Statistics Class 10th Mathematics Tamilnadu Board Solution

Class 10th Mathematics Tamilnadu Board Solution
Exercise 11.1
  1. 59, 46, 30, 23, 27, 40, 52, 35, 29 Find the range and coefficient of range of…
  2. 41.2, 33.7, 29.1, 34.5, 25.7, 24.8, 56.5, 12.5 Find the range and coefficient…
  3. The smallest value of a collection of data is 12 and the range is 59. Find the…
  4. The largest of 50 measurements is 3.84 kg. If the range is 0.46 kg, find the…
  5. The standard deviation of 20 observations is √5. If each observation is…
  6. Calculate the standard deviation of the first 13 natural numbers.…
  7. 10, 20, 15, 8, 3, 4 Calculate the standard deviation of the following data.…
  8. 38, 40, 34 ,31, 28, 26, 34 Calculate the standard deviation of the following…
  9. |c|c|c|c|c|c| x&3&8&13&18&23 y&7&10&15&10&8 Calculate the standard deviation of…
  10. The number of books bought at a book fair by 200 students from a school are…
  11. Calculate the variance of the following data |c|c|c|c|c|c|c|c|c|
  12. The time (in seconds) taken by a group of people to walk across a pedestrian…
  13. A group of 45 house owners contributed money towards green environment of…
  14. Find the variance of the following distribution. |c|c|c|c|c|c|c|c|…
  15. Mean of 100 items is 48 and their standard deviation is 10. Find the sum of…
  16. The mean and standard deviation of 20 items are found to be 10 and 2…
  17. If n = 10 , bar x = 12 and sum x^2 = 1530 , then calculate the coefficient of…
  18. Calculate the coefficient of variation of the following data: 20, 18, 32, 24,…
  19. If the coefficient of variation of a collection of data is 57 and its S.D is…
  20. A group of 100 candidates have their average height 163.8 cm with coefficient…
  21. Given sum x = 99 , n = 9 and sum (x-10)^2 = 79 . Find sum x^2sum (x - bar x)^2…
  22. The marks scored by two students A, B in a class are given below.…
Exercise 11.2
  1. The range of the first 10 prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 isA.…
  2. The least value in a collection of data is 14.1. If the range of the collection…
  3. The greatest value of a collection of data is 72 and the least value is 28.…
  4. For a collection of 11 items, sum x = 132 , then the arithmetic mean isA. 11 B.…
  5. For any collection of n items, sum (x - bar x) =A. sum x B. bar x C. i bar x D.…
  6. For any collection of n items, (sum x) - bar x =A. i bar x B. (11-2) bar x C.…
  7. If t is the standard deviation of x, y. z, then the standard deviation of x +…
  8. If the standard deviation of a set of data is 1.6, then the variance isA. 0.4…
  9. If the variance of a data is 12.25, then the S.D isA. 3.5 B. 3 C. 2.5 D. 3.25…
  10. Variance of the first 11 natural numbers isA. root 5 B. root 10 C. 5 root 2 D.…
  11. The variance of 10, 10, 10, 10, 10 isA. 10 B. root 10 C. 5 D. 0
  12. If the variance of 14, 18, 22, 26, 30 is 32, then the variance of 28,…
  13. Standard deviation of a collection of data is 2√2. If each value is multiplied…
  14. Given sum (x - bar x) = 48 , bar x = 20 and n = 12. The coefficient of…
  15. Mean and standard deviation of a data are 48 and 12 respectively. The…

Exercise 11.1
Question 1.

Find the range and coefficient of range of the following data.

59, 46, 30, 23, 27, 40, 52, 35, 29


Answer:

Given the set of data 59, 46, 30, 23, 27, 40, 52, 35, 29.


Let the maximum value in the data set be denoted by xmax and minimum by xmin


Range = Maximum value in the data set – Minimum value in the data set


= xmax - xmin


= 59 – 23


= 36


∴ Range R = 36


Let the Coefficient of Range be denoted by s.


s = 




= 0.439≈0.44


∴ s = 0.44



Question 2.

Find the range and coefficient of range of the following data.

41.2, 33.7, 29.1, 34.5, 25.7, 24.8, 56.5, 12.5


Answer:

Given the set of data 41.2, 33.7, 29.1, 34.5, 25.7, 24.8, 56.5, 12.5.


Let the maximum value in the data set be denoted by xmax and minimum by xmin


Range = Maximum value in the data set – Minimum value in the data set


= xmax - xmin


= 56.5 – 12.5


= 44


∴ Range R = 44


Let the Coefficient of Range be denoted by s.


s = 




= 0.637≈0.64


∴ s = 0.64



Question 3.

The smallest value of a collection of data is 12 and the range is 59. Find the largest value of the collection of data.


Answer:

Given xmin = 12, R = 59


Range R = xmax - xmin


Substituting the given terms into the above formula, we get


R = xmax - xmin


59 = xmax - 12


 xmax = 59 + 12 = 71



Question 4.

The largest of 50 measurements is 3.84 kg. If the range is 0.46 kg, find the smallest measurement.


Answer:

Given xmax = 3.84kg, R = 0.46kg


Range R = xmax - xmin


Substituting the given terms into the above formula, we get


R = xmax - xmin


0.46 = 3.84- xmin


 xmin = 3.84 - 0.46 = 3.38kg



Question 5.

The standard deviation of 20 observations is √5. If each observation is multiplied by 2, find the standard deviation and variance of the resulting observations.


Answer:

Let the observations be x1, x2, x3….x20


Given SD = 


Formula for SD = 


Where N = 20, SD = 


If each observation is multiplied by 2, we get new set of observations.


Let the new observations be y1, y2, y3….y20


New Standard Deviation SD’ = 


We know that  and 


But yi = 2xi


Therefore, 


Substituting in the SD’ formula,


SD’ = 



= 2 × SD = 2


Formula relating SD and Variance is Variance = SD2


So Variance = (22 = 20



Question 6.

Calculate the standard deviation of the first 13 natural numbers.


Answer:

To find 


Formula for 




SD = 


3.74



Question 7.

Calculate the standard deviation of the following data.

10, 20, 15, 8, 3, 4


Answer:

To find 

Formula for 




SD = 


5.97



Question 8.

Calculate the standard deviation of the following data.

38, 40, 34 ,31, 28, 26, 34


Answer:

To find 

Formula for 




SD = 


4.69



Question 9.

Calculate the standard deviation of the following data.



Answer:


To find 


Formula for 



SD = 



= 6.32



Question 10.

The number of books bought at a book fair by 200 students from a school are given in the following table.



Calculate the standard deviation.


Answer:


To find 


Formula for 



SD = 



= 1.107



Question 11.

Calculate the variance of the following data



Answer:


To find 


Formula for 



SD = 



= 3.88


To find Variance


Variance = SD2


= 3.882 = 15.058



Question 12.

The time (in seconds) taken by a group of people to walk across a pedestrian crossing is given in the table below.



Calculate the variance and standard deviation of the data.


Answer:


To find 


Formula for 



SD = 



= 6.063


To find Variance


Variance = SD2


= 6.0632 = 36.76



Question 13.

A group of 45 house owners contributed money towards green environment of their street. The amount of money collected is shown in the table below.



Calculate the variance and standard deviation.


Answer:


To find 


Formula for 



SD = 



= 20.39


To find Variance


Variance = SD2


= 20.392 = 416



Question 14.

Find the variance of the following distribution.



Answer:


To find 


Formula for 



SD = 



= 7.361


To find Variance


Variance = SD2


= 7.3612 = 54.19



Question 15.

Mean of 100 items is 48 and their standard deviation is 10. Find the sum of all the items and the sum of the squares of all the items.


Answer:

Mean of 100 items  = 48


Standard deviation SD = 10


To find : 


To find :


Formula for 


= 48 = 


Sum of all the items 


To find:


SD2 = 


100 = 482


240400



Question 16.

The mean and standard deviation of 20 items are found to be 10 and 2 respectively. At the time of checking it was found that an item 12 was wrongly entered as 8. Calculate the correct mean and standard deviation.


Answer:

Mean of 20 items  = 10


SD of 10 items = 2


Wrong Value = 8


Correct Value = 12


To find Mean:


Formula for 


= 10 = 


Sum of all the items 200


This value is wrong since item 12 was wrongly entered as 8.


Let  be the corrected sum


200 + (Correct Value) - (Wrong Value)


= 200 + 12 - 8 = 204


Corrected mean will be  = 10.2


To find SD:


SD2 = 


4 = 


 = 2080


This value is wrong since item 12 was wrongly entered as 8.


Let  be the corrected sum of squares.


 = 2080 + (Correct Value)2 - (Wrong Value)2


= 2080 + 122 - 82


= 2160


Corrected SD:


SD2 = 


SD2 = 


SD = 1.99


The corrected mean is 10.2, corrected SD is 1.99



Question 17.

If  and , then calculate the coefficient of variation.


Answer:

SD2 = 


SD2 = 


SD = 3


Coefficient of Variation


CV = 


 = 25



Question 18.

Calculate the coefficient of variation of the following data: 20, 18, 32, 24, 26.


Answer:

To find 

Formula for 


24



SD = 


4.89


CV = 


20.41



Question 19.

If the coefficient of variation of a collection of data is 57 and its S.D is 6.84, then find the mean.


Answer:

Given SD = 6.84 and CV = 57, to find 

CV = 


57


12



Question 20.

A group of 100 candidates have their average height 163.8 cm with coefficient of variation 3.2. What is the standard deviation of their heights?


Answer:

Given N = 100, CV = 3.2,  = 163.8 cm

CV = 


3.2


5.24



Question 21.

Given and . Find .


Answer:

SD = 


Or SD2 = 


To find 


 = 


 SD2 =  = 


Expanding the numerator in the form of (a - b)2 = a2 + b2 - 2ab,


SD2 =  = 


Substituting all the known terms in the above equation,


SD2 =  = 7.77


We know that, SD2 = 


7.77 = 


 = 1159


SD2 =  = 7.77


 = 70



Question 22.

The marks scored by two students A, B in a class are given below.



Who is more consistent?


Answer:

SD of A


 = 60



SD = 


SDA = 5.4


SD of B


 = 64



SD = 


SDB = 6.7


5.4<6.7


SD of A is less than SD of B.


 A is more consistent than B




Exercise 11.2
Question 1.

The range of the first 10 prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 is
A. 28

B. 26

C. 29

D. 27


Answer:

Range R = xmax - xmin

R = 29 - 2 = 27


So the correct option is D.


Question 2.

The least value in a collection of data is 14.1. If the range of the collection is 28.4, then the greatest value of the collection is
A. 42.5

B. 43.5

C. 42.4

D. 42.1


Answer:

Given xmin = 14.1, R = 28.4

Range R = xmax - xmin


28.4 = xmax - 14.1


 xmax = 42.5


So the correct option is A.


Question 3.

The greatest value of a collection of data is 72 and the least value is 28. Then the coefficient of range is
A. 44

B. 0.72

C. 0.44

D. 0.28


Answer:

Given xmax = 72, xmin = 28

Coefficient of range s = 



So the correct option is C.


Question 4.

For a collection of 11 items, , then the arithmetic mean is
A. 11

B. 12

C. 14

D. 13


Answer:

N = 11

AM 


So the correct option is B.


Question 5.

For any collection of n items,  =
A. 

B. 

C. 

D. 0


Answer:

 can be rewritten as 

We know that  so 


Also,  because  is just a constant term.


Substituting these in , we get  = 0


So the correct option is D.


Question 6.

For any collection of n items,  =
A. 

B. 

C. 

D. 0


Answer:

We know that  so 


Substituting this in the question, we have



So the correct option is C.


Question 7.

If t is the standard deviation of x, y. z, then the standard deviation of x + 5, y + 5, z + 5 is
A. 

B. t + 5

C. t

D. x y z


Answer:

It is a direct consequence of the theory of standard deviation that, the same constant term added to the terms in a set of data doesn’t change the value of standard deviation.

So the correct answer is t and correct option is C.


Question 8.

If the standard deviation of a set of data is 1.6, then the variance is
A. 0.4

B. 2.56

C. 1.96

D. 0.04


Answer:

Variance = SD2


So Variance = 1.62 = 2.56


Therefore, the correct option is B.


Question 9.

If the variance of a data is 12.25, then the S.D is
A. 3.5

B. 3

C. 2.5

D. 3.25


Answer:

Variance = SD2


Or SD = 


SD = 3.5


So the correct option is A.


Question 10.

Variance of the first 11 natural numbers is
A. 

B. 

C. 

D. 10


Answer:

Variance of first n natural numbers is given by 


V = 10


So the correct option is D.


Question 11.

The variance of 10, 10, 10, 10, 10 is
A. 10

B. 

C. 5

D. 0


Answer:

Standard deviation of a constant set of data is 0.


We know that Variance = SD2


Variance = 0


So the correct option is D.


Question 12.

If the variance of 14, 18, 22, 26, 30 is 32, then the variance of 28, 36,44,52,60 is
A. 64

B. 128

C. 32√2

D. 32


Answer:

Two data sets are given.


Set A : 14, 18, 22, 26, 30 with V = 32


Set B : 28, 36, 44, 52, 60 with V = ?


We notice that each data entry in set B is twice the corresponding data entry in set A.


SD of set B = 2 × SD of set A.


We know that Variance = SD2


Variance of Set B = 22 × Variance of Set A.


= 4 × 32 = 128.


So the correct option is B.


Question 13.

Standard deviation of a collection of data is 2√2. If each value is multiplied by 3, then the standard deviation of the new data is
A. 

B. 4√2

C. 6√2

D. 9√2


Answer:

SD of set B = n × SD of set A, where n is the data multiplier

Here, n = 3.


So, SD of set B = 3 × SD of set A = 3 × 2 = 6


So, the correct option is C.


Question 14.

Given  and n = 12. The coefficient of variation is
A. 25

B. 20

C. 30

D. 10


Answer:

Insufficient data.


Question 15.

Mean and standard deviation of a data are 48 and 12 respectively. The coefficient of variation is
A. 42

B. 25

C. 28

D. 48


Answer:

CV = 


CV = 


So, the correct option is B.