error bar represents the range that the population mean would exist.

(the population mean usually denoted by the Greek word μ).

this is the same as a confidence interval.

Xbar ～ N(μ, σ^2/n)

=> μ = Xbar ± z・σ/root(n)

where z = (Xbar - μ)/(σ/root(n)).

considering

T = X1+X2+・・・+Xn

Xk = 0 if false

or 1 if true

(k=1,2,・・・,n),

T depends on a binomial distribution.

moreover, if n approaches infinity, T depends on a normal distribution.

T ～ B(n,p) ～ N(np, np(1-p)) ( as n is infinity)

where p is the population rate.

=> np = T ± z・root(np(1-p))

where z = (T - np)/(root(np(1-p))).

therefore

p = T/n ± z・root(p(1-p)/n)

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