The first constraint is the number of vacancies available for women to be appointed to. For example, if a department has a turnover of 5% per year and the number of academic staff is growing at 2% per year then the overall vacancy rate is 7% per year.

The next most important constraint is the proportion of women in the pool of potential applicants. If the pool of potential applicants is 50% women and no women leave the department then a department with a vacancy rate of 7% could achieve an increase in the proportion of women of 3.5 percentage points per year. In these circumstances a department could get from 25% to 50% women in 7-8 years. However, if turnover was around 3% and the number of academics was static then the best the department could achieve, if no women leave, is a growth in the proportion of women of 1.5 percentage points per year. At that rate it would take 17 years to get from 25% to 50%. Of course, this might well be an underestimate since it is unlikely that no women would leave over a seventeen year period.

There is an additional complication. Suppose a department is able to make six appointments over a three year period and it makes the appointments from a pool that is 50% women. If recruitment is fair with respect to gender then the probability distribution for the number of women appointed will be a binomial distribution with N=6 and p=0.5. This gives a probability of 0.31 of appointing exactly three women, a probability of 0.34 of appointing two or fewer women and a probability of 0.34 of appointing four or more women (probabilities do not add to one due to rounding). Hence for time periods in which a small number of appointments are made fair recruitment processes could easily result in apparent growth rates between 2/3 and 4/3 times the expected rate.

**Conclusions**:

1. There are limits to how fast the proportion of women among academic staff in STEM can increase. Growth rates of a few percentage points per year are not unreasonable.

2. Estimating the long-term growth rate from measurements made over short periods is futile.

The second conclusion implies that simply collecting data on the proportion of women among new appointees is unlikely to reveal inadvertent bias in the recruitment process. While these data are necessary it is also necessary to assess the recruitment process against best practice established by large studies, such as that described in the US National Academies Report Gender Differences at Critical Transitions in the Careers of Science, Engineering and Mathematics Faculty. (A briefing on the report is available from the pages of the National Academies Committee on Women in Science, Engineering and Medicine.)

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