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NCERT Class XI Mathematics - Statistics - Solutions

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Question : 34
Total: 34
The mean and standard deviation of a 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 observations are omitted.
Solution:  
Here n = 100, Incorrect mean (
−
x
) = 20 and Incorrect S.D. (σ) = 3
−
x
 =
1
n
Σxi ⇒ Σxi = n ×
−
x
 = 100 × 20 = 2000
∴ Incorrect Σxi = 2000
Now
1
n
Σxi2−(
−
x
)2 = 9
⇒
1
100
Σxi2−(20)2 = 9 ⇒ Σxi2 = 40900
∴ Incorrect Σxi2 = 40900
When wrong items 21, 21 and 18 are omitted from the data then we have 97 observations.
∴ Correct Σxi = Incorrect Σxi – 21 – 21 – 18 = 2000 – 21 – 21 – 18 = 1940
∴ Correct mean =
1940
97
 = 20
Also correct Σxi2 = Incorrect Σxi2 - (21)2−(21)2−(18)2 = 40900 - 441 - 441 - 324 = 39694
∴ Correct variance =
1
97
(correct‌Σ‌xi2)−(correct‌mean)2
=
1
97
×39694−(20)2 = 409.22 - 400 = 9.22
Correct S.D. = √9.22 = 3.036
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