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Question
Let a, b, c, d, e be the observations with mean m and standard deviation s. The standard deviation of the observations a + k, b + k, c + k, d + k, e + k is ______.
Options
s
ks
s + k
`s/k`
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Solution
Let a, b, c, d, e be the observations with mean m and standard deviation s. The standard deviation of the observations a + k, b + k, c + k, d + k, e + k is s.
Explanation:
Given observation are a, b, c, d and e
∴ Mean = m = `(a + b + c + d + e)/5`
∴ `sumx_i` = 5m
Now mean of a + k, b + k, c + k, d + k and e + k is
= `(a + k + b + k + c + k + d + k + e + k)/5`
= `((a + b + c + d + e) + 5k)/5`
= `(5m + 5k)/5`
= m + k
∴ S.D. = `sqrt((sum(x_i + k)^2)/n - [(sumx_i + k)/n]^2`
= `sqrt((sum(x_i^2 + k^2 + 2x_ik))/n - (m + k)^2`
= `sqrt((sumx_i^2)/n + (sumk^2)/n + (2ksumx_i)/n - m^2 - k^2 - 2mk)`
= `sqrt((sumx_i^2)/n + k^2 + 2km - m^2 - k^2 - 2mk)`
= `sqrt((sumx_i^2)/n - m^2)` .....`[because (sumx_i)/n = m]`
= s
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