Warning: this post contains mathematics. I apologise to those of you who find this intimidating but I make statements that follow mathematically from the definition of the gender pay gap and it is important that I give my reasoning so that those who are not put off by a bit of straightforward algebra can check it. If you can not read the equations and want to, try this link.

The important conclusion is that an institutional gender pay gap is an incomplete and ambiguous measure of inequality. It is incomplete because the gender pay gap can be small or zero even when the overall proportion of women in the workplace is low. We therefore need to know the proportion of women in the workplace as well as the gender pay gap. It is ambiguous because while if the proportion of women in each salary interval falls as the salary increases then the pay gap is non-zero it is possible for the pay gap to be small or zero and the proportion of women by salary interval to still have undesirable features such as a lack of women at the highest levels. In addition, because the gender pay gap compounds structural factors that are common to both men and women, namely, the salary scale and the number of people in each salary interval, with an inequality factor, namely, how the proportion of women varies with salary, it is difficult to compare different workplaces unless the structural factors are similar.

For these reasons the minimum information required to make sense of a gender pay gap is

- the number of women and the number of men (number rather than proportion since the proportion can be calculated from the numbers and the numbers give an idea of whether the measured gap reflects underlying inequality or just a fluctuation or contingency)
- the average salary for women
- the average salary for men
- the proportion of women by grade or salary

**Myth 1: The gender pay gap measures the extent to which women are paid less than men for doing the same job.**

There are three contributions to the gender pay gap:

1. Occupational segregation: there are more women in low paid occupations and occupations in which women predominate attract lower pay.

2. Vertical segregation: within an occupation there are more women at lower levels.

3. In some cases women are paid less than men for doing the same job or for work of equivalent value, which is illegal.

There are numerous causes of the gender pay gap, for example, research commissioned by the

__Government Equalities Office__in the UK identifies several factors including differences in years of full time work and the negative effect on wages of having worked part time or taken time out of the labour market to care for a family.

**Myth 2: The gender pay gap is a useful indicator of inequality.**

The gender pay gap is defined by:

*gap = (average pay for men - average pay for women)/(average pay for men)*

The average pay for women can be written as:

and the average pay for men as

where is the number of people with salary and is the proportion of them who are women. The number of different salaries (or salary categories) is . The symbol means add all the terms from 1 to together. These formulas work when people are paid on a salary scale or when there are enough people that it makes sense to make a histogram of the number of people in each salary interval.

The total number of people is . The total number of women is , the total number of men is and the overall proportion of women is . This implies that the difference between the average pay for men and the average pay for women can be written as

and the gender pay gap as

.

So, the difference between average pay for men and average pay for women depends on two structural factors, namely, the salary scale and the proportion of jobs at each salary scale point, and an inequality factor, namely, the way in which the proportion of women at each scale point varies with scale point. One implication is that the pay gap will be zero whenever the proportion of women is constant with scale point regardless of what that proportion is. Hence, the pay gap is an incomplete measure of inequality. A workplace with a zero pay gap that has only 10% women is hardly a shining example of gender equality.

From a mathematical point of view we have two equations

, which defines , and

which defines the gap. If we want then we have two equations in unknowns. This is an under-determined system, unless there are only two steps on the salary scale, so there is the possibility of finding other solutions that give a zero gap besides for all

*i*. This means that while if is constant then the gap is zero and if falls systematically as

*i*increases then the gap will be non-zero there could be solutions which have a small or zero gap that nevertheless have undesirable features such as a lack of women at the highest salary levels. The figure below shows an example, which has 220 men and 180 women (45% women among a staff of 400) on an eleven point scale where each point is has a salary 5% greater than the one below starting from £20,000. The average salary for men is £24,923.41 and the average salary for women is £24,901.72, which is a gap of 0.09%. Nevertheless, only 35% of the posts in the top three grades are held by women and only 20% of the posts in the highest grade are held by women. Click here to view the spreadsheet I used to create this figure. The spreadsheet itself is available at this link.

So, as a measure of inequality the gender pay gap is both incomplete and ambiguous.

**Myth 3: The overall national pay gap will be eliminated if each workplace eliminates its own pay gap.**

Suppose Employer A has a largely female, largely relatively unskilled workforce while Employer B has a largely male, largely skilled or professional workforce. Both employers could eliminate their pay gaps but Employer A would still be paying their predominantly female workforce less on average than Employer B was paying their predominantly male workforce.

**Myth 4: The gender pay gap provides a means of comparing inequality across workplaces**

As noted in under Myth 2, the gender pay gap depends on two structural factors and an inequality factor. Unless the workplaces have the same salary scale and the same proportion of jobs at each salary scale point it is very difficult to draw conclusions about differences in equality in different workplaces. It would also be helpful if there was agreement on whether to divide by the average salary for men or the average salary for women in the expression for the pay gap. It could also be the case that in the example discussed under Myth 3 that Employer B has a gender pay gap that is hard to eliminate due to a shortage of women with the necessary professional qualifications, for example, in engineering, while Employer A is able to eliminate their gap despite the fact that women working for Employer B have higher average salaries than women working for employer B.