Question

# Which of the following can be an alternate representation of variance, where xi being the midpoint of class intervals, fi being frequency of class interval and ¯x the mean,N = ∑ni=1fi

A

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B

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C

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D

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Solution

## The correct options are A B Variance σ2 = 1N(∑ni=1fi(xi−¯x)2 = 1N∑ni=1fi(x2i+¯x2−2¯xxi) =1N[∑ni=1fix2i+¯x2∑ni=1fi−2¯x∑ni=1fixifi] =1N[∑ni=1fixi+¯x2N−2¯xN¯x] Because, 1N∑ni=1xifi=¯x = 1N∑ni=1fix2i+¯x2−2¯x2 = 1N∑ni=1fix2i−¯x2 = 1N∑ni=1fix2i - (∑ni=1fixi)2N)2 = 1N2[N∑ni=1fix2i−(∑ni=1fixi)2] ∴ option a and b are correct.

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