# Does this plot tell anything about gender disparities---and further, discrimination---in academia?

I stumbled upon this plot on Twitter, and many users thought they saw a shocking pattern in it---which is emphasized by coloring it with red:


However, this plot does not say what people think---or want---to say
In contrast, it is quite useless.
Here's why.

## Setting the scene for proportions

Assume that in a population of $N$ individuals at at a given time $T = t$ (say, $T=2021$ for last year), every individual with a gender $G = m$ (for man) or $G = w$ (for woman) has one of the following college degrees (denoted by $D$):

1. Science, technology, engineering, and medicine (STEM): $D=s$;
2. Humanities, arts, and literature: $D=h$; or
3. No degree at all and considered uneducated: $D = u$.

In an ideal scenario, at any given year $t$, we know the gender and education level of all individuals in the population:

| Individual | Gender | Education |
| ---------- | ------ | --------- |
| 1 | w | s |
| 2 | w | s |
| 3 | m | u |
| 4 | w | h |
| 6 | m | s |
| 7 | w | u |
| $\vdots$ | $\vdots$ | $\vdots$ |
| N | m | h |

This table may be summarized in different ways, which are mostly expressed in terms of proportions: proportion of women in the population, proportion of men without a college degree, etc.
To work with these *proportions*, which are modeled by probabilities, we need to use the Bayes' theorem.

## The Bayes's Theorem

The Bayes' theorem (or the *Bayes' rule*) rule states that the conditional probability of $A$ given $B$---$P(A|B)$, that is, the probability of $A$ if we know the value of $B$---can be calculated from the joint probability $P(A,B)$---that is, the probability of $A$ and $B$ at the same time---and the marginal probability of $P(B)$---that is, the probability of $B$ regardless of the values $A$ might take---via

$$P(A|B) = \frac{P(A, B)}{P(B)} = \frac{P(B|A) \times P(A)}{P(B)},$$

or equivalently,

$$P(A|B) \times P(B) = P(B|A) \times P(A).$$

For notation simplicity, we denote the probabilities as $P_{A|B}$, $P_{A,B}$, $P_{A}$, etc.

Having that in mind, our gender groups may be different in various ways, which may be expressed in terms of proportions:

Based on the way we summarize the data, we may calculate the following probabilities, which are basically the proportion of people having certain values for each variable:

#### 1. Demographics: What are the proportion of men to women at year $t$?

## Back to proportions

We are mostly interested in the following proportions (or probabilities):

### 1. Demographics

Let us denote the proportion of people with gender $G=g$ at year $T=t$ with $P^{T=t}_{G=g}$, or more simply, $P^t_{g}$.

In general, the number of men and women are not the same, i.e., $P^t_{m} \neq P^t_{w}$.
However, the proportions add up to one, thus $P^t_{m} = 1 - P^t_{w}$.

### 2. Education levels of the whole population

We may summarize the table by expressing the proportion of individuals that (do not) have certain degrees at a given year.
The proportions can be expressed in terms of probabilities with $P^t_s$, $P^t_h$, $P^t_u$.
Again, the proportions add up to one:

$$P^t_s + P^t_h + P^t_u = 1$$

### 3. Joint probabilities: If you pick a random individual out of the population, what is the probability that it's, say, a man who has no college degree?

This probability is quite hard to intuitively make sense of, specially because one cannot easily make a sensible comparison between different joint probabilities ($P^{t}_{g,d} = P^{T=t}_{G=g, D=d}$).
This stems from the fact that multiple joint probabilities add up to one:

$$P^{t}_{m,s} + P^{t}_{m,s} + P^{t}_{m,u} + P^{t}_{w,s} + P^{t}_{w,h} + P^{t}_{w,u} = 1.$$
Thus, unsurprisingly, said probability ($P^{t}_{m,u}$) cannot be calculated from another probability, say, $P^{t}_{w,u} = 0.1$.

### 4. Among those with a certain education level, what proportion of them are, say, women?

In many contexts, we are interested in knowing what percentage of people with a certain degree have a certain gender ($G=w$).

This proportion can be expressed in terms of conditional probabilities ($P^{t}_{g|d} = P^{T=t}_{G=g|D=d}$) and it frequently surfaces in the public discourse---perhaps most importantly, the participation of women in STEM ($P^{t}_{w|s}$) is of interest.

Making use of the Bayes' rule, we have

$$\begin{aligned}
P^{t}_{g|d} &= \frac{P^{t}_{g,d}}{P^{t}_{d}} \\
&= \frac{P^{t}_{d|g}\times P^{t}_{g}}{P^{t}_{d}}
\end{aligned}$$

### 5. How do the educational "preferences" differ among men and women?

This proportion is of great scientific importance and many people mistake it for other proportions mentioned above.
Expressing it in conditional probabilities, if you know someone's gender, what is the probability that they have attained a certain college degree:
$P^{t}_{d|g} = P^{T=t}_{D=d|G=g}$.

You here about this when you it is mentioned that, say, most girls pursue a degree in humanities and arts (say, $P^{t}_{h|w} = 0.6$).

**Note that** I put the word "preference" in quotation marks, as it does not necessarily reflect one's *true* psychological preference in pursuing a degree; in some cases (e.g., among the unprivileged groups) it is likely that environmental or societal factors prevent them from discovering or pursuing their preferred paths in life.

Applying the Bayes' rule yields

$$\begin{aligned}
P^{t}_{d|g} &= \frac{P^{t}_{d,g}}{P^{t}_{g}} \\
&= \frac{P^{t}_{g|d}\times P^{t}_{d}}{P^{t}_{g}}
\end{aligned}$$

## The problems with how the proportions are conventionally used and perceived in communications

Regardless of the conclusions we may draw from the proportions (discussed below), there are vital matters that are left out when comparing (the temporal trends of) the proportions.
They have great implications for policy making---or at a less elite level, political orientation of citizens and "activists"---that result in misguided (mental) efforts.

### We are bad at math (and Thomas Bayes hates us)

The proportions are often mistaken: Almost always it is $P^{t}_{g|d}$ that catches our eyes such that we (try to) equate it with other proportions; and related to this,

### We miss the bigger picture

Quite often, the temporal change in a proportions are studies in isolation:

If our data has a higher resolution that distinguishes between more than one educational degree---say, $D$ can take values $p$, $e$, $m$, and $l$(for physics, engineering, medicine, and literature, respectively)---the changes in $P^{t}_{g|d}$ is indicative of nothing if we do not consider it for all possible educational degrees at the same time.

Failing to do so often blinds us to the fact that, over time and generations, preferences of individuals change; some members of a group find out that they flourish better in another field of study.
This shortsightedness, combined with our problems with Thomas Bayes, results in great misunderstandings.

### We simply are blinded by what is shown to us

In line with the above issues, most people do not take into account that the number of individuals seeking a degree changes over time, and to a lesser degree, the demographic proportions are not 1:1.

Importantly, almost all such plots and articles do not report how many---or what proportion of---individuals were actively seeking a degree; that is, $D=u$, the "uneducated" people, are not counted or reported.

## How does it all have to do with "gender equality" and its change over time?

In the highly politicized public discourse, the differences in the above proportions are considered as evidence for/against the existence of either (a) gender equality in a society or lack thereof, or (b) gender differences---determined by a mix of internal (biological) or external (societal) factors---in "preferences" or lack thereof.

### Camp A: We need more justice

The argumentation of the first camp boils down to the following:

The innate (biological) group differences among men and women, in comparison to societal factors, is negligible, i.e., $P^{t}_{d|w} \approx P^{t}_{d|m}$.

Thus, in an equal society---in which everyone is given equal opportunities regardless of their group membership, e.g., gender or ethnic background---one expects to see the same proportion of men and women in with each degree as the demographic proportions (which is almost 1:1).
Expressed mathematically,

$$\begin{aligned}
\frac{P^{t}_{w|d}}{P^{t}_{m|d}} &\approx \frac{P^{t}_{w}}{P^{t}_{m}} \approx 1.
\end{aligned}$$
### Camp B: We need to embrace it

The latter camp see the disparity from a different lens ,and a radical view of which can be expressed as follows:

The observed differences informs us about the presence of some---perhaps mystics---factors manifested as "preferences".
That is, since we undoubtedly know that $\frac{P^{t}_{w|d}}{P^{t}_{m|d}} \neq \frac{P^{t}_{w}}{P^{t}_{m}}$, then observed disparities in the outcome are not something *to be ashamed of*---instead, a "realist" must embrace this reality and conclude that the premise of the former group is basically not true, and, indeed, $P^{t}_{d|w} \approx P^{t}_{d|m}$.

## False dichotomy? Join Camp Centrists

Every decent (social) scientist knows that the reality is always too complex to be tethered by some simplistic, often reductionist, explanation---we ought to be extremely modest about the conclusions we conjecture from incomplete, noisy data and parsimonious models.

Thus, to acknowledge our limitations, we must meet halfway the two camps: be open to accepting the possible "innate" (biological) differences among groups---and give it the credit that it deserves, no more and no less.

At the same time, give members of each group equal opportunities---by, e.g., making a less sexist society---and try to minimize the societal factors that influence what we have so far called "preferences", letting the individuals exercise their own free will in choosing the trajectory that they desire and meets their goals and values best.

## Concluding remarks

To be written later (on how to measure/estimate societal factors using $P^{t}_{u|g}$ as a proxy.)