Statistics

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation.

Formula Used: Mean Deviation =

Explanation: Here, Observations 3011, 2780, 3020, 2354, 3541, 4150, 5000 are Given.

Since, Median is the middle number of all the observation,

So, To Find the Median, Arrange the numbers in Ascending order, we get

2354, 2780, 3011, 3020, 3541, 4150, 5000

Therefore, The Median = 3020

Deviation |d| = |x-Median|

Now, The Mean Deviation is

Mean Deviation =

Hence, The Mean Deviation is 649.42

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation.

Formula Used: Mean Deviation =

Explanation: Here, Observations 38, 70, 48, 34, 42, 55, 63, 46, 54, 44 are Given.

Since, Median is the middle number of all the observation,

So, To Find the Median, Arrange the numbers in Ascending order, we get

34, 38,42,44,46,48,54,55,63,70

Here the Number of observations are Even then Median =

Therefore, The Median = 47

Deviation |d| = |x-Median|

And, The number of observations is 10.

Now, The Mean Deviation is

Mean Deviation =

Hence, The Mean Deviation is 8.6

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation.

Formula Used: Mean Deviation =

Explanation: Here, Observations 34, 66, 30, 38, 44, 50, 40, 60, 42, 51 are Given.

Since, Median is the middle number of all the observation,

So, To Find the Median, Arrange the numbers in Ascending order, we get

30, 34, 38, 40, 42, 44, 50,51, 60, 66

Here the Number of observations are Even then the middle terms are 42 and 44

Therefore, The Median = = 43

Deviation |d| = |x-Median|

And, The number of observations is 10.

Now, The Mean Deviation is

Mean Deviation =

Hence, The Mean Deviation is 8.7

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation.

Formula Used: Mean Deviation =

Explanation: Here, Observations 22, 24, 30, 27, 29, 31, 25, 28, 41, 42are Given.

Since, Median is the middle number of all the observation,

So, To Find the Median, Arrange the numbers in Ascending order, we get

22, 24, 25, 27, 28, 29, 30, 31, 41, 42

Here the Number of observations are Even then the middle terms are 28 and 29

Therefore, The Median = = 28.5

Deviation |d| = |x-Median|

And, The number of observations is 10.

Now, The Mean Deviation is

Mean Deviation =

Hence, The Mean Deviation is 8.7

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation.

Formula Used: Mean Deviation =

Explanation: Here, Observations 38, 70, 48, 34, 63, 42, 55, 44, 53, 47 are Given.

Since, Median is the middle number of all the observation,

So, To Find the Median, Arrange the numbers in Ascending order, we get

34, 38, 43, 44, 47, 48, 53, 55, 63, 70

Here the Number of observations are Even then the middle terms are 47 and 48.

Therefore, The Median = = 47.5

Deviation |d| = |x-Median|

And, The number of observations are 10.

Now, The Mean Deviation is

Mean Deviation =

Hence, The Mean Deviation is 8.4

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from mean.

Formula Used:

Explanation: Here, Observations 4, 7, 8, 9, 10, 12, 13, 17 are Given.

Deviation |d| = |x-Mean|

Mean =

Mean of the Given Observations =

And, The number of observations is 8.

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation of the given Observations =

Hence, The Mean Deviation is 3

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from Mean.

Formula Used: Mean Deviation =

Explanation: Here, Observations 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17 are Given.

Deviation |d| = |x-Mean|

Mean =

Mean of the Given Observations =

And, The number of observations is 12.

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation of the given Observations =

Hence, The Mean Deviation is 2.33

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation.

Formula Used: Mean Deviation =

Explanation: Here, Observations 38, 70, 48, 40, 42, 55, 63, 46, 54, 44 are Given.

Deviation |d| = |x-Mean|

Mean =

Mean of the Given Observations =

And, The number of observations is 10.

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation of the given Observations =

Hence, The Mean Deviation is 8.4

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation.

Formula Used: Mean Deviation =

Explanation: Here, Observations 36, 72, 46, 42, 60, 45, 53, 46, 51, 49 are Given.

Deviation |d| = |x-Mean|

Mean =

Mean of the Given Observations =

And, The number of observations is 10

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation of the given Observations =

Hence, The Mean Deviation is 7.4

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation.

Formula Used: Mean Deviation =

Explanation: Here, Observations 57, 64, 43, 67, 49, 59, 44, 47, 61, 59 are Given.

Deviation |d| = |x-Mean|

Mean =

Mean of the Given Observations =

And, The number of observations is 10

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation of the given Observations =

Hence, The Mean Deviation is 7.4

Given, Numbers of observations are given in two groups.

To Find: Calculate the Mean Deviation from their Median.

Formula Used: Mean Deviation =

For Group 1: Since, Median is the middle number of all the observation,

So, To Find the Median, Arrange the Income of Group 1 in Ascending order, we get

4000, 4200, 4400, 4600, 4800

Therefore, The Median = 4400

Deviation |d| = |x-Median|

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation Of Group 1 =

For Group 2: Since, Median is the middle number of all the observation,

So, To Find the Median, Arrange the Income of Group 2 in Ascending order, we get

3800,4000,4200,4400,4600,4800,5800

Therefore, The Median = 4400

Deviation |d| = |x-Median|

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation Of Group 2 =

Hence, The Mean Deviation of Group 1 is 240 and Group 2 is 457.14

(i) Given, Numbers of observations are given .

To Find: Calculate the Mean Deviation from their Median.

Formula Used: Mean Deviation =

Explanation: Since, Median is the middle number of all the observation,

So, To Find the Median, Arrange the given length of shops in Ascending order, we get

15.2, 27.9, 30.2, 32.5, 40.0, 52.3, 52.8, 55.2, 72.9, 79.0

Here the Number of observations are Even then Median =

Therefore, The Median = 46.15

Deviation |d| = |x-Median|

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation From Median =

Hence, The Mean Deviation is 16.74

(ii) Here, Observations 15.2, 27.9, 30.2, 32.5, 40.0, 52.3, 52.8, 55.2, 72.9, 79.0

are Given.

Deviation |d| = |x-Mean|

Mean =

Mean of the Given Observations =

And, The number of observations is 10

Now, The Mean Deviation is

Mean Deviation =

Mean Deviation of the given Observations =

Hence, The Mean Deviation is 16.64

(iii) 34, 66, 30, 38, 44, 50, 40, 60, 42, 51

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from Mean

Formula Used: Mean Deviation =

Explanation: Here, Observations 34, 66, 30, 38, 44, 50, 40, 60, 42, 51 are Given.

Mean =

Mean for given data

Deviation |d| = |x-Mean|

And, The number of observations is 10.

Now, The Mean Deviation is

Mean Deviation M.D=

Now,

- M.D = 45.5-9=36.5

+ M.D = 45.5+9=54.5

So, There are total 6 observation between - M.D and + M.D

(iv)Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from Mean

Formula Used: Mean Deviation =

Explanation: Numbers of observations are 22,24,30,27,29,31,25,28,41,42.

Mean =

Mean for given data

Deviation |d| = |x-Mean|

And, The number of observations is 10.

Now, The Mean Deviation is

Mean Deviation M.D=

Now,

- M.D = 29.9-4.88= 25.02

+ M.D = 29.9+4.88=34.78

So, There are total 6 observation between - M.D and + M.D

(V)Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from Mean

Formula Used: Mean Deviation =

Explanation: Numbers of observations are 38,70,48,34,63,42,55,44,53,47.

Mean =

Mean for given data

Deviation |d| = |x-Mean|

And, The number of observations is 10.

Now, The Mean Deviation is

Mean Deviation M.D=

Now,

- M.D = 49.4-8.68=40.72

+ M.D = 49.4+8.68=58.08

So, There are total 6 observation between - M.D and + M.D

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

We know, Median is the Middle term,

Therefore, Median = 61

Let x_i =Heights in inches

And, f_i = Number of students

N=197

Mean deviation=

Hence, The mean deviation is 1.70.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

We know, Median in the even terms ,

Therefore, Median = 4

Let x_i =Number of calls

And, f_i = Frequency

N = 245

Mean Deviation =

Mean deviation for given data =

Hence, The mean deviation is 1.49

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

Here, N =50

Then, N/2 =

SO, The median Corresponding to 25 is 13

N = 50

Mean Deviation =

Mean deviation for given data =

Hence, The mean Deviation is 2.72.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from the mean.

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from Mean So,

Mean =

Mean = =9

Mean Deviation =

Mean deviation for given data =

Hence, The mean Deviation is 3.3.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from the mean.

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from Mean So,

Mean =

Mean = =14

Mean Deviation =

Mean deviation for given data =

Hence, The mean Deviation is 6.32.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from the mean.

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from Mean So,

Mean =

Mean = =14

Mean Deviation =

Mean deviation for given data =

Hence, The mean Deviation is 6.32.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from the mean.

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from Mean So,

Mean =

Mean of given Data = =50

Mean Deviation =

Mean deviation for given data =

Hence, The mean Deviation is 16

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation from the mean.

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from Mean So,

Mean =

Mean of the given data is =21.65

Mean Deviation =

Mean deviation for given data is

Hence, The mean Deviation is 1.25.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

N = 21

SO, The median Corresponding to 10.5 is 27

Mean Deviation =

Mean deviation for given data =

Hence, The Mean Deviation is 4.80.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

N = 50

SO, The median Corresponding to 12.5 is 74

Mean Deviation =

Mean deviation for given data =

Hence, The Mean Deviation is 12.5.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

N = 20

SO, The median Corresponding to 10 is 12

Mean Deviation =

Mean deviation for given data

Hence, The Mean Deviation is 1.25.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

Median is the middle term of the X_i,

Here, The middle term is 25

Therefore, Median = 25

Mean Deviation =

Mean deviation for given data

Hence, The Mean Deviation is 9

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the mean. So,

Mean =

Here, Mean =

Therefore, Mean = 358

Mean Deviation =

Mean deviation for given data

Hence, The Mean Deviation is 157.92

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the mean. So,

Mean =

Here, Mean =

Therefore, Mean = 49

Mean Deviation =

Mean deviation for given data

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the mean. So,

Mean =

Here, Mean =

Therefore, Mean = 27

Mean Deviation =

Mean deviation for given data

Hence, The Mean Deviation is 10.24

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the mean. So,

Mean =

Here, Mean =

Therefore, Mean = 49

Mean Deviation =

Mean deviation for given data

Hence, The Mean Deviation is 14.95

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

N = 96

SO, The cumulative frequency just greater than 48 is 59, and the corresponding value of x is 38.25

Median = 38.25

Mean Deviation =

Mean deviation for given data

Hence, The Mean Deviation is 10.21.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

N = 50

SO, The cumulative frequency just greater than 25 is 58, and the corresponding value of x is 28

Median = 28

Now, Mean =

Mean = 27

Mean deviation from Median

And, Mean deviation from Median

Hence, The Mean Deviation from the median is 9.56 and from mean is 9.44.

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation. Here, The Given class interval does not continue, So, we subtract 0.5 from a lower limit of the class and add 0.5 to the upper limit of the class,By this Class interval remain same while the function becomes continues.

Here we have to calculate the mean deviation from the median. So,

N = 100

Thus, the cumulative frequency slightly greater than 50 is 63 and lie under class 35.5-40.5.

Median =

Median = 38

Mean Deviation =

Mean deviation for given data

Hence, The Mean Deviation is 73.5

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the mean. So,

Mean =

Here, Mean =

Therefore, Mean = 9.2

Mean Deviation =

Mean deviation for given data

Hence, The Mean Deviation is 3.84

Given, Numbers of observations are given.

To Find: Calculate the Mean Deviation

Formula Used: Mean Deviation =

Explanation.

Here we have to calculate the mean deviation from the median. So,

N = 20

So, the cumulative frequency just greater than 10 is 12, and the corresponding value of x is 15

Median = 15

Now, Mean =

Mean = 14.1

Mean deviation from Median

And, Mean deviation from Median

Hence, The Mean Deviation from the median is 6.9 and from mean is 6.99.

Explanation: Here, Mean

And, n=6

Variance (X) =

Variance = 23.33

Standard deviation =

Standard deviation = 4.83

Explanation: Here, Mean

And, n=8

Variance (X) =

Variance = 9.25

Standard deviation =

Standard deviation = 3.04

Explanation: Here, Mean

And, n=10

Variance (X) =

Variance = 15394.9

Standard deviation =

Standard deviation = 39.24

Explanation: Here, Mean

And, n=8

Variance (X) =

Variance = 38.75

Standard deviation =

Standard deviation = 6.22

Given, The variance of 20 observations is 4.

To Find: Find the variance of resulting observations.

Explanation: Let Assume, x₁,x₂,x₃,…,x₂₀ be the given observations.

So, Variance (X) = 5 (Given)

X =

Now, Let u₁,u₂,…u₂₀ be the new observation,

When we multiply the new observation by 2, then

U_i=2x_i (for i=1,2,3…,20) ---(i)

Now,

Mean =

Mean =

Since, u_i- = 2x_i

2(x_i)

Now, (u_i-)² = (2(x_i))²

4(x_i)²

Comparing Both the observations,

Variance (U) =4×Variance (X)

4×5

20

Hence, The variance of new observations is 20.

Given, The variance of 20 observations is 4.

To Find: Find the variance of resulting observations.

Explanation: Let Assume, x₁,x₂,x₃,…,x₁₅ be the given observations.

So, Variance (X) = 4 (Given)

X =

Now, Let u₁,u₂,…u₂₀ be the new observation,

When new observation increase by 9, then

U_i=x_i+9(for i=1,2,3…,20) ---(i)

Now,

Variance (U) =60

Hence, The variance of new observations is 60.

Given, Mean of 5 observation is 4.4 and variance is 8.24.

To Find: Find the other two observation

Assumption: Let x and y be the other two observation. And Mean is 4.4

Here, Mean =

9+x+y=22

X+y=13 ……(1)

Now, Let Variance (X) be the variance of this observation which is to be 8.24

If is the mean than we get,

8.24 =

8.24 =

8.24 =

X²+y² =97 ……(2)

(x+y)²+(x-y)²=2(x²+y²)

By Subsititute the value we get,

13²+(x-y)²=2×97

(x-y)²=194-169

(x-y)²=25

x-y =±5 ……(3)

On solving equations (1) and (3) we get,

2x = 18

X = 9

And, y =4

Hence, The other two observations are 9 and 4.

Given, The mean is 8 and SD is 4 for 6 observation

To Find: Find a new mean and new standard Deviation.

Formula Used: Standard Deviation ()

Explanation:

Let Assume, x₁,x₂,x₃,…,x₆ be the given observations.

So, Variance (X) = 8 (Given)

n=6

and (SD)

X =

Now, Let u₁,u₂,…u₂₀ be the new observation,

When we multiply the new observation by 3, then

U_i=3x_i (for i=1,2,3…,6) ……(i)

Now,

3×8 = 25

U = 24

Therefore, The Mean of new observation is 24

Now,

Standard Deviation

Since, Variance (X) =16

Variance (U) =

9×16

Hence, The mean of new observation is 24 and Standard deviation of new data is 12.

Given, Mean of 8 observation is 9 and variance is 9.25.

To Find: Find the other two observation

Assumption: Let x and y be the other two observation. And Mean is 9

Here, Mean =

60+x+y=72

X+y=12 ……(1)

Now, Let Variance (X) be the variance of this observation which is to be 9.25

If is the mean than we get,

9.25 =

9.25 =

642+x²+y²=722

X²+y² =80 ---(2)

(x+y)²+(x-y)²=2(x²+y²)

By Subsititute the value we get,

12²+(x-y)²=2×80

(x-y)²=160-144

(x-y)²=14

x-y =±4 ……(3)

On solving equations (1) and (3) we get,

X= 8, 4

And y = 4,8

Hence, The other two observations are 8 and 4.

To Find: Find the correct mean and standard deviation.

Explanation: Here, n=200,

X =

Since, the score was incorrect,

Now, The sum is incorrect

Corrected

8000-7

The correct score is 7993

So, The mean of correct score =

Mean = 39.95

Now, Standard variance = 15

Since, Variance =

Variance = 255

X =

255 =

255 =

Now, the correct
-34²-53²+43²+35²

365000-1156-2809+1849+1225

Corrected Variance =

1820.54-1596.40

Corrected variance =224.14

Now, Corrected Standard Deviation =

Correct Deviation is 14.97

To Find: Find the correct mean and standard deviation.

Explanation: Here, n=100,

X =

Now,

Corrected

3990

So, The mean of correct score =

Mean = 39.9

Now, Standard variance = 5.1

Since, Variance =

Variance = 26.01

X =

26.01 =

26.01 =

Corrected -50+40

Corrected

Corrected Variance =

1625.91-1592.01

Corrected variance =34 (Approx)

Now, Corrected Standard Deviation =

Correct Deviation is 5.83

Hence, The correct Mean is 39.9 and Correct SD is 5.83

Given: Mean = 10

And, standard deviation = 2

We know that,

When wrong item is omitted,

Corrected mean

When it is replaced by 12,

Corrected mean

Now, for standard deviation,

Variance = (2)² = 4

And we know that,

Now, omitting the wrong observation, we get,

Corrected Deviation

Now, replacing the observation

Corrected Deviation

To Find: Find the correct mean and standard deviation.

Explanation: Here, n=100,

X =

So, Corrected

Corrected

Now, Standard variance = 3

Since, Variance =

Variance = 9

X =

9 =

9 =

Corrected

Corrected

Correct mean =

Corrected Variance =

409.22-400

Corrected variance =9.22

Now, Corrected Standard Deviation =

Correct Deviation is 3.04

Hence, The correct Mean is 20 and Correct SD is 3.04

We know,

On Diving both sides by , we get

Taking square root both side

Hence, Proved

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation: Here, Mean =

Mean =

Now, N=70, ,

100[1.857-0.0987]

100[1.7583]

Standard Deviation

SD = 13.26

Hence, The standard deviation is 13.26

Given, The data is given in table

To Find: Find the mean and variance

Explanation: Mean =

Mean =

Now, N=70

Hence, The mean is 3.88 and variance is 3.63

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation:

Now, N=109, ,

Mean

100×2.396

Variance = 239.6

Standard Deviation

SD = 15.47 years

Hence, The standard deviation is 15.47

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation:

Now, N=40, ,

Mean

Variance = 8.12

Standard Deviation

SD = 2.85 years

Hence, The standard deviation is 2.85

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation:

Variance = 37.44

Standard Deviation

SD = 6.12

Hence, The standard deviation is 6.12

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation:

Now, N=60, ,

Mean

Variance = 1.88

Standard Deviation

SD = 1.37

Hence, The standard deviation is 1.37

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation:

Mean

Mean = 26.11

Variance = 165.47

Standard Deviation

SD = 12.86

Hence, The standard deviation is 12.86

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation:

Now, N=300, ,

Mean

Variance = 1807.31

Standard Deviation

SD = 42.51

Hence, The standard deviation is 42.51

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation:

Now, N=79, ,

Mean

Variance = 305.20

Standard Deviation

SD = 17.47

Hence, The standard deviation is 17.47

Given, Uncorrected mean is 40 and corrected SD is 5.1 and N = 100

To Find: Find the correct mean and SD

Explanation: Here,

Then,

The corrected sum of observation

So,

Now, Given Incorrect SD = 5.1

Corrected

Correct SD is 5

Hence, Correct mean is 39.90 and correct SD is 5

Given, The data is given in table/

To Find: Find the standard deviation

The formulaused: SD =

Explanation:

Now, N=50, ,

Mean

Variance = 58

Standard Deviation

SD = 7.62

Hence, The standard deviation is 7.62

Given, The data is given in table

To Find: Find the mean and, the variance of the frequency

Explanation: Here, the class interval is not continues frequency distribution, So we have to convert into continues frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit then we get,

Now, N=16, ,

Mean

Variance = 3.74

Hence, The variance is 3.74

Given, The data is given in table

To Find: Find the mean and, the variance of the frequency

Explanation: Here, the class interval in not continues frequency distribution, So we have to convert into continues frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit then we get,

Now, N=70, ,

Mean

Variance = 0.98

Standard Deviation

SD = 0.99

Hence, The standard deviation is 0.99

Given, Uncorrected mean is 40 and corrected SD is 10 and N = 100

To Find: Find the correct mean and SD

Explanation: Here,

Then,

The corrected sum of observation

So,

Variance =100

10000+160000

Incorrect

Correct

Correct

Correct SD,

Correct SD,

Hence, SD = 10.24

Given, Uncorrected mean is 40 and corrected SD is 10 and N = 100

To Find: Find the correct mean and SD

Explanation: Here,

Then,

The corrected sum of observation

So,

Variance =16

Incorrect

Correct

Correct

Correct SD,

Correct SD,

SD = 6.62

Variance = 6.62×6.62

Hence, Variance = 43.82

Given, The data is given in table

To Find: Find the standard deviation of the frequency

Explanation:

Now, N=70, ,

Mean

Variance = 100(3.57-1.96)

Variance = 161

Standard Deviation

SD = 12.7

Hence, The standard deviation is 12.7

Variation of the distribution of wages in plant A(=18)

So, Standard deviation of the distribution A()

Similarly, the Variation of the distribution of wages in plant B(=100)

So, Standard deviation of the distribution B()

And, Average monthly wages in both the plants is 2500,

Since The plant with a greater value of SD will have more variability in salary.

Hence, Plant B has more variability in individual wages than plant A

Given, The mean and SD is given of 50 students.

To Find: which shows more variability, height and weight.

The formulaused: Coefficient of variations =

Explanation:

The coefficient of variations in weights =

The coefficient of variations in weights =

As results clearly show that Cv heights is greater than Cv in weights.

Hence, Heights will show more variability than weights

Here, the Coefficient of variation for the first distribution is 60

And, Coefficient of variation for the first distribution is 70

SD =21 and SD = 16

We know that, Coefficients variation =

So, Mean

For first distribution

Mean = 35

For the second distribution

Mean = 22.86

Hence, Means are 35 and 22.86 .

Given, The data is given in table

To Find: Find the standard deviation of the frequency

Explanation:

Now, N=120,

Mean

Variance = 1076332.64

Standard Deviation

SD = 1037.47

Coefficients variation =

Cv=32.08

Hence, The coefficient variation is 332.08

(i) Average weekly wages =

Total weekly wages = (Avg weekly wages)×(No. of workers)

Total weekly wages of Firm A = 52.5×586 = Rs 30765

Total weekly wages of Firm B = 47.5×648 = Rs 30780

Firm B pays a larger amount as Firm A

(ii) Here SD(firm A) 10 and SD (Firm B) = 11

Coefficient variance (Firm A) =

Cv (Firm A) = 19.04

Coefficient variance (Firm B) =

Cv (Firm B) = 23.15

Hence, Cv of firm B is greater that that of firm A, Firm B has greater variability in individual wages.

Here SD(Boys) is 3 and SD (girls) = 2

Coefficient variability =

Coefficient variance (Boys) =

Cv (Boys) = 5

Coefficient variance (Girls) =

Cv (Girls) = 4.4

Hence, Cv Boys is greater than Cv girls,then the distribution of weights of boys is more variable than that of girls

Coefficient variability =

Cv (Maths) =

Cv (Maths) = 28.57

Cv (physics) =

So, Cv (physics) = 46.87

Cv (chemistry) =

So, Cv (chemistry) = 48.89

Hence, Cv of chemistry is greatest, the variability of marks in chemistry is highest and that of Mathematics is lowest.

Given, The data is given in the table.

To find: Check which group is more variable G1 and G2

Explanation: Let’s find the coefficient of the variable for group G1

Here, n=150 a= 45

Mean

Variance = 227.84

Standard Deviation

Standard Deviation

The coefficient of variation =

So, Coefficient of variation =

Cv = 33.83

Now find the coefficient of variable for group G2

Here, n=150 a= 45

Mean

Variance = 243.84

Standard Deviation

Standard Deviation

The coefficient of variation =

So, Coefficient of variation =

Cv = 35.02

Since, G2 has a high coefficient of variance,

Hence, Group G2 is more variable.

Given, The data is given in table

To Find: Find the coefficient variation.

Explanation:

Now, N=100,

Mean

Variance = 37.34

Standard Deviation

SD = 6.11

Coefficients variation =

Cv=21.9

Hence, The coefficient variation is 21.9

Given, Data is given in the form of two table

To Find: Find out which one is more stable in value

Explanation: Let’s Find for Value X

Now, N=70

=48

SD (X)=

SD (X)==9.9

Coefficient of variation =

Coefficient of variation = = 20.6

Let’s Find for Value Y

Now, N=10

=105

SD (Y)=

SD (Y)==2

Coefficient of variation =

The coefficient of variation = = 1.90

Hence, Cv for Y is smaller that Cv of X, So X is more stable that y.

For Factory A

Now, N=120,

Mean

Variance = 12097

Standard Deviation

SD = 109.98

Coefficients variation =

Cv=13.61

For Factory B

Now, N=120,

Mean

Variance = 11400

Standard Deviation

SD = 106.77

Coefficients variation =

Cv=14.29

Since, the coefficient of variation of factory B is greater than the coefficient of variation of factory A,

Hence, This means bulbs of factory A are more consistent from the point of view of the length of life.

Given, Marks obtained by two students in 10 tests are given in Table.

To Find: Who is more intelligent and who is more consistent?

Explanation: Marks obtained by Ravi

Mean,

SD ()=

SD ()=

SD ()=

SD ()=13.003

Coefficient of variation =

Cv =

For Hashima

Mean,

SD ()=

SD ()=

SD ()=

SD ()=25.16

Coefficient of variation =

Cv =

Since, The coefficient of variation in mark obtained by Hashima is greater than the coefficient of Variation in mark obtained by Ravi,

Hence, Hasima is more consistent and intelligent

Let the numbers be 1,2,3,…,n

Sum of First n natural numbers is

Mean

Hence,

Given, The sum of the squares of derivations for 10 observation, and mean is 2.5.

To Find: Find the standard derivation

Explanation: Here, N=10 and mean = 2.5

The square of each division =

Standard deviation

Hence, standard deviation is 0.5

And, y_i=ax₁+b

Hence, proved

Given,

To Find: Write the expression for the standard deviation of Y.

Explanation: We have

Mean (y) =

We can write as Mean (y) =

Mean (y) =

Var(X) =

But, Var(X)=

Then, Var(Y) =

Now, Substitute the value of y_iand Y, then we get

Hence, Proved

Given, n=20, d_i=x_i-a

Where

K=±2

We know,

Standard deviation

And, mean

Now, multiply by a in x_i

New standard deviation,

Hence, Proved

we know, mean x =

Hence,

The general formula of Mean is

For mean deviation, d = (x-mean)

M.D =

We know,

M.D =

Variance =

SD,

Hence,

If v is the variance and is the standard deviation, then

We know that the formula of standard variance is

So,

equal to that measured from the another value

Given, n=10,

We know, Variance =

Variance =

Variance = (153-144)

SD = 3

Coefficient of variance =

Coefficient of variance =

Cv = 25

Hence, Cv = 15

Let us calculate the mean first,

Mean

Standard deviation

Given, Series is a, a+d, a+2d,…,+a+2n

Mean (X) =

Now, deviation from mean is x_i-X

M.D =

M.D =

M.D =

Hence, M.D =

Mean (X)=

Here, N= 10

Mean deviation =

Hence, MD is 0

The mean deviation of the numbers 3, 4, 5, 6, 7

Mean

=5

Mean deviation =

Mean deviation =

Hence, Mean deviation is 0

Given, n=10 mean 250

SD,

SD = 5

Now, Coefficient of variance =

Cv =

Cv = 50

Hence, Coefficient of Variation is 10

we have given, y_i=ax_i+b

Mean (Y) =

Mean (y) =

Then, Var(Y) =

And, Var(X) =

Hence, Var(Y) = a² Var(X)

We have

Mean (X) =

We can write as: Mean (X) =

Mean (X) =

Var(X) =

Now, Substitute the value of y_iand Y, then we get

Hence, Proved

Let take two observation 16 and 32

Now,

SD = 8

Now, If we divide each observation then SD is

SD = 4

Hence, SD will also be half.

Given,

To Find: Write the expression for the standard deviation of Y.

Explanation: We have

Mean (y) =

We can write as: Mean (y) =

Mean (y) =

Var(X) =

Then, Var(Y) =

Now, Substitute the value of y_iand Y, then we get

Given, N=60,,

Variance = 44

Hence, variance = 44

Let a, b, c,d,e be the observation and mean is m

Let suppose new mean be m₁

M₁ =m+k

Now, The standard deviation

So, The standard deviation for new observation

Now, we can compare both observation

a+k-n=a+k-(m+k)

a+k–n = a+k–m- k

a+k-n=a-m

Similary

b+k-n=b-m

c+k-n=c-m

d+k-n=d-m

e+k-n=e-m

when we substitute the values, we get,

S₁ = S

Hence, The Sd is S

First 10 natural numbers are 1,2,3,4,5,6,7,8,9,10

So, N =10

Hence, SD is 2.87

Consider 10 positive integer

Let Assume, 1,2,3,4,5,6,7,8,9,10

Now, If we multiply by -1 in each number we get,

-1,-2,-3,-4,-5,-6,-7,-8,-9,-10

And then we add 1 in each number

0,-1,-2,-3,-4,-5,-6,-7,-8,-9

Now,

Standard deviation

Hence, variance is 8.25

Consider numbers are 1,2,3,4,5,6,7,8,9,10

If one is added to each number then, numbers will be

Let say x_i = 2,3,4,5,6,7,8,9,10,11

So, N= 10

Standard deviation

Hence, the variance is 8.25

Given,

=50×100

=5000

Now,

Mean(y) = a.mean(x) + b

Therefore,

55 = a.48 + b

We can see that only first option satisfies this equation. Therefore, a is the correct answer.

Explanation: Mean

Mean Deviation =

Mean Deviation =

Hence, The MD is

Let x₁,x₂,…x_n be n observation

And X is the aithemetic mean then,

We know, Standard deviation

So,

Let x₁,x₂,…x_n be n observation

And X is the arithmetic mean then,

We know, Standard deviation

So,

The standard deviation of the observations 6, 5, 9, 13, 12, 8, 10 is

And, N=7

Standard deviation