Help interpreting standard deviation

I am looking at the 1979 cohort of the National Longitudinal Survey of Youth from the BLS. See here : https://www.bls.gov/nls/nlsy79.htm

I am just having some trouble interpreting the values for standard deviation.

Suppose I am looking at log of wages (natural log of wages).

Suppose my mean is 2.5, and standard deviation is 0.5.

How can I interpret my SD? Do I need to assume that the population distribution for log wages is normally distributed with zero mean and standard deviation 1?

I am just getting a bit confused here because standard deviation is a relational measure. We need to relate it to something, perhaps some standardized values, in order to conclude whether it is relatively high or low. Is it possible to do this without assuming anything about the population distribution? (such as zero mean or standard deviation 1)

Thanks.

• Is this link relevant to your question? nlsinfo.org/content/cohorts/nlsy79/… Jan 29 '21 at 1:43
• @SystemTheory Yes! that was great. thanks Feb 13 '21 at 15:29

First, there are substantial reasons to believe the data is not log-normal so it would be improper to assume log-normality. So, no you should not do that.

You can research the literature on youth wages, it may have information on distributions. However, that is probably more than is necessary.

There are two primary ways to estimate the standard deviation in economics. There are more than two ways.

Both of those ways understand the idea of an estimator as being "good" by finding an optimal procedure by integrating the product of a utility function and a probability distribution to create a risk function. The method with the "best" risk function for a given utility function "wins."

And you probably just thought it was a formula. There is more than one formula.

The two methods have a slightly different interpretation. If the method of maximum likelihood was used, then the mean and the standard deviation are the most likely values for the observed data. All other methods are methods of mediocre likelihood after all. If the Frequentist method was used, which is the most likely case, then it is an unbiased estimator, so that if you were to repeatedly make measurements, then the value it gives you will be correct, on average, though not necessarily even close for a specific case.

Since you do not know the distribution, you probably used the Frequentist method.

Now as to how to interpret it, you don't really. It is an estimator. Without more information, it just sits as is. Of course, you do have more information. You have the sample mean and you have Chebyshev's inequality. You know that 75% of the values must be within two standard deviations and 88.88% must be within three standard deviations. You can choose any arbitrary set of valuations that you like. You can look at 2.5 standard deviations, or you can reverse it and solve for a fixed percent.

That will give you a feel for how wide wages really are, in logarithms.

It is here that the two interpretations become important. If you decide to reverse the logarithmic transformation, the Frequentist estimator is no longer an unbiased estimator. That is because the logarithm is concave. You just cannot plug the values in and then reverse the process. Nonetheless, at this scale, unless you are actually doing scientific work, they are probably not that far off.

The Maximum Likelihood Estimator is invariant to transformation, so you can plug it into the exponent of the base of natural logarithms and get a correct estimate.