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$k' = 1 / \sum{p^2}$ is the "effective" number of buckets over which the distribution is uniform. $1 \le k' \le k =$ the # of buckets.

###Edit###

1. Thanks for adding the markup. Also, I'm new to this site, and I don't know how to reply to a comment, as opposed to replying to a post.

2. I developed k' myself, sometime in the 1970's, in response to a student's request for a way of measuring the variability across categories of admission diagnoses to a mental hospital, the question being whether there was more variability at certain times than at others. Recently, I discovered that it had previously been proposed by E.H. Simpson in a short article: Measurement of Diversity, Nature, 163 (1949), 688.

k' is not restricted to uniform distributions. Calling it the effective number of categories over which the distribution is uniformly distributed is just a way of establishing the scale it's on. It's analogous to the df in repeated measures anova after correcting for non-sphericity, with the normalized eigenvalues of the covariance matrix replacing the p's.

$k' = 1 / \sum{p^2}$ is the "effective" number of buckets over which the distribution is uniform. $1 \le k' \le k =$ the # of buckets.

Edit

1. Thanks for adding the markup. Also, I'm new to this site, and I don't know how to reply to a comment, as opposed to replying to a post.

2. I developed k' myself, sometime in the 1970's, in response to a student's request for a way of measuring the variability across categories of admission diagnoses to a mental hospital, the question being whether there was more variability at certain times than at others. Recently, I discovered that it had previously been proposed by E.H. Simpson in a short article: Measurement of Diversity, Nature, 163 (1949), 688.

k' is not restricted to uniform distributions. Calling it the effective number of categories over which the distribution is uniformly distributed is just a way of establishing the scale it's on. It's analogous to the df in repeated measures anova after correcting for non-sphericity, with the normalized eigenvalues of the covariance matrix replacing the p's.

$k' = 1 / \sum{p^2}$ is the "effective" number of buckets over which the distribution is uniform. $1 \le k' \le k =$ the # of buckets.

$k' = 1 / \sum{p^2}$ is the "effective" number of buckets over which the distribution is uniform. $1 \le k' \le k =$ the # of buckets.

###Edit###

1. Thanks for adding the markup. Also, I'm new to this site, and I don't know how to reply to a comment, as opposed to replying to a post.

2. I developed k' myself, sometime in the 1970's, in response to a student's request for a way of measuring the variability across categories of admission diagnoses to a mental hospital, the question being whether there was more variability at certain times than at others. Recently, I discovered that it had previously been proposed by E.H. Simpson in a short article: Measurement of Diversity, Nature, 163 (1949), 688.

k' is not restricted to uniform distributions. Calling it the effective number of categories over which the distribution is uniformly distributed is just a way of establishing the scale it's on. It's analogous to the df in repeated measures anova after correcting for non-sphericity, with the normalized eigenvalues of the covariance matrix replacing the p's.

k' = 1 / sum p^2 is the "effective" number of buckets over which the distribution is uniform. 1 <= k' <= k = the # of buckets. (Apologies, but I don't know the markup language.)

$k' = 1 / \sum{p^2}$ is the "effective" number of buckets over which the distribution is uniform. $1 \le k' \le k =$ the # of buckets.