# Measure spread of non normal distribution?

I have sentiment data for customer reviews drawn from a larger population of reviews. Each product has a number of customer reviews. Each review has a sentiment (opinion/feeling) between 0 and 1 where 0 is very negative and 1 is very positive.

Customers seldom write reviews when one is indifferent. They are either positive or negative. So the distribution of sentiments is not normal but more bimodal with some skew to the middle, so there are lots of negative reviews and lots of positive reviews but not much in the middle.

How can I measure the spread of data and compare them between different products? For example, a product can have a mean sentiment of 0.8 when it has most reviews around that value. Another product can also have a mean of 0.8 but have wildly positive reviews and some very negative. The latter product would have a larger spread of the sentiment. The products with the largest spread of sentiments is likely marketed wrongly, so it would be important to identify them: people may buy them and think they will do something it doesn't.

I assume standard deviation is out of the picture since the distribution isn't normal or t-distributed. Are there other measures of spread for this kind of bimodal distribution?

If you look at introductions to the beta distribution (e.g. this Wikipedia article) you will see that standard deviations (SDs) can certainly be defined for distributions that are bounded (e.g. confined to the interval $$[0, 1]$$) and may be far from normal in shape (e.g. bimodal and in your case possibly U-shaped). The standard deviation is also well defined for many distributions that can be far from symmetric such as the Poisson or the lognormal. So, any idea that the SD is defined and useful only for near-normal distributions is unfounded.

What is certain is that you can't carry over ideas which work only for normal distributions (e.g. that the SD is the distance between the mean and each inflection on the density function) and rules of thumb from normal distributions about the fraction of observations within $$\pm 1, \pm 2, \dots$$ SD of the mean may break down.

Your data may well be messier than the beta distribution and even show small gaps, bumps and spikes. But I see no barrier to your using the SD as a descriptive measure, nor indeed to also using the interquartile range or IQR. Just keep plotting your data so that you get a sense of how any measure works with your data and any circumstances where it is misleading.

You already have a strong sense that a given mean can correspond to different distributions. That is also going to be true for SDs.

• Thanks a lot for this! It's going to take a while to digest that wikipedia article about beta distribution but it looks interesting. Good point on using the SD but being careful with conclusions that is only relevant for ND. I'll graph some IQD and see how it compares to the SD. Very grateful. Oct 18, 2018 at 9:32
• I can't promise that beta distributions will be useful for your analysis. I just cited them as respectable citizens in the distribution community showing up that SDs make sense for bounded measured variables too. Oct 18, 2018 at 11:51

You properly say that the distribution is bimodal. A perhaps slightly more articulated analysis would be to try to measure the relative contribution of the two subsamples (promoters and detractors) and corresponding parameters. They will likely be two steeply falling distributions.

The size of the two subsamples, their means and the slope of the two curves might tell you more about how polarising is the product - and maybe help detect fake reviews.

• You are right - text edited. Not sure what would be the best distribution, trying to code a solution with some distribution I am familiar with. Oct 18, 2018 at 21:58
• Interesting. How would you measure the relative contribution of the two subsamples? It sounds like a good idea but I can't figure out how to quantify it so I can compare different products or product categories. Oct 19, 2018 at 9:35