error bar

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)