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I have a categorical variable with three levels

Cu
Cu000
Cu000
Cu035
Cu175
Cu000 
Cu175
Cu035
.
.
.

The exact distribution of the three values (Cu000, Cu035, Cu175) is as shown below.

Cu000 Cu035 Cu175 
  251   275   263

I know that if the variable is binary then variance is p*(1-p), not sure how do I estimate the variance within a 3 level categorical variable? Any suggestion is apricated thanks.

Science11
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    What do you want the variance of? There are three categories here, any one of which is (obviously) binary. Thus, you must be looking for something other than the three separate variances, but what exactly would that be? – whuber Aug 20 '21 at 21:54
  • @whuber, I am sorry the question was not clear. I have updated my question now. Hope this is a bit clear. I am looking for variance within a categorical variable with 3 levels. – Science11 Aug 21 '21 at 00:05
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    The variance of what quantity? The phrase "within a categorical variable with 3 levels" gives conditions but not what to calculate. Are you looking for the variances of each proportion for example? – Glen_b Aug 21 '21 at 01:55
  • @Glen_b, "variance of a categorical variable with 3 levels", – Science11 Aug 21 '21 at 02:11
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    In the sense that the asker is asking, there isn't one. As I understand, the asker is looking for a single number to capture how much the categorical variable deviates from having all probability mass on a single outcome. (doesn't exist—maybe entropy or some qualitative variation measure will suffice, but I don't know the asker's purpose) But you CAN look at each individual outcome (there are 3) as a Bernoulli r.v. to compute its variance. – Arya McCarthy Aug 21 '21 at 03:29
  • @AryaMcCarthy, that is exactly what I was looking for, a single number like Variance, for a categorical variable with more than 2 levels. Thanks Arya. Please post this in the answer section and I will accept this. – Science11 Aug 21 '21 at 03:36
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    I second the idea by @AryaMcCarthy about entropy. – Dave Aug 21 '21 at 03:58
  • I worry that we have led the OP into a solution that prior to suggesting it was not implied by the words they were using. It would be unfortunate if that guess is not actually what is required. – Glen_b Aug 21 '21 at 05:18
  • I’m not posting this as an answer because it’s not an answer. We don’t know enough about your problem for my carelessly suggested idea to make sense. – Arya McCarthy Aug 21 '21 at 12:35

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