Defining a Symmetry Statistic to test for Normality?

[WWGD]

Hi,
Just curious. One of the requirements for a data set to be normally-distributed is that of symmetry.

Question: Is there or could one define a statistic to this effect, meaning to determine the level of symmetry on the data set and use that to determine at certain confidence level if the data set is symmetric or the underlying population that gave rise to the data is symmetric?

Of course this is necessary but not sufficient I am thinking as a start to find the difference between points above the sample mean minus thosebelow the sample mean and dividing by the number of data values.

I know there is a related Chi-squared that computes the number of values within 1,2,3 deviations from the mean and compares them with those expected according to the 68-95-99 rule, but just curious how one could just use the above test.

Thanks.

 

[andrewkirk]

The most used measure of asymmetry in a distribution is the coefficient of skewness. I haven't ever done tests for asymmetry but googling 'confidence intervals for skewness' comes up with a lot of links. I expect that amongst those are ways to estimate confidence intervals, thereby allowing a test of the null hypothesis that the population skewness is zero - ie that the population is distributed symmetrically about its mean.

If the hypothesis of interest is normality rather than just asymmetry there are standard tests for normality. Eight are listed here.

 

[WWGD]

The most used measure of asymmetry in a distribution is the coefficient of skewness. I haven't ever done tests for asymmetry but googling 'confidence intervals for skewness' comes up with a lot of links. I expect that amongst those are ways to estimate confidence intervals, thereby allowing a test of the null hypothesis that the population skewness is zero - ie that the population is distributed symmetrically about its mean.

If the hypothesis of interest is normality rather than just asymmetry there are standard tests for normality. Eight are listed here.

Thanks, but I was trying to use sample data. The skewness idea is helpful, but sample data is not likely to have 0 skewness. I guess I can look at the distribution of skewness for sample data for normal populations see if I can construct confidence intervals to test for symmetry of the population data..

 

[andrewkirk]

Thanks, but I was trying to use sample data. The skewness idea is helpful, but sample data is not likely to have 0 skewness. I guess I can look at the distribution of skewness for sample data for normal populations see if I can construct confidence intervals to test for symmetry of the population data..

What you need to do is use one of those procedures for constructing a confidence interval for sample skewness (google 'confidence intervals for skewness' to get a list of procedures), based on a null hypothesis that the population skewness is zero. Then if the constructed interval does not contain the sample skewness, the null hypothesis that the population distribution is symmetric can be rejected at whatever confidence level was used to construct the interval.

It's essentially the same as using a sample mean to test a hypothesis that a population mean is zero, except that it is applied to population skewness rather than population mean.

 

[WWGD]

What you need to do is use one of those procedures for constructing a confidence interval for sample skewness (google 'confidence intervals for skewness' to get a list of procedures), based on a null hypothesis that the population skewness is zero. Then if the constructed interval does not contain the sample skewness, the null hypothesis that the population distribution is symmetric can be rejected at whatever confidence level was used to construct the interval.

It's essentially the same as using a sample mean to test a hypothesis that a population mean is zero, except that it is applied to population skewness rather than population mean.

Yes, thanks, that word Skewness, I can't ever remember it. I wish I had a pet , would name it Skewness, so I can remember it better. I may adopt a dog...