# Theorem name? Skewness of repeated samples = skewness of the population

Is there a theorem (etc.) that states the average skewness of repeated samples will approach the skewness of the population? If so, I need the name.

I confirmed my suspicion that this is true. I created a dataset (n=30,000) by joining normal distributions, such that is looks "normal-like," with a skew of about -1. 100 repeated samplings of 500 has an average skewness of -1. Also, the means of the samples are a normal distribution - of course (central limit theorem).

• Do you really need such a theorem? Assume you have a skewed sample from a skewed parent population. However, a power transforms converts the sample data to approximately normal and the parent also becomes nearly normal. If you have convergence to normality, then the untransformed data converges to your skewed parent population. – AJKOER Jul 2 at 22:49
• are you after en.wikipedia.org/wiki/Law_of_large_numbers – seanv507 Jul 2 at 22:55
• @AJKOER There's nothing correct about any of those remarks: there will not always be such a power transformation (it isn't even defined when any data are negative) and "convergence to normality" doesn't occur either for the skewness or the data distribution. – whuber Jul 3 at 14:24

## 1 Answer

It sounds like you are interested in whether the sample skewness converges to the true skewness as you take $$n \rightarrow \infty$$ (i.e., whether the sample skewness is a consistent estimator of the skewness parameter). The theorems you would be looking for are theorems that demonstrate consistency of the estimator.

• Thanks for pointing me in the direction of consistent estimators. Now I know the term I need to look for. I haven't found the answer yet. I ran a simulation using the skewness function in the moments package (R), taking random samples. With 200 samples of size 100 (repeated), the average skewness is consistently about 5% low. When I increase the sample size to 200, it drops to being consistently about 3% low. The dataset I'm sampling has only 1300 rows. It appears this estimator is biased, and the bias decreases with sample size. The dataset is near-normal, but right-skewed (.8). – user179810 Jul 4 at 2:11
• The Law of Large Numbers (LLN) applies to the averages of the skewness statistics: Sample n observations, repeat N times, then take the average of those N skewnesses (each comprised of n observations). The LLN states that as N --> infinity (n fixed), the average will approach the expected value of the skewness statistic based on the n observations, differing from the "population" skewness (bias). Finite moments are needed, which is true if your "population" is an existing data set; in this case the "population" kurtosis involves only "n" divisors; no "n-1" or other types of correction factors. – BigBendRegion Jul 13 at 20:00