Class 11 RD Sharma Solutions - Chapter 32 Statistics - Exercise 32.4

Question 1. Find the mean, variance and standard deviation for the following data:

(i) 2, 4, 5, 6, 8, 17

(ii) 6, 7, 10, 12, 13, 4, 8, 12

(iii) 227, 235, 255, 269, 292, 299, 312, 321, 333, 348

(iv) 15, 22, 27, 11, 9, 21, 14,9

Solution: 

(i)

x	d = (x - Mean)	d2
2	-5	25
4	-3	9
5	-2	4
6	-1	1
8	1	1
17	10	100
Total = 42	Total = 140

= 1/6[42] = 7

= 1/6[140] = 23.33

Standard deviation = √Var(x) = √23.33 = 4.8

(ii) 

x	d = (x - Mean)	d2
6	-3	9
7	-2	4
10	1	1
12	3	9
13	4	16
4	-5	25
8	-1	1
12	3	9
Total = 72	Total = 74

Mean = 

= 1/8[72] = 9

= 1/8[74] = 9.25

Standard deviation = √Var(x) = √9.25 = 3.04

(iii)

xi	di = xi - 299	di2
227	-72	5184
235	-64	4096
255	-44	1936
269	-30	900
292	-7	49
299	0	0
312	13	169
321	22	484
333	34	1156
348	49	2401
Total = -99	Total = 16375

Mean = = 299 + (-99/10) = 289.1

= 16375/10 - (-99/10)² 

= 1637.5 - 98.01 

= 1539.49

Standard deviation = √Var(x) = √1539.49 = 39.24

(iv)

xi	di = xi - 15	di2
15	0	0
22	7	49
27	12	144
11	-4	16
9	-6	36
21	6	36
14	-1	1
9	-6	36
Total = 8	Total = 318

Mean = = 15 + 8/8 = 16

= 318/8 - 1 = 38.75

Standard deviation = √Var(x) = √38.75 = 6.22

Question 2. The variance of 20 observations is 4. If each observation is multiplied by 2, find the variance of the resulting observations.

Solution:

Given: n = 20, and 

Now multiply each observation by 2, we get

Suppose X = 2x be the new data.

 =5

So, for the new data, we have

= 4 × 5 

= 20

Question 3. The variance of 15 observations is 4. If each observation is increased by 9, find the variance of the resulting observations.

Solution:

Given: n = 15, and 

Now increase each observation by 9, we get

Suppose X = x + 9 be the new data.

So for the new data:

= 4

Question 4. The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2, and 6, find the other two observations.

Solution:

Let us considered the other two observations are x and y

Given: The mean of 5 observations is 4.4 and their variance is 8.24

So, 

Mean = 1 + 2 + 6 + x + y = 5 × 4.4 

= x + y = 13

Variance = [(1 - 4.4)² + (2 - 4.4)² + (6 - 4.4)² + (x - 4.4)² + (y - 4.4)³]

11.56 + 5.76 + 2.56 + (x - 4.4)² + (y - 4.4)² = 41.2

(x - 4.4)² + (y - 4.4)² = 21.32

On solving this equation, we get 

(x - 4.4)² + (13 - x - 4.4)² = 21.32

(x - 4.4)² + (8.6 - x)² = 21.32

x² - 8.8x + 19.36 + 73.96 - 17.2x + x² = 21.32

2x² - 26x + 72 = 0

x² - 13x + 36 = 0

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

x = 4 or x = 9

So, the other two observation are 4 and 9. 

Question 5. The mean and standard deviation of 6 observations are 8 and 4 respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

Solution:

Given: Mean of 6 observations = 8

Standard Deviation of 6 observation = 4

k = 3

So, let us considered mean and Standard Deviation of the observation are  and 

then the mean and Standard Deviation of the observation multiplied by a constant 'k' are

So, the new mean = 8 × 3 = 24

New Standard Deviation = 4 × 3 = 12

Question 6. The mean and variance of 8 observations are 9 and 9.25 respectively. If six of the observations are 6, 7, 10, 12, 12, and 13, find the remaining two observations.

Solution:

Given: Mean of 8 observations = 9

Standard Deviation of 8 observations = 9.25

Observations = 6, 7, 10, 12, 12, and 13

So, let us considered the other two observations are x and y

Mean = (6 + 7 + 10 + 12 + 12 + 13 + x + y)/8 = 9

= 60 + x + y =72

= x + y = 12            -(1)

Variance = 1/8(6² + 7² + 10² + 12² + 12² + 13² + x² + y²) - (81)² = 9.25

= 642 + x² + y² = 722

= x² + y² = 80            -(2)

Now, (x + y)² + (x - y)² = 2(x² + y²)

= 144 + (x - y)² = 2 × 80

= (x - y)² = 16

= x - y = ±4

If x - y = 4, then x + y = 12 and x - y = 4

So, x = 8, y = 4

If x - y = -4 then x + y = 12 and x - y = -4

So, x = 4, y = 8

Hence, the remaining two observations are 4 and 8.

Question 7. For a group of 200 candidates, the mean and standard deviations of scores were found to be 40 and 15 respectively. Later on it was discovered that the scores of 43 and 35 were misread as 34 and 53 respectively. Find the correct mean and standard deviation.

Solution:

Given: n = 200, 

 = 200 × 40 = 8000

Corrected  = Incorrect  - (sum of incorrect values) + (sum of correct values)

= 8000 - 34 - 53 + 43 + 35 = 7991

Corrected mean = 

= 7991/200 = 39.955

= 200 × 1825 = 365000

Incorrect  = 36500

Corrected  = (incorrect ) - (sum of squares of incorrect value) +

                                        (sum of squares of correct values)

= 365000 - (34)² - 53² + (43)² + 35² = 364109

So, Corrected 

= 14.97

Question 8. The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and standard deviation?

Solution:

Given: n = 100, 

 = 100 × 1626.01 = 162601

Incorrect = 162601

Corrected = (incorrect ) - (sum of squares of incorrect values) + (sum of squares of correct values)

= 162601 - (50)² + (40)² = 161701

So, Corrected 

Question 9. The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:

(i) If wrong item is omitted.

(ii) If it is replaced by 12.

Solution:

Given: n = 20, 

(i) If we remove 8 from the given observation then 19 observation are left.

Now, Incorrect  = 200

⇒ Corrected  + 8 = 200

⇒ Corrected  = 192

and,

⇒ Incorrect  = 2080

⇒ Corrected  + 8² = 2080

⇒ Corrected  = 2080 - 64

⇒  Corrected  = 2016

Therefore,

Corrected mean =  = 10.10

⇒ So, corrected variance = 

= 2016/19 - (192/19)² 

= (38304 -36864)/361

= 1440/361 

So, the corrected standard deviation =  = 1.997

(ii) If we replace the incorrect observation(i.e., 8) by 12 

Given: Incorrect  = 200

Therefore, Corrected  = 200 - 8 + 12 = 204

Incorrect  = 2080

Therefore, Corrected   = 2080 - 8² + 12² = 2160

Now, Corrected mean = 204/20 = 10.2

Corrected variance = 

= 2016/20 - (204/20)²   

= 

= 

= 1584/400

So, the corrected standard deviation = 

= 19.899/10 = 1.9899

Question 10. The mean and standard deviation of group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21, and 18. Find the mean and standard deviation if the incorrect observation were omitted.

Solution:

(i) Given: n = 100, 

Mean = 

= 20 × 100 = 2000

Incorrect  = 2000

and, 

Incorrect  = 40900.

When the incorrect observations 21, 21, 18 are removed from the data

then the total number of observation are n = 97

Now,

Incorrect = 2000

Corrected  = 2000 - 21 - 21 - 18 = 1940

and,

Incorrect  = 40900

Corrected  = 40900 - 21² - 21² - 18²

= 40900 - 1206

= 39694

Therefore, Corrected mean = 1940/97 = 20

Corrected variance = 

 = (39694/97) - (20)² = 409.22 - 400 = 9.22

So, the corrected standard deviation = √9.22 = 3.04

Question 11. Show that the two formulae for the standard deviation of ungrouped data

are equivalent, where 

Solution:

Given:

On dividing both the sides by n we get,

Now, taking square root on both the sides, we get