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Question
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
`overline x = (sumx_i)/n`
∴ `sum x_i = n overline x`
= 100 × 20
= 2000
New `sumx_i = 2000 - 21 - 21 - 18`
= 1940
New (corrected) mean = `1940/97`
= 20
σ = `1/n sqrt(n sum x_i^2 - (sumx_i)^2)`
∴ `sum x_i^2 = (n^2 σ^2 + (sumx_i)^2)/n`
= `((100)^2 xx 9 + (2000)^2)/100`
= 900 + 20 × 2000
= 900 + 40000
= 40900
Exact value of `sumx_i^2` = 40900 − (21)2 − (21)2 − (18)2
= 40900 − 441 − 441 − 324
= 39694
New (corrected) standard deviation = `1/97 sqrt(97 xx 39634 - (1940)^2)`
= `1/97 sqrt(3850318 - 3763600)`
= `1/97 xx sqrt86718`
= 3.036
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