If I have a sample of 100, who give responses to a set of multiple response variables, say for example, the brands they have purchased in the last 12 months.

How can I calculate the confidence intervals around the percentage for the share of all purchases for a particular brand?

I can use standard confidence interval formula for CI around "%" but can the same be said for CI around "Share %" as displayed below?

Brand   |  N   |   %   | Share %

Brand1  |  20  |  20%  |  9%

Brand2  |  40  |  40%  | 19%

Brand3  |  45  |  45%  | 21%

Brand4  |  35  |  35%  | 16%

Brand5  |  75  |  70%  | 35%
Total   | 215  |       | 100%

1 Answer 1


I am not 100% confident but I would personally use the sum of gaussian and ratio of gaussian. Let S1 the variable which represents Share1. S1=X1/(X1+X2+X3+X4+X5) where Xi are independent gaussian distributions which represents each Brand. X1+X2+X3+X4+X5 behaves like a gaussian distribution with (using probabilities, not %) :

  • A standard error of sqrt(s1²+s2²+s3²+s4²+s5²), where si=sqrt(pi*(1-pi)/N). For example : s1=sqrt(0.6*(0.4)/100)=0.049
  • A mean of p1+p2+p3+p4+p5=2.15

Once it's computed, the last step is to compute the confidence interval of a ratio of normal distributions. This question seems to have been properly answered there How to compute the confidence interval of the ratio of two normal means an online calculator has been put in place by the first respondent using Fieller's method.


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