Trying to calculate industry averages for female employees I have a dataset that lists employers in a particular industry, the number of employees they have and the breakdown of employees as male or female.  I'm trying to get a number that reveals something about the industry's percentage of female employees - and that we can use as a comparison to gauge the diversity of specific employers or specific subsets of employers in the dataset.  
What's the best way to do this? Do I total all the female employees in this dataset and divide by the total employees in the dataset to get a percentage? (for example = 33% of workers in this dataset/industry are female) Or do I take an average of the percentages of female employees from all the companies across the dataset? (So, for example, if company A has 22% female employees, company B has 36% female employees, and company C has 40 % female employees - the average of the percentages is 33%)  
Ideally, I want to get a number that creates an illuminating comparison. So if company A has 22% female employees - I want to be able to show how they measure up. 
 A: You might want to read through this, even though it is long.  There could be some decent stuff at the end.  
Short question:
How do I compare central tendency and tendency of variation for employers of female employees.
Basic math questions behind the question:


*

*What is "typical"?  

*How do I measure distance away from "typicalness".  Is 9 units really different than 9.1 units?


Business questions behind the question:    


*

*how does diversity fit into making a company highly valuable and/or amazingly productive?  


Basic math thoughts:
They say "typical" but you have to think about what that means.  If 19 firms have 1/100 female and 1 has 100/100 female, then is the "typical" really 101/20 = 5 female employees per firm?  A median is going to say tat the "typical" is 1.  I might look at the mean and the median and see how different they are, but this begs the next question.
Standard deviation helps us make sense of "how far is far" when we have numbers.  Three standard deviations from the mean, while strange language, gives a decent defense for "atypical" with some basic assumptions.  Revisiting the previous example (with variation) - if 19 firms have 1/100 female or 2/100 and 1 has 100/100 female, then what is the "typical", and how "atypical" is the 100/100? I like to use software at this point, because it is a lot easier, and can be harder to get wrong.
code:
set.seed(207125)
n <- 19 #random draws

#make all samples
nf <- c(sample(c(2,1),19,replace=T),100)

#compute stats
mean(nf)
sd(nf)

yields:
> mean(nf)
[1] 6.5

> sd(nf)
[1] 22.01315

I don't think that 6.5 makes for a reasonable expectation per firm.  Similarly I don't think natural variation among firms is 22.0.
In this case I try the "pseudo-sigma", a robust inter-quartile approximation to the standard deviation, one that isn't as easily pushed around by outliers.
#more robust stats
median(nf)
IQR(nf)/1.35

results:
> median(nf)
[1] 2
> IQR(nf)/1.35
[1] 0.7407407

So we want to find out "far" the outlier is away from the "typical" we subtract the typical and divide by the variation:
far <- (100-median(nf))/(IQR(nf)/1.35)
far

> far
[1] 132.3

If you don't know where to start these are some very very basics that can get you going.
Basic business thoughts:
Air is important for life, but just because you have air doesn't mean life is there too.  There is more to the organic than the oxygen.  Diversity has meaning in an ecosystem, and without that context, its power as an indicator is lost.
You have to take into account the "physics" of the system you are analyzing.  You could try and understand why the challenger disaster happened by looking at fuel and never get an answer.  The answer you are trying to find has to be in the data, and the approach has to be able to pick it out.
You need to know more than count here, to look at the deeper things. 
"Measure up" is a baggage rich term.  If you stack 100 teams randomly all male or all female, then measure their business performance in terms that really live at the heart and soul of a business - things like productivity, value creation, great place to work - the gender composition of the team is going to be stunningly less informative than the character of the individual people in the mix.  You can assemble all-star teams that are all men or all women and you can assemble homogeneous train-wreck teams in an identical  manner.  
Effective diversity is not just about a check-box.  It isn't merely the count is on a particular line.  It is about the culture.  It is about the process of integration.  Like every career, it is about the trajectory over time.  A personal career trajectory gets its meaning from the opportunities and choices made within the context of the economic climate in our lives.  Similarly the trajectory of culture within an organization has a context.  The most powerful and effective changes are going to make in consideration of that context, not necessarily in disregard of them.
If a company has 50% females, but they are paid half as much for the same work as males, and none of them are above the midpoint in the chain of command, is that "healthy diversity"?  No.  Not even a little.  
The math so far starts with the data and moves to the question.  You should either start with the question and let it drive the data, or let the data tell you what it can and not say what it cant.
A great fit at work is like dating, with HR being the matchmaker for the whole team.  Read this (link).  Now, ask what make a great work climate that delivers great things for the company?  We know that sexism reduces the candidate talent pool by half, and introduces a stack of unintended biases and unintended consequences into the decision making process.  There are costs for institutionalized bigotry.  That is what to avoid, but not what to major on.  Lack of poison isn't the same thing as abundance of life.  If all we did was fill check-boxes with tokens, then likely we missed out on the things that make "amazingness".  Likely all we did was "engineer bureaucracy" - which is profoundly dangerous for the big, messy, humany things that are critical for success. 
Think about what makes for amazing health, and how diversity fits within that ecosystem, then you can evaluate not only the tallies of numbers, but the organizational health and future of the business, and you can be much more correct, and much more valuable for those sorts of answers.
A decent starting place, it is klugy but beautiful, is stock evaluation.  The information indicating value is in there.  (link)  It isn't necessarily all in there, but it likely is more than the gender tallies.  That kind of an extended context, especially viewed over longer time, might be nicely informative.
What if your question was of the form:
How does company value over time relate to health of diversity over time?
One could put some of this into a science fair as math, if they did it well, and have a very strong entry.  Social engineering isn't theory, it started in economics with Adam Smith and has been grown since.  
