# How to estimate the correlation between a property of two populations given several algorithms

Editing question to be more specific:

I'm relatively new to statistics, and hoping I can get some help. The brilliance that is Stack Exchange has been immensely helpful in the past.

In my case, I have a bunch of data for the vocabulary of parents and their children. I want to test the correlation between the vocab size of children to the vocab size of their parents. There are different ways to calculate "vocab size," and I'd like to determine which method most strongly correlates child vocab to parent vocab. Here is a sample of my data:

Child Vocab Method 1    Parent Vocab Method 1   Child Vocab Method 2    Parent Vocab Method 2
0.160626116          0.118348938             0.693759899             0.742152507
0.120358694          0.118333458             0.694389147             0.7404324
0.115148036          0.103091141             0.636671881             0.750822137
0.098036288          0.123344877             0.692421132             0.757787103
0.182142187          0.116444253             0.684319515             0.723742968
0.165819982          0.155696351             0.699473063             0.757061947
0.108361775          0.164021164             0.649796093             0.767774552
0.122965306          0.163835682             0.672889844             0.79362656
0.128326546          0.148130042             0.666237275             0.758969757


Thanks so much

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I think you meant to compare the "vocab size of parents to the vocab size of their children"? Also, why are you correlating the sd of the size, rather than the plain size? – Robert Kubrick May 12 '12 at 14:47
Thanks @Robert, I made the edit. The reason I want to compare SDs is that my ultimate goal is to compare the effectiveness of various methods of calculating "vocab size." The typical way is "# of unique words/# – Adam_G May 12 '12 at 14:50
Thanks @Robert, I made the edit. The reason I want to compare SDs is because my ultimate goal is to compare the effectiveness of various methods for calculating "vocab size." Vocab size is generally "# of unique words"/"total # of words". I'm exploring different versions of this. In all of them, we come up with proportions. Rather than compare the proportions themselves, I want to compare, e.g., "If parent X's vocab is in the 90th percentile, is child X's vocab also in the 90th percentile?" Does that make more sense? – Adam_G May 12 '12 at 14:55
Well, you don't know the population standard deviation, do you? – Macro May 12 '12 at 15:04
ok, I see at least 2 different questions in your last comment. To use this website effectively (and get more answers), the first rule is to make your question as specific as possible, and ask them one at the time. Illustrating the question with an example of what you're doing also helps the contributors. – Robert Kubrick May 12 '12 at 15:05

You may be looking for something as simple as a correlation coefficient.

I would start my analysis by plotting the two different methods and looking for which one seems to show a relationship between child and parent vocab. Then I would also look at correlation coefficients; and models predicting the child vocab from the parent vocab. But see here for a recent q&a on the relative merits of correlation coefficients v. other options.

The nine points of data for each method you have posted do no look promising, however. There is little obvious relationship between the scores for parent and child on either method. Correlations are slightly negative (opposite of what you'd expect) but not significantly different from zero. Obviously, more data will help.

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 Thanks @peter. The 9 points are out of 210, so it's not surprising that they didn't yield much. – Adam_G May 13 '12 at 2:54

I think I understood the question to be what terminology to use for a standard deviation. Your example gives you a sample from which you can estimate a standard deviation for an assumed infinite or finite population. The population has a standard deviation called the population standard deviation which you do not know but can estimate from the data. Under certain assumptions you can ascribe statistical properties to the estimate. For example the sample variance with n-1 used instead of n is unbiased as long as the population has a finite variance and your samples are drawn independently from a infinite population. The difference with a finite population you can actually calculate the population variance (or population standard deviation) by enumerating all the values from the finite population. Then there are formulae for the population parameters (e.g. mean, variance and standard deviation) that will look similar to the sample estimates. The main difference is that the population number N is used in place of the sample size n. The estimates have slightly different properties than in the infinite population case. There is what is called the finite population correction that is often used to take the result for infinite populations to get the corresponding result for the finite population. Finite population arise in survey sampling and Cochran's classic book "Sampling Techniques" describes all this very clearly.

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 After the question was modified my answer above does not seem to apply. – Michael Chernick May 13 '12 at 14:10

Since it is now clear that you want to compare the correlation coefficients between different methods of determining vocabulary size I think you could pairwise test that two correlation coefficients are equal based on their sample correlations (assuming bivariate normality for parent and child volcabulary. If you are comparing several methods multiplicity should be taken into account.

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 Thanks @michael!. – Adam_G May 13 '12 at 20:45