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NCERT Class XI Mathematics - Statistics - Solutions

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Question : 32
Total: 34
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:  
Here n = 20, Incorrect mean (x) = 10, Incorrect S.D. (s) = 2
Now,
−
x
 =
1
n
Σxi ⇒ Σxi = n ×
−
x
 = 20 × 10 = 200
∴ Incorrect Σxi = 200
Also ,
1
n
Σxi2−(
−
x
)2 = 4
⇒
1
20
Σxi2−(10)2 = 4 ⇒ Σxi2 = 2080
∴ Incorrect Σxi2 = 2080
(i) When wrong item 8 is omitted from the data then we have 19 observations.
∴ Correct Σxi = Incorrect Σxi - 8
Correct Σxi = 200 - 8 = 192
∴ Correct mean =
192
19
 = 10.1
Also, correct Σxi2 = Incorrect Σi2−(8)2
⇒ Correct Σxi2 = 2080 - 64 = 2016
∴ Correct variance =
1
19
 (correct Σxi2) - (correct‌mean)2
=
1
19
 × 2016 - (
192
19
)2 =
2016
19
−
36864
361
 =
38304−36864
361
 =
1440
361
∴ Correct S.D. = √
1440
361
 = √3.98 = 1.99
(ii) If wrong item 8 is replaced by 12
Correct Σxi = Incorrect Σxi – 8 + 12 = 200 – 8 + 12 = 204
∴ Correct mean =
204
20
 = 10.2
Also correct Σxi2 = Incorrect Σxi2 - (8)2+(12)2 = 2080 - 64 + 144 = 2160
∴ Correct variance =
1
20
(correct‌Σ‌xi2)−(correct‌mean)2
=
2160
20
−(
204
20
)2 =
2160
20
−
41616
400
 =
43200−41616
400
 =
1584
400
∴ Correct S.D. = √
1584
400
 = √3.96 = 1.98
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