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:

Number of observations = 100

Incorrect mean (x) = 20

Incorrect standard deviation (σ) = 3

20 = 1/100 ∑¹⁰⁰_{i = 1}x_i

∑²⁰_{i = 1}x_i= 20 × 100

= 2000

That is, incorrect sum of observations

= 2000

Correct sum of observations

= 2000 - 21 - 21 - 18

= 2000 - 60

= 1940

Therefore,

correct mean = correct sum/(100 - 3) = 1940/97 = 20

Standard deviation,

σ = √ 1 / N ∑N_i = 1 (X_i² − μ²)

3 = √ 1 / 100 ∑N_i = 1 (X_i² − μ²)

= Incorrect ∑ⁿ_{i = 1} x_i² = 100 (9 + 400)

= 40900

Correct ∑ⁿ_{i = 1} x_i² = 4

= Incorrect ∑ⁿ_{i = 1} x_i² - (21)² - (21)² - (18)²

= 40900 - 441 - 441 - 324

= 39694

Correct standard deviation

= √39694/97 - (20)²

= √409.216 - 400

= √9.216

= 3.036

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NCERT Solutions Class 11 Maths Chapter 15 Exercise ME Question 7

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.

Summary:

Given that the mean and standard deviation of a group of 100 observations were found to be 20 and 3. Correct mean and correct standard deviation are 20 and 3.036