# How to convert mean and standard deviation to a single meaningful and quantifiable value?

I look at mean and standard deviation of various map like data and each give meaningful information but sometimes they provide different information.

For example, let's take states in a country. One state has higher mean than others but the standard deviation by itself looks OK since individual points from the whole state tend to be higher. In another case, the mean looks average but the state has both higher and lower values equally distributed so the standard deviation is higher.

Now how can you quantify both mean and standard deviation to a single meaningful value? Let say, the map color looks hot both when mean of each state is deviating from the mean of the country as well as standard deviation of the state is high.

## 1 Answer

It depends on the distribution / type of data you are looking at. If the data are normally distributed, the mean and SD are (typically / can be) independent of each other, in which case they cannot be meaningfully combined. If the data are of a different type, for example, binary data that follow the binomial distribution, the variance (and thus SD) are a function of the mean. That implies, once you have the mean, you don't need to indicate the SD; it provides no new information.

I'm guessing your data are at least normal-ish, at least in the sense that the SD is not a function of the mean. (You could check this with, say, a scatterplot.) Even when data are normal, the variability is often related to the mean, for example, there are many cases where the variance grows with the mean such that there is a constant coefficient of variation. If your data do have a SD that varies systematically with the mean, you could use a variance stabilizing transformation, such as taking the log or square root of your data.

If the SDs vary independently of the means or you want to keep the data in its raw form, you will need to present both sets of information. This could be done by presenting two maps side by side, or by representing the mean with one attribute (say, color) and the SD with another attribute (say, texture; e.g., lines that rotate from horizontal with low SDs to vertical with high SDs).

• I understand your reference to on normally distributed data, the mean and SD are independent of each other. I was thinking something along this line... Take abs(mean of individual state - mean of country) and then somehow normalize that to standard deviation and then show a combined number which has the max between the two. In effect, I want to see which state has some form of abnormality. – Joel Apr 16 '14 at 17:52
• You can certainly turn state means into a $z$-score (or something loosely analogous, in your suggestion), but this does not address the additional information in each state's SD (assuming there is). If there is independent info in the state SDs, that will need to be represented somehow in addition. – gung - Reinstate Monica Apr 16 '14 at 18:05