There's a large literature for that, e.g.,
E. L. Lehmann, Testing Statistical Hypotheses.
E. L. Lehmann, Nonparametrics: Statistical Methods Based on Ranks.
Sidney Siegel, Nonparametric Statistics for the Behavioral Sciences.
In this case, PG will be more interested in the non-parametric case, i.e., distribution-free where we make no assumptions about probability distributions.
We start an hypothesis test with an hypothesis, commonly called the null hypothesis which is an assumption that there is no effect or, in PG's case, no bias. Then with that assumption, we are able to do some probability calculations.
Then we look at the real data and calculate the probability of, say, the evidence of bias being as large as we observed. If that probability is small, say, less than 1%, then we reject the null hypothesis, that is, reject the assumption of no bias, and conclude that the null hypothesis is false and that there is bias. The role of the assumption about the sample is so that we know that the problem is bias and not something about the sample.
In hypothesis testing, about all that matters are just two numbers -- the probability of Type I error and that of Type II error. We want both probabilities to be as low as possible.
Type I Error: We reject the null hypothesis when it is true, e.g., we conclude bias when there is none.
Type II Error: We fail to reject (i.e., we accept) the null hypothesis when it is false.
When looking for bias, Type I error can be called a false alarm of bias, and Type II error can be called a missed detection of bias.
In PGs case, suppose we have 100 startups and five of those have women founders. Suppose for each of the startups we have the data from "their subsequent performance is measured".
Our null hypothesis is that the expected performance of the women is the same as that of the men.
So, let's find those two averages and take the difference, say, the average of the women less the average of the men.
PG says if this difference is positive, then there was bias, but PG has not given us any estimate of the probability of Type I error, that is, of the probability (or rate) of a false alarm.
I mean we don't want to get First Round Capital in trouble with Betty Friedan, Gloria Steinem, Marissa Mayer, Sheryl Sandberg, Hillary Clinton, Ivanka Trump, or Lady Gaga unjustly! :-).
Let's call this difference our test statistic.
So, let's find the probability of a false alarm:
So, let's put all 100 measurements in a pot, stir the pot vigorously (we can use a computer for this), pull out five numbers and average, pull out the other 95 numbers and average, take the difference in the two averages, that of the five less that of the 95, and do this, say, 1000 times. Ah, computers are cheap; let's be generous and do this 10,000 times.
For a random number, how about starting with a 32 bit integer, with appropriately long precision arithmetic multiply by 5^15, add 1, take modulo 2^47, and scale as we want?
So, we get an empirical distribution of these differences, from the five less the 95. Looking at the distribution, we see what the probability is of getting a difference as high or high or higher than our test statistic. If that probability is low, say, 1% or less, then we reject the null hypothesis of no bias and conclude bias with our estimate of probability of Type I error 1% or less.
If with the 1% we reject, then it looks like First Round has done a transgression, will get retribution from Betty, et al., and needs to seek redemption and Betty, et al., are happy to have their suspicions confirmed. Else First Round looks like the good guys, are "certified statistically fair to women", may get more deal flow from women, and Betty, et al., can be happy that First Round is so nice!
Notice that either way Betty, et al., are "happy". That's called "happy women, happy life"! Or, heads, the women win, tails they lose, and in no event is there a huge crowd of angry women in front of First Round's offices with a bonfire of lingerie screaming "bias"!
When we reject the null hypothesis, we want to know that the reason was men versus women and not something else, e.g., a biased sample. So here is where we use our assumption of independence with the same mean.
Now we have a handle on Type I error.
Here we have done a non-parametric statistical hypothesis test, i.e., have made no assumptions, except the means, about the distributions of the male/female CEO performance measurements.
And we can select our desired false alarm rate in advance and get that rate almost exactly.
For Type II error, that is more difficult.
Bottom line, what we really want is, for whatever rate of false alarms we are willing to tolerate, the lowest rate of missed detections we can get.
Can we do that? With enough more data, yup. There is a classic result due to J. Neyman (long at Berkeley) and K. Pearson (early in statistics) that shows how.
How? Regard false alarm rate as money and think of investing in SF real estate. We put our money done on the opportunities with highest expected ROI until we have spent all our money. Done. For details, an unusually general proof can follow from the Hahn decomposition from the Radon-Nikodym theorem in measure theory, e.g., Rudin, Real and Complex Analysis. Right, in the discrete case, we have a knapsack problem, known to be in NP-complete.
What we have done with our pot stirring is called resampling, and for more such look for B. Efron, long at Yale, and P. Diaconis, once at Harvard, now long at Stanford.
Tom, with a reputation as a hacker, likes to work late, say, till 2 AM. So, we look at the intrusion alerts each minute between 2 AM and 3 AM (something like the performance of the women) and compare with those of the other minutes of 24 hours (like the performance of the men) much as above and ask if Tom is trying to hack the servers.
Or, we have a server farm and/or a network, and we want to detect problems never seen before, e.g., zero day problems. So, we have no data at all on the problems we are trying to detect because we have never seen any of those before.
So, to do a good job, let's pick some system we want to monitor and for that system, get data on, say, each of 10 variables at, say, 20 times a second. Now what?
Our work with bias in women venture applications used just one number for our measurement and test statistic. So we were uni-dimensional. Here we have 10 numbers and need to be multi-dimensional.
Well, in principle we should be able to do much better (pair of Type I and Type II error rates) with 10 numbers than just one. The usual ways will require us to have, with our null hypothesis, the probability distribution of the 10 numbers, but can only get something like that from smoking funny stuff -- not even big data is that big.
So, we want to need no assumptions about distribution, that is, be distribution-free.
So, we want some statistical a hypothesis test that is both multi-dimensional and distribution free.
Can we do that? Yup.
"You mean you can select false alarm rate in advance and get that rate essentially exactly, as in PG's bias example?" Yup.
"Could that be used in a real server farm or network to detect zero day problems -- security, performance, hard/software failures, system management errors?" Yup -- just what it was invented for.
"Attempted credit card fraud?" Ah, once a guy in an audience thought so!
How? Ah, sadly there is no more room in this post!
What else might we do with hypothesis tests? Well, look around at, right, big data or just small data.
Do we have a case of big data analytics or artificial intelligence (AI)?
Ah, I've given a sweetheart outline of statistical hypothesis testing, and now you are suggesting some things really low grade? Where did I go wrong to deserve such an insult?