# Categorical variable standard deviation

Suppose in a situation where there is one questionnaire question which allows N participant to give response from a scale of 1 to 10.

To find the standard deviation from this case we can first calculate the mean from all N participant. Then for each i participant's observation we can find how far this observation is to the mean, taking the square and we have the variance.

If we sums up all participant's variance, divided by N-1 participant, taking the square root, we get the standard deviation in this case. This is how we understand how spread the answer is.

Now what if my question now sets to either good or bad. In a sense it is an indicator variable now where +1 is good, -1 is bad.

How do you find out the "Standard Deviation" or to say to find out how far each observation/participant's answers is to the average? Is there any metric measure other than standard deviation which does not exists in categorical data to find out how spread all participant opinions are?

Thanks.

• In the case of a binary random variable, the distribution is completely described by the mean ... It is also very easy to just tabulate the distribution. – Andreas Dzemski Feb 27 '18 at 13:12
• Hi, what do you mean by tabulate the distribution? You mean to bootstrap to get the distribution and find a distribution that close to the bootstrap one and find the mean and stdev? – Gabriel Feb 27 '18 at 14:17
• Tabulate = say how many have -1 and 1 in sample. Maybe you can edit your question to say what you want to accomplish. – Andreas Dzemski Feb 27 '18 at 15:22