

Adım Adım Çözüm

1

Calculate the mean of the combined 10-number dataset.

The combined mean is 22.

The mean of the combined dataset is needed to compute individual deviations.

2

Compute the squared deviation of each data point from the combined mean and average them to determine the variance.

The variance is 36.

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

3

Calculate the square root of the variance to find the standard deviation.

The standard deviation is 6.

Standard deviation is the non-negative square root of variance.

Anahtar Kavram

Standard deviation of a combined dataset

Soru 17Soru →

A dataset of 25 student test scores has a mean of $70$ , a standard deviation of $8$ , and an interquartile range ( $\text{IQR}$ ) of $12$ . A instructor creates a modified dataset by multiplying each original test score by $1.5$ and then adding $10$ to the result. What is the standard deviation of the modified dataset?

8

12

18

22

28

Cevabı ve açıklamayı göster

Cevap: $12$

Cevap

12

For any dataset, if each value

x

is transformed into

a \cdot x + b

, where

a

and

b

are constants, the new standard deviation is given by

|a| \cdot \sigma_{\text{old}}

. In this problem,

a = 1.5

and

b = 10

. Therefore, the new standard deviation is

1.5 \times 8 = 12

. The addition of

10

affects measures of center (such as the mean and median) but has no effect on measures of dispersion (such as standard deviation, IQR, and range).

Adım Adım Çözüm

1

Analyze the effect of multiplying each data point by a constant on standard deviation.

Multiplying every value in a dataset by a constant

c = 1.5

multiplies the standard deviation

\sigma

by

|c|

. The new intermediate standard deviation becomes

1.5 \times 8 = 12

.

Standard deviation measures dispersion around the mean, so scaling all data points by a factor scales the distance between each point and the mean by that same factor.

2

Analyze the effect of adding a constant to each data point on standard deviation.

Adding a constant

k = 10

to every value shifts the entire distribution without changing the distances between data points or the relative spread around the mean. Thus, standard deviation remains

12

.

Adding a constant shifts both the individual values and the mean by the same amount, leaving

(x_i - \bar{x})

unchanged.

Anahtar Kavram

Linear Transformations on Measures of Dispersion

Soru 18Soru →

A logistics company recorded the daily delivery processing times (in minutes) for a warehouse over a given period, forming Dataset $X$ . Dataset $X$ has a range of $40$ minutes, an interquartile range ( $\text{IQR}$ ) of $15$ minutes, and a standard deviation of $8.5$ minutes. A new dataset, Dataset $Y$ , is created by transforming each processing time $x$ in Dataset $X$ according to the formula $y = 1.5x - 10$ . Which of the following statements regarding the measures of dispersion for Dataset $Y$ must be true? Select all such statements.

Geçerli olan tümünü seçin

The range of Dataset

Y

is

60

minutes.

The standard deviation of Dataset

Y

is

12.75

minutes.

The interquartile range of Dataset

Y

is

12.5

minutes.

The standard deviation of Dataset

Y

is

2.75

minutes.

The interquartile range of Dataset

Y

is equal to the interquartile range of Dataset

X

.

Cevabı ve açıklamayı göster

Cevap: The range of Dataset $Y$ is $60$ minutes.; The standard deviation of Dataset $Y$ is $12.75$ minutes.

Cevap

The statements asserting that the range of Dataset Y is 60 minutes and that the standard deviation of Dataset Y is 12.75 minutes are both correct.

Under a transformation of the form

y = ax + b

(where

a > 0

), any measure of dispersion

D

transforms according to

D_Y = a \cdot D_X

. The constant

b

does not affect spread. Therefore, the range becomes

1.5 \times 40 = 60

minutes and the standard deviation becomes

1.5 \times 8.5 = 12.75

minutes.

Adım Adım Çözüm

1

Recall the effect of a linear transformation

y = ax + b

on measures of dispersion.

Measures of dispersion (range, IQR, standard deviation) are scaled by

|a|

and are completely unaffected by the constant addition/subtraction

b

.

Adding or subtracting a constant shifts all data points by the exact same amount without altering the relative distances between data points.

2

Calculate the range for Dataset Y.

\text{Range}_Y = 1.5 \times \text{Range}_X = 1.5 \times 40 = 60

minutes.

The multiplicative factor is

a = 1.5

.

3

Calculate the standard deviation for Dataset Y.

S_Y = 1.5 \times S_X = 1.5 \times 8.5 = 12.75

minutes.

The standard deviation scales proportionally by

1.5

.

4

Calculate the interquartile range (IQR) for Dataset Y.

\text{IQR}_Y = 1.5 \times \text{IQR}_X = 1.5 \times 15 = 22.5

minutes.

The IQR also scales proportionally by

1.5

.

Anahtar Kavram

Linear Transformations of Dispersion Measures

Soru 19Soru →

A researcher recorded the annual rainfall totals (in inches) for a specific region over a 20-year period, forming Dataset $P$ . Dataset $P$ has a range of $18$ inches and an interquartile range (IQR) of $8$ inches. A second dataset, Dataset $Q$ , is created by multiplying each rainfall total in Dataset $P$ by $1.2$ and then subtracting $3$ inches. What is the sum of the range and the interquartile range (IQR) of Dataset $Q$ ?

25.2

26.0

28.2

31.2

37.2

Cevabı ve açıklamayı göster

Cevap: $31.2$

Cevap

31.2

For any linear transformation of the form

Y = aX + b

, measures of spread such as range and interquartile range (IQR) are multiplied by

|a|

, while the constant term

b

has no effect. Multiplying the original range (

18

) and IQR (

8

) by

1.2

yields a new range of

21.6

and a new IQR of

9.6

. Summing these values gives

21.6 + 9.6 = 31.2

.

Adım Adım Çözüm

1

Determine the impact of a linear transformation

Y = aX + b

on measures of dispersion.

Measures of dispersion (Range, IQR, Standard Deviation) scale by

|a|

and are unaffected by the additive constant

b

.

Adding or subtracting a constant shifts all data points by the same amount, leaving the distance between points unchanged, whereas multiplying by a factor scales all distances between points.

2

Calculate the range of Dataset

Q

.

\text{Range}_Q = 1.2 \times \text{Range}_P = 1.2 \times 18 = 21.6

The range scales by the multiplier

1.2

and is not affected by subtracting

3

.

3

Calculate the interquartile range (IQR) of Dataset

Q

.

\text{IQR}_Q = 1.2 \times \text{IQR}_P = 1.2 \times 8 = 9.6

The IQR scales by the multiplier

1.2

and is not affected by subtracting

3

.

4

Compute the sum of the range and the IQR of Dataset

Q

.

\text{Sum} = 21.6 + 9.6 = 31.2

Adding the newly calculated Range and IQR gives the required total spread measure.

Anahtar Kavram

Linear Transformations on Dispersion Metrics

Tahmini Süre:1m 30s

Soru 20Soru →

In a meteorological study, the daily peak wind speeds (in kilometers per hour) recorded over 7 consecutive days were $14, 18, 21, 24, 27, 31,$ and $37$ . As part of a data calibration process, each recorded wind speed is increased by $15\%$ and then increased by an additional constant of $5\text{ km/h}$ . What is the range, in kilometers per hour, of the calibrated daily peak wind speeds?

Cevabı ve açıklamayı göster

Cevap: 26.45

Cevap

The range of the calibrated daily peak wind speeds is 26.45 km/h.

For any dataset transformed linearly according to

y = a \cdot x + b

where

a > 0

, the range of the transformed dataset is given by

\text{Range}(Y) = a \cdot \text{Range}(X)

. Here, increasing each value by

15\%

corresponds to

a = 1 + 0.15 = 1.15

, and adding

5

corresponds to

b = 5

. The original maximum is

37

and the original minimum is

14

, giving an original range of

37 - 14 = 23

. Therefore, the new range is

1.15 \times 23 = 26.45\text{ km/h}

. The constant shift of

5\text{ km/h}

affects the center (mean, median) but has zero effect on the dispersion.

Adım Adım Çözüm

1

Determine the range of the original set of wind speeds.

Original Range = 37 - 14 = 23 km/h.

The range is defined as the difference between the maximum and minimum values in a dataset.

2

Apply the properties of linear transformations to measures of dispersion.

The transformation is y = 1.15x + 5. The range is scaled by 1.15 and unaffected by the addition of 5.

Adding a constant shift to all data points shifts the entire distribution without changing the spread (dispersion), whereas multiplying all data points by a constant factor 'a' scales all measures of dispersion by |a|.

3

Compute the calibrated range.

Calibrated Range = 1.15 * 23 = 26.45 km/h.

Multiplying the original range of 23 by the scale factor of 1.15 gives the exact range of the transformed dataset.

Anahtar Kavram

Effect of linear transformations on measures of dispersion (range, standard deviation, IQR)

Tahmini Süre:1m 30s

Measures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

Cevap

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