Measures of Dispersion Alıştırma Soruları — JAMB UTME

What is the mean deviation of the data set $4, 7, 8, 11, 15$ ?

Cevabı ve açıklamayı göster

Cevap: 3.2

Cevap

The mean deviation of the data set is 3.2.

To find the mean deviation, first calculate the mean of the dataset, which is 9. Then, compute the absolute difference of each number from 9, obtaining values of 5, 2, 1, 2, and 6. Finally, divide the sum of these absolute values (16) by the total number of items (5) to get 3.2.

Adım Adım Çözüm

1

Calculate the arithmetic mean of the given data set.

The mean

\bar{x} = 9

.

The mean is needed to evaluate how far each data point deviates from the central value.

2

Find the absolute difference between each value and the mean.

The absolute deviations are 5, 2, 1, 2, and 6.

Mean deviation measures dispersion using absolute distances, ignoring negative signs.

3

Calculate the mean of the absolute deviations.

Mean deviation = 3.2.

Dividing the total sum of absolute deviations (16) by the number of observations (5) gives the mean deviation.

Anahtar Kavram

Mean Deviation for Ungrouped Data

Soru 2Soru →

The set of numbers $k - 2$ , $k$ , $k + 1$ , and $k + 5$ is given, where $k$ is any real constant. What is the variance of this set of numbers?

Cevabı ve açıklamayı göster

Cevap: 6.5

Cevap

The variance of the given set of numbers is 6.5.

The mean of the set is

\bar{x} = k + 1

. Subtracting the mean from each data point gives deviations of

-3

,

-1

,

0

, and

4

. The squares of these deviations are

9

,

1

,

0

, and

16

, which sum to

26

. Dividing this sum by

4

gives a variance of

6.5

. A key statistical property illustrated here is that adding or subtracting a constant

k

from every value in a dataset shifts the mean by

k

but leaves measures of dispersion (such as variance and standard deviation) unchanged.

Adım Adım Çözüm

1

Find the mean (\bar{x}) of the given set {k - 2, k, k + 1, k + 5}.

\bar{x} = \frac{(k - 2) + k + (k + 1) + (k + 5)}{4} = \frac{4k + 4}{4} = k + 1

The mean is calculated by summing all values and dividing by the total count of numbers.

2

Determine the deviation of each value from the mean, (x_i - \bar{x}).

(k - 2) - (k + 1) = -3, k - (k + 1) = -1, (k + 1) - (k + 1) = 0, (k + 5) - (k + 1) = 4

Deviations measure how far each data value lies from the mean.

3

Square each individual deviation and sum the results.

(-3)^2 + (-1)^2 + 0^2 + 4^2 = 9 + 1 + 0 + 16 = 26

Squaring converts all deviations into non-negative values.

4

Divide the sum of squared deviations by the total number of observations (N = 4) to find the variance.

\text{Variance} = \frac{26}{4} = 6.5

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

Anahtar Kavram

Variance and Invariance under Constant Translation

Soru 3Soru →

The table below shows the frequency distribution of marks obtained by a group of students in a mathematics test:

Mark ( $x$ )	2	4	6	8	10
Frequency ( $f$ )	2	1	4	1	2

Find the mean deviation of the distribution.

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The mean deviation of the distribution is 2.

To find the mean deviation, first calculate the mean

\bar{x} = \frac{\sum fx}{\sum f} = \frac{60}{10} = 6

. Next, sum the absolute deviations multiplied by their frequencies:

\sum f|x - \bar{x}| = 2(4) + 1(2) + 4(0) + 1(2) + 2(4) = 20

. Dividing this total by the sum of frequencies

10

yields a mean deviation of

2

.

Adım Adım Çözüm

1

Calculate the arithmetic mean of the distribution

\bar{x} = \frac{\sum f x}{\sum f} = \frac{(2 \times 2) + (1 \times 4) + (4 \times 6) + (1 \times 8) + (2 \times 10)}{2 + 1 + 4 + 1 + 2} = \frac{60}{10} = 6

The mean is required as the central benchmark from which individual deviations are measured.

2

Calculate the sum of absolute deviations weighted by frequency

\sum f |x - \bar{x}| = 2|2 - 6| + 1|4 - 6| + 4|6 - 6| + 1|8 - 6| + 2|10 - 6| = 8 + 2 + 0 + 2 + 8 = 20

Each absolute difference from the mean must be multiplied by its frequency to account for the total deviation.

3

Divide the total absolute deviation by the total frequency

\text{Mean Deviation} = \frac{\sum f |x - \bar{x}|}{\sum f} = \frac{20}{10} = 2

The mean deviation represents the average distance of all observations from the arithmetic mean.

Anahtar Kavram

Mean Deviation of a Frequency Distribution

Tahmini Süre:1m 30s

Soru 4Soru →

The test scores of five students in a mathematics quiz are $5, 8, 11, 12,$ and $14$ . What is the variance of these test scores?

10

\sqrt{10}

50

2.8

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The variance of the test scores is

10

.

The mean of the scores is

10

. Subtracting

10

from each score gives deviations of

-5, -2, 1, 2,

and

4

. Squaring these gives

25, 4, 1, 4,

and

16

, which sum to

50

. Dividing this sum by the number of data points (

5

) yields a variance of

10

.

Adım Adım Çözüm

1

Calculate the mean (

\bar{x}

) of the data set

\bar{x} = \frac{5 + 8 + 11 + 12 + 14}{5} = \frac{50}{5} = 10

The mean is required to find the deviation of each score from the central value.

2

Compute the squared deviations from the mean

(x_i - \bar{x})^2

(5-10)^2 = 25

,

(8-10)^2 = 4

,

(11-10)^2 = 1

,

(12-10)^2 = 4

,

(14-10)^2 = 16

Squaring deviations ensures all negative differences become positive.

3

Find the sum of all squared deviations

\sum (x_i - \bar{x})^2 = 25 + 4 + 1 + 4 + 16 = 50

Summing the squared deviations measures total variation around the mean.

4

Divide the total sum of squared deviations by the number of data values (

N = 5

)

\text{Variance } (\sigma^2) = \frac{50}{5} = 10

Variance is defined as the average of the squared deviations.

Anahtar Kavram

Variance of Ungrouped Data

Tahmini Süre:1m 30s

Soru 5Soru →

A small business recorded the number of customer inquiries received per day over six consecutive days as follows: $4$ , $7$ , $8$ , $11$ , $13$ , and $17$ . What is the variance of the daily customer inquiries?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The variance of the daily customer inquiries is

18

.

To find the variance of the data set

\{4, 7, 8, 11, 13, 17\}

, first calculate the mean:

\bar{x} = \frac{4+7+8+11+13+17}{6} = 10

. Next, compute the squared deviation of each data point from the mean:

(4-10)^2 = 36

,

(7-10)^2 = 9

,

(8-10)^2 = 4

,

(11-10)^2 = 1

,

(13-10)^2 = 9

, and

(17-10)^2 = 49

. Summing these squared deviations gives

108

. Dividing this total by the number of observations (

6

) yields a variance of

18

.

Adım Adım Çözüm

1

Calculate the arithmetic mean of the given data set.

\bar{x} = 10

The mean is required as the central point from which deviations are calculated.

2

Determine the squared deviation of each data value from the mean.

(-6)^2 = 36

,

(-3)^2 = 9

,

(-2)^2 = 4

,

1^2 = 1

,

3^2 = 9

,

7^2 = 49

Variance measures the average squared distance of data points from the mean.

3

Sum all calculated squared deviations.

\sum (x - \bar{x})^2 = 36 + 9 + 4 + 1 + 9 + 49 = 108

This provides the total sum of squares for the data set.

4

Divide the total sum of squares by the number of observations (

n = 6

).

\sigma^2 = \frac{108}{6} = 18

Population variance formula is

\sigma^2 = \frac{\sum (x - \bar{x})^2}{n}

.

Anahtar Kavram

Variance of Ungrouped Data

Tahmini Süre:1m 30s

Soru 6Soru →

Five daily rainfall measurements (in mm) recorded in a city are $3, 6, 7, 9,$ and $15$ . What is the standard deviation of these rainfall measurements?

3.2

4

16

20

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The standard deviation of the rainfall measurements is

4

.

First compute the mean of the data set:

\bar{x} = \frac{3 + 6 + 7 + 9 + 15}{5} = 8

. Next, sum the squared deviations from the mean:

(3-8)^2 + (6-8)^2 + (7-8)^2 + (9-8)^2 + (15-8)^2 = 25 + 4 + 1 + 1 + 49 = 80

. Dividing by the number of observations (

5

) yields the variance

\sigma^2 = \frac{80}{5} = 16

. Taking the square root gives the standard deviation

\sigma = \sqrt{16} = 4

.

Adım Adım Çözüm

1

Calculate the arithmetic mean (

\bar{x}

) of the dataset

\bar{x} = \frac{3 + 6 + 7 + 9 + 15}{5} = \frac{40}{5} = 8

The mean is needed to find deviations for each data point.

2

Find the deviation of each number from the mean and square it

(3-8)^2 = 25

,

(6-8)^2 = 4

,

(7-8)^2 = 1

,

(9-8)^2 = 1

,

(15-8)^2 = 49

Squaring ensures all deviation values are non-negative.

3

Sum the squared deviations and divide by the total number of items (

n = 5

) to find the variance

\text{Variance } (\sigma^2) = \frac{25 + 4 + 1 + 1 + 49}{5} = \frac{80}{5} = 16

Variance measures the average squared distance from the mean.

4

Take the square root of the variance to obtain standard deviation

\text{Standard Deviation } (\sigma) = \sqrt{16} = 4

Standard deviation returns the dispersion measure back to the original units.

Anahtar Kavram

Standard Deviation of Ungrouped Data

Soru 7Soru →

The masses, in kilograms, of five wooden crates stored in a warehouse are $3$ , $7$ , $8$ , $10$ , and $12$ . What is the variance of the masses of these crates?

9.2

2.4

11.5

46.0

Cevabı ve açıklamayı göster

Cevap: $9.2$

Cevap

The variance of the masses is

9.2

.

The mean of the numbers is

8

. The squared deviations are

25, 1, 0, 4,

and

16

. The sum of these squared deviations is

46

. Dividing by the count of numbers (

5

) gives

9.2

.

Adım Adım Çözüm

1

Calculate the mean (

\bar{x}

) of the dataset.

\bar{x} = \frac{3 + 7 + 8 + 10 + 12}{5} = \frac{40}{5} = 8

The mean is required to find individual deviations.

2

Compute the deviation of each observation from the mean,

(x - \bar{x})

.

3 - 8 = -5

,

7 - 8 = -1

,

8 - 8 = 0

,

10 - 8 = 2

,

12 - 8 = 4

Deviations quantify how far each data point lies from the central mean.

3

Square each deviation,

(x - \bar{x})^2

.

(-5)^2 = 25

,

(-1)^2 = 1

,

0^2 = 0

,

2^2 = 4

,

4^2 = 16

Squaring eliminates negative signs and weights larger deviations.

4

Sum the squared deviations and divide by the total number of observations

N = 5

.

Variance

\sigma^2 = \frac{25 + 1 + 0 + 4 + 16}{5} = \frac{46}{5} = 9.2

Variance is defined as the mean of the squared deviations.

Anahtar Kavram

Variance of Ungrouped Data

Tahmini Süre:1m 30s

Soru 8Soru →

The arithmetic mean of five observations $1$ , $3$ , $x$ , $9$ , and $12$ is $6$ . What is the variance of this set of observations?

3.6

4

16

20

Cevabı ve açıklamayı göster

Cevap: $16$

Cevap

The variance of the set of observations is

16

.

First, the missing value is found by equating the sum of the five numbers to

5 \times 6 = 30

, which gives

x = 5

. The data set is

1, 3, 5, 9, 12

. The deviations from the mean

6

are

-5, -3, -1, 3, 6

, and their squares are

25, 9, 1, 9, 36

. Summing these squared deviations gives

80

. Dividing by the number of observations (

5

) yields the variance of

16

.

Adım Adım Çözüm

1

Find the missing value

x

using the arithmetic mean formula.

x = 5

The mean of

N = 5

numbers is

\bar{x} = 6

, so the total sum is

5 \times 6 = 30

. Summing the known numbers gives

1 + 3 + x + 9 + 12 = 30 \implies 25 + x = 30 \implies x = 5

.

2

Calculate the deviations

(x_i - \bar{x})

for each data value from the mean

\bar{x} = 6

.

Deviations are

-5, -3, -1, 3, 6

Subtracting

6

from each value

(1, 3, 5, 9, 12)

gives

1-6=-5

,

3-6=-3

,

5-6=-1

,

9-6=3

, and

12-6=6

.

3

Square each deviation.

Squared deviations are

25, 9, 1, 9, 36

Squaring each deviation yields

(-5)^2 = 25

,

(-3)^2 = 9

,

(-1)^2 = 1

,

3^2 = 9

, and

6^2 = 36

.

4

Compute the population variance

\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}

.

\sigma^2 = 16

The sum of squared deviations is

25 + 9 + 1 + 9 + 36 = 80

. Dividing by

N = 5

gives

\frac{80}{5} = 16

.

Anahtar Kavram

Variance of Ungrouped Data with a Missing Observation

Soru 9Soru →

The daily temperature readings, in degrees Celsius, recorded over four consecutive days are $10$ , $12$ , $14$ , and $16$ . What is the mean deviation of these temperatures?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The mean deviation of the temperatures is 2 °C.

To find the mean deviation, first calculate the mean of the data values: (10 + 12 + 14 + 16) / 4 = 13. Next, find the absolute difference of each value from the mean: |10 - 13| = 3, |12 - 13| = 1, |14 - 13| = 1, and |16 - 13| = 3. Finally, average these absolute differences: (3 + 1 + 1 + 3) / 4 = 8 / 4 = 2.

Adım Adım Çözüm

1

Calculate the arithmetic mean (average) of the data set

$\bar{x} = \frac{10 + 12 + 14 + 16}{4} = 13$

Mean deviation measures dispersion relative to the mean, so the mean must be calculated first.

2

Compute the absolute deviation of each value from the mean

$|10 - 13| = 3$, $|12 - 13| = 1$, $|14 - 13| = 1$, $|16 - 13| = 3$

Mean deviation requires non-negative distances of each observation from the mean.

3

Sum the absolute deviations and divide by the sample size (N = 4)

$\text{Mean Deviation} = \frac{3 + 1 + 1 + 3}{4} = \frac{8}{4} = 2$

The mean deviation is the average of the absolute deviations.

Anahtar Kavram

Mean Deviation for Ungrouped Data

Soru 10Soru →

A football coach records the number of goals scored by a team in six consecutive matches as $1, 3, 4, 6, 7,$ and $9$ . What is the variance of the goals scored?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The variance of the goals scored is 7.

To find the variance of the goals scored, first calculate the mean:

\bar{x} = \frac{1+3+4+6+7+9}{6} = \frac{30}{6} = 5

. Next, determine the sum of the squared deviations from the mean:

(1-5)^2 + (3-5)^2 + (4-5)^2 + (6-5)^2 + (7-5)^2 + (9-5)^2 = 16 + 4 + 1 + 1 + 4 + 16 = 42

. Dividing this total by the number of data values (

N = 6

) yields

\frac{42}{6} = 7

.

Adım Adım Çözüm

1

Calculate the mean of the dataset

\bar{x} = 5

The mean is needed as the reference point for computing deviations.

2

Compute the squared deviation of each data point from the mean

Squared deviations are 16, 4, 1, 1, 4, and 16

Variance measures the average squared distance from the mean.

3

Sum the squared deviations and divide by the number of observations N = 6

Variance = 7

The formula for variance of ungrouped data is \sigma^2 = \frac{\sum (x - \bar{x})^2}{N}.

Anahtar Kavram

Variance of Ungrouped Data

Soru 11Soru →

The heights (in cm) of a seedling recorded over five consecutive weeks are $9$ , $13$ , $15$ , $17$ , and $21$ . What is the standard deviation of the heights?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The standard deviation of the seedling heights is 4 cm.

To find the standard deviation, first compute the mean: (9 + 13 + 15 + 17 + 21) / 5 = 15. Next, calculate the sum of squared deviations: (-6)^2 + (-2)^2 + 0^2 + 2^2 + 6^2 = 36 + 4 + 0 + 4 + 36 = 80. Divide by 5 to find the variance of 16. Finally, taking the square root of 16 gives the standard deviation of 4 cm.

Adım Adım Çözüm

1

Calculate the mean of the data values

Mean = 15 cm

The mean is necessary to calculate the deviations of each data point.

2

Calculate the squared deviations from the mean

Squared deviations are 36, 4, 0, 4, and 36 (sum = 80)

Variance measures the average of squared deviations from the mean.

3

Calculate the population variance

Variance = 80 / 5 = 16

Dividing the sum of squared deviations by N gives the variance.

4

Calculate the standard deviation

Standard deviation = sqrt(16) = 4 cm

The standard deviation is the square root of the variance.

Anahtar Kavram

Standard Deviation of Ungrouped Data

Soru 12Soru →

The table below shows the frequency distribution of marks obtained by a group of students in a short test:

Mark ( $x$ )	1	2	3	4
Frequency ( $f$ )	2	3	3	2

What is the mean deviation of the distribution?

0.9

1.0

1.05

2.25

Cevabı ve açıklamayı göster

Cevap: $0.9$

Cevap

The mean deviation of the distribution is

0.9

.

The mean of the data is

\bar{x} = \frac{25}{10} = 2.5

. Calculating the sum of weighted absolute deviations yields

2(1.5) + 3(0.5) + 3(0.5) + 2(1.5) = 9.0

. Dividing by total frequency

10

gives the mean deviation as

0.9

.

Adım Adım Çözüm

1

Calculate the mean (

\bar{x}

) of the frequency distribution.

\sum f = 2 + 3 + 3 + 2 = 10

,

\sum fx = (2 \times 1) + (3 \times 2) + (3 \times 3) + (2 \times 4) = 2 + 6 + 9 + 8 = 25

. Therefore,

\bar{x} = \frac{25}{10} = 2.5

.

The mean is required to determine the deviations of each score value.

2

Find the absolute deviation

|x - \bar{x}|

for each score value.

For

x = 1

:

|1 - 2.5| = 1.5

; for

x = 2

:

|2 - 2.5| = 0.5

; for

x = 3

:

|3 - 2.5| = 0.5

; for

x = 4

:

|4 - 2.5| = 1.5

.

Mean deviation measures average distance from the mean, ignoring signs.

3

Multiply each absolute deviation by its corresponding frequency and sum them.

\sum f|x - \bar{x}| = (2 \times 1.5) + (3 \times 0.5) + (3 \times 0.5) + (2 \times 1.5) = 3 + 1.5 + 1.5 + 3 = 9.0

.

Frequencies reflect how many times each deviation occurs in the dataset.

4

Compute the mean deviation by dividing the weighted sum by the total frequency

\sum f

.

\text{Mean Deviation} = \frac{\sum f|x - \bar{x}|}{\sum f} = \frac{9.0}{10} = 0.9.

The mean deviation is the average of these absolute deviations across all observations.

Anahtar Kavram

Mean Deviation for Discrete Frequency Distribution

Soru 13Soru →

The ages, in years, of five participants in a workshop are $2, 4, 5, 7,$ and $12$ . What is the mean deviation of the ages?

2.8

0.0

3.5

6.0

Cevabı ve açıklamayı göster

Cevap: $2.8$

Cevap

2.8

The mean deviation is calculated by taking the average of the absolute differences between each data point and the mean. For the dataset

2, 4, 5, 7,

and

12

, the mean is

6

. The absolute deviations from

6

are

4, 2, 1, 1,

and

6

, which sum to

14

. Dividing

14

by

5

gives

2.8

.

Adım Adım Çözüm

1

Calculate the arithmetic mean (

\bar{x}

) of the dataset.

\bar{x} = \frac{2 + 4 + 5 + 7 + 12}{5} = \frac{30}{5} = 6

The mean is needed as the reference point to calculate deviations.

2

Calculate the absolute deviation

|x - \bar{x}|

for each data item.

|2 - 6| = 4, \quad |4 - 6| = 2, \quad |5 - 6| = 1, \quad |7 - 6| = 1, \quad |12 - 6| = 6

Mean deviation measures distance from the mean, so absolute values are taken.

3

Find the average of these absolute deviations.

\text{Mean Deviation} = \frac{4 + 2 + 1 + 1 + 6}{5} = \frac{14}{5} = 2.8

The formula for mean deviation of ungrouped data is

\frac{\sum |x - \bar{x}|}{n}

.

Anahtar Kavram

Mean Deviation for Ungrouped Data

Soru 14Soru →

The test scores of six students in a mathematics quiz are $12$ , $18$ , $25$ , $31$ , $47$ , and $50$ . What is the range of these scores?

38

50

30.5

28

Cevabı ve açıklamayı göster

Cevap: $38$

Cevap

The range of the scores is

38

.

The range is a measure of dispersion defined as the difference between the maximum value (

50

) and the minimum value (

12

). Subtracting

12

from

50

gives

38

.

Adım Adım Çözüm

1

Identify the maximum and minimum values in the dataset.

Maximum score =

50

, Minimum score =

12

.

Range measures the spread between the highest and lowest values in a distribution.

2

Calculate the range using the formula

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

.

\text{Range} = 50 - 12 = 38

.

Subtracting the minimum value from the maximum value yields the measure of dispersion known as range.

Anahtar Kavram

Range of a Dataset

Soru 15Soru →

A library recorded the number of books borrowed over five consecutive days as $2$ , $4$ , $5$ , $6$ , and $8$ . What is the standard deviation of the number of books borrowed?

2

4

5

6

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The standard deviation of the number of books borrowed is

2

.

The mean of the five data points is

5

. The sum of the squared deviations from the mean is

9 + 1 + 0 + 1 + 9 = 20

. Dividing by

5

gives a variance of

4

. Taking the square root of

4

yields the standard deviation of

2

.

Adım Adım Çözüm

1

Calculate the arithmetic mean (

\bar{x}

) of the given dataset.

\bar{x} = \frac{2 + 4 + 5 + 6 + 8}{5} = \frac{25}{5} = 5

The mean is required to determine the deviations of each data point.

2

Find the deviations from the mean

(x - \bar{x})

and square each deviation

(x - \bar{x})^2

.

(2-5)^2 = 9, \quad (4-5)^2 = 1, \quad (5-5)^2 = 0, \quad (6-5)^2 = 1, \quad (8-5)^2 = 9

Squaring ensures all deviation values are positive before summation.

3

Calculate the variance (mean of squared deviations).

\text{Variance } (\sigma^2) = \frac{9 + 1 + 0 + 1 + 9}{5} = \frac{20}{5} = 4

Variance measures the average squared spread around the mean.

4

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

\text{Standard Deviation } (\sigma) = \sqrt{4} = 2

Standard deviation expresses dispersion in the original unit of measurement.

Anahtar Kavram

Standard Deviation of Ungrouped Data

Soru 16Soru →

The set of four numbers $\{x, 6, 8, 10\}$ has a variance of $5$ . Given that $x < 6$ , find the value of $x$ .

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x

is 4.

By expressing the mean as

\bar{x} = \frac{x+24}{4}

and setting the sum of squared deviations divided by 4 equal to 5, we arrive at the quadratic equation

x^2 - 16x + 48 = 0

. Solving this equation gives two possible values,

x = 4

and

x = 12

. Applying the restriction

x < 6

uniquely identifies

x = 4

as the correct solution.

Adım Adım Çözüm

1

Calculate the mean of the dataset in terms of

x

\bar{x} = \frac{x + 24}{4}

The mean of a dataset is the sum of all values divided by the total number of items.

2

Write the variance equation using the formula

\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}

\frac{(x - \bar{x})^2 + (6 - \bar{x})^2 + (8 - \bar{x})^2 + (10 - \bar{x})^2}{4} = 5

The given variance is 5 and the number of observations

N = 4

.

3

Substitute

\bar{x} = \frac{x + 24}{4}

and expand the squared deviations

\frac{3}{16}x^2 - 3x + 14 = 5 \implies x^2 - 16x + 48 = 0

Expanding the squared terms and multiplying through by

\frac{16}{3}

yields a standard quadratic equation.

4

Solve the quadratic equation

x^2 - 16x + 48 = 0

for

x

(x - 4)(x - 12) = 0 \implies x = 4 \text{ or } x = 12

Factoring the quadratic equation gives two possible roots.

5

Apply the given condition

x < 6

x = 4

Since

x < 6

, we reject

x = 12

and select

x = 4

.

Anahtar Kavram

Variance of Ungrouped Data and Algebraic Problem Solving

Soru 17Soru →

Five packages delivered by a courier service have masses of $5\text{ kg}$ , $8\text{ kg}$ , $11\text{ kg}$ , $12\text{ kg}$ , and $14\text{ kg}$ . What is the mean deviation of the masses of these packages?

0.0\text{ kg}

2.8\text{ kg}

10.0\text{ kg}

14.0\text{ kg}

Cevabı ve açıklamayı göster

Cevap: $2.8\text{ kg}$

Cevap

2.8\text{ kg}

The mean of the data set is

10\text{ kg}

. The distances of each data value from the mean are

5

,

2

,

1

,

2

, and

4

. The sum of these distances is

14

, and dividing by

5

yields a mean deviation of

2.8\text{ kg}

.

Adım Adım Çözüm

1

Calculate the arithmetic mean

(\bar{x})

of the dataset

\bar{x} = \frac{5 + 8 + 11 + 12 + 14}{5} = \frac{50}{5} = 10\text{ kg}

Mean deviation requires the central mean value as a reference point for all deviations.

2

Find the absolute deviation

|x - \bar{x}|

for each value in the dataset

|5 - 10| = 5, |8 - 10| = 2, |11 - 10| = 1, |12 - 10| = 2, |14 - 10| = 4

Mean deviation measures the average distance of values from the mean regardless of sign.

3

Sum the absolute deviations and divide by the sample size

(n = 5)

\text{Mean Deviation} = \frac{5 + 2 + 1 + 2 + 4}{5} = \frac{14}{5} = 2.8\text{ kg}

Dividing total absolute deviation by the total count yields the mean deviation.

Anahtar Kavram

Mean Deviation

Soru 18Soru →

A meteorologist recorded the rainfall (in millimeters) in a town over five days as $2$ , $5$ , $6$ , $7$ , and $10$ . What is the mean deviation of the rainfall data?

2

0

10

6

Cevabı ve açıklamayı göster

Cevap: $2$

Cevap

The mean deviation of the rainfall data is

2

.

To find the mean deviation, first calculate the mean of the data:

(2 + 5 + 6 + 7 + 10) / 5 = 6

. Next, calculate the absolute difference of each value from 6, which yields 4, 1, 0, 1, and 4. The average of these absolute differences is

(4 + 1 + 0 + 1 + 4) / 5 = 10 / 5 = 2

.

Adım Adım Çözüm

1

Calculate the arithmetic mean (

\bar{x}

) of the dataset.

\bar{x} = \frac{2 + 5 + 6 + 7 + 10}{5} = \frac{30}{5} = 6

The mean is required to compute the deviation of each observation from the central value.

2

Find the absolute deviation

|x - \bar{x}|

for each data point.

|2 - 6| = 4, \quad |5 - 6| = 1, \quad |6 - 6| = 0, \quad |7 - 6| = 1, \quad |10 - 6| = 4

Mean deviation measures the average absolute distance of data values from the mean.

3

Sum the absolute deviations and divide by the total number of observations (

N = 5

).

\text{Mean Deviation} = \frac{4 + 1 + 0 + 1 + 4}{5} = \frac{10}{5} = 2

Dividing the sum of absolute deviations by

N

gives the mean deviation.

Anahtar Kavram

Mean Deviation

Soru 19Soru →

The mean of five numbers $3, 5, 7, x,$ and $y$ is $6$ . Given that the variance of the numbers is $8$ and $x < y$ , calculate the value of $y$ .

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

The total sum of the five numbers is

5 \times 6 = 30

, giving

x + y = 15

. The sum of squared deviations from the mean

6

is

5 \times 8 = 40

. The known numbers

3, 5, 7

contribute

(-3)^2 + (-1)^2 + 1^2 = 11

to this sum, leaving

(x-6)^2 + (y-6)^2 = 29

. Substituting

y = 15 - x

yields

2x^2 - 30x + 88 = 0 \implies x^2 - 15x + 44 = 0

. Factoring gives roots

4

and

11

. Since

x < y

, we find

y = 11

.

Adım Adım Çözüm

1

Use the definition of the arithmetic mean to write a linear relationship between

x

and

y

.

x + y = 15

, or

y = 15 - x

.

The total sum of 5 numbers with a mean of 6 is

5 \times 6 = 30

. Subtracting the known numbers

3 + 5 + 7 = 15

leaves

x + y = 15

.

2

Apply the variance formula for population data.

\sum (x_i - 6)^2 = 40

.

Variance is the mean of squared deviations from the mean:

\frac{\sum (x_i - \bar{x})^2}{5} = 8 \implies \sum (x_i - 6)^2 = 40

.

3

Compute the sum of squared deviations for the known elements and simplify the variance equation.

(x - 6)^2 + (y - 6)^2 = 29

.

The squared deviations for

3, 5, 7

are

(-3)^2 = 9

,

(-1)^2 = 1

, and

1^2 = 1

. Subtracting

9 + 1 + 1 = 11

from

40

gives

29

.

4

Substitute

y = 15 - x

into the simplified equation and solve the resulting quadratic equation.

x = 4

or

x = 11

.

Substituting

y = 15 - x

gives

(x - 6)^2 + (9 - x)^2 = 29 \implies 2x^2 - 30x + 88 = 0 \implies x^2 - 15x + 44 = 0 \implies (x - 4)(x - 11) = 0

.

5

Select the correct pair

(x, y)

using the condition

x < y

.

x = 4

and

y = 11

.

Since

x < y

,

x

must be the smaller value (

4

) and

y

must be the larger value (

11

).

Anahtar Kavram

Calculation of variance and mean for ungrouped data containing unknown elements

Soru 20Soru →

The table below shows the distribution of scores obtained by a group of candidates in an examination:

Score ( $x$ )	2	4	6	8	10
Frequency ( $f$ )	1	2	6	2	1

Calculate the variance of the scores.

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The variance of the scores is 4.

The mean score is

\bar{x} = 6

. Summing the weighted squared deviations gives

\sum f(x - 6)^2 = 48

. Dividing by the total frequency

N = 12

results in a variance of

\frac{48}{12} = 4

.

Adım Adım Çözüm

1

Find the total frequency and calculate the mean score.

Total frequency

N = 12

and mean

\bar{x} = 6

.

The mean is given by

\bar{x} = \frac{\sum f x}{\sum f} = \frac{2 + 8 + 36 + 16 + 10}{12} = \frac{72}{12} = 6

.

2

Determine the sum of squared deviations multiplied by their frequencies.

\sum f(x - \bar{x})^2 = 48.

Evaluating each term:

1(2-6)^2 = 16

,

2(4-6)^2 = 8

,

6(6-6)^2 = 0

,

2(8-6)^2 = 8

, and

1(10-6)^2 = 16

. Summing these yields

16 + 8 + 0 + 8 + 16 = 48

.

3

Compute the variance by dividing the total squared deviations by the total frequency.

Variance = 4.

Variance is

\sigma^2 = \frac{\sum f(x - \bar{x})^2}{\sum f} = \frac{48}{12} = 4

.

Anahtar Kavram

Variance of a Frequency Distribution

Measures of Dispersion

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