Five months ago, I introduced the gender swap number as a superior statistic to the median gender pay gap. Feedback from clients and others since then has confirmed my hope they would find it more intuitive to use and interpret. This encouraged me to call on Parliament to amend the UK gender pay gap legislation so that swap numbers could be published for employers with 250 or more employees. I now want to build on this by showing a couple of simple calculations that can turn a swap number into an estimate of how long it will take an employer to close its pay gap.

**What do these 8 employers have in common?**

All have published their 2020 gender pay gap data (which you can find here) and all stated they had between 250 & 499 employees. For the purposes of this article, I am going to assume each employer here has 400 employees.

The answer is that they all have a **Gender Swap Number per 1,000 Employees (GSN1K)** of 40, -40 in the case of Nestle Waters and +40 for the other 7 employers. Even though the median woman earn very different amounts per hour at these employers, the amount of work needed to close their pay gaps to zero is the same on average. I will demonstrate shortly a simple formula that you can use to estimate that on average, it will take each employer 27 years to reduce their GSN1K to below 2.5 assuming their average annual turnover of employees is 10%.

Why does the median gender pay gap vary so much across these 8 employers even if they have the same GSN1K? The short answer is that they have different pay scales and the long answer can be found under recommendation 5 of my 7+5 recommendations for improving pay gap reporting.

**How to calculate a Swap Number for any employer**

You can skip this section if you already know how to calculate swap numbers.

I explain the full calculation in this article and demonstrate it using Ryanair as an example but here is a reminder of how to calculate the swap number for any pair of employee categories. I will use men & women as the two categories here but the swap number calculation can be used for any two categories of employees e.g. you could replace men & women below with able-bodied & disabled.

**Count the number of men and women in each of the 4 pay quarters.**- You will have already done this calculation if you are legally obliged to submit gender pay gap data to the government.

**Count the number of men and women in each of the 2 pay halves**by combining the relevant pay quarters i.e.**M(lh)**= Lower Half Men = Number of men in lower pay quarter + number of men in lower middle pay quarter.**F(lh)**= Lower Half Women = Number of women in lower pay quarter + number of women in lower middle pay quarter.**M(uh)**= Upper Half Men = Number of men in upper pay quarter + number of men in upper middle pay quarter.**F(uh)**= Upper Half Women = Number of women in upper pay quarter + number of women in upper middle pay quarter.**N**= Total number of employees =**M(lh)**+**M(uh)**+**F(lh)**+**F(uh)**

**Calculate the Swap Numbers separately for each gender**:-**MSN**= Male Swap Number = ( M(lh) – M(uh) )/2 = ( number of men in lower pay half – number of men in upper pay half )/2**FSN**= Female Swap Number = ( F(lh) – F(uh) )/2 = ( number of women in lower pay half – number of women in upper pay half )/2- By definition
**MSN**&**FSN**must have opposite signs unless one or both are equal to zero e.g. if**FSN**is +10 then**MSN**should be -10 but can be -9 or -11 if one gender has an even number of employees and the other gender has an odd number of employees.

**Calculate the Gender Swap Number (Rate per 1000 employees)****GSN = Gender Swap Number**= (FSN – MSN)/2 = we subtract one from the other because MSN & FSN have opposite signs as explained above.**GSN1****K**=**Gender Swap Number per 1000 Employees**=**GSN*** 1000 /**N =**multiply GSN by 1,000 and divide by total number of employees.

So if you’ve followed the calculations correctly and assumed the 8 employers above have 400 employees each, then by step 3, you should have a **Gender Swap Number (GSN**) of +16 for the 7 employers that have a majority of their female employees in the lower pay half and -16 for Nestle Waters who have a majority of their female employees in the upper pay half. In other words, to eliminate the median gender pay gap, 16 men in the upper pay half have to swap places with 16 women in the lower pay half for 7 employers whereas at Nestle Waters, 16 women in the upper half have to swap places with 16 men in the lower pay half. When you proceed to step 4, these GSNs become **Gender Swap Number per 1000 Employees (GSN1K)** of +40 for 7 employers and -40 for Nestle Waters.

**Why will it take 27 years to close the pay gap?**

I will demonstrate with Martins Craft Bakery but you can repeat everything I do here for the other 8 employers and get the same answer.

The 27 year estimate with 10% annual employee turnover requires the following assumptions to be made. It is unlikely that all will hold for all 8 employers.

**The total number of employees remains unchanged**– I am assuming that Martins Craft Bakery has 400 employees today and that will never change.**Men & women are equally likely to leave the employer each year**– If I assume that 10% of employees leave Martins Craft Bakery every year then I also assuming that 10% of the men and 10% of the women will leave every year.**The gender ratio of new employees will be the same as the current gender ratio**– In 2020, 80% of employees at Martins Craft Bakery were women and again I assume that 80% of new employees in every year going forward will also be women.**The gender ratio of new employees will be the same for each pay quarter**– in addition to assuming 80% of new employees will be female going forward, I also assume this will be the case in each pay quarter.**The annual turnover of employees will be the same in every year**– since I am assuming that Martins Craft Bakery has 400 employees, if I then assume 10% of employees are leaving and being replaced every year, I am assuming that this turnover is always 10% of 400 i.e. every year 40 employees at Martins Craft Bakery leave and are replaced.

With these assumptions, you should be able to work out that I am assuming that 8 men and 32 women leave Martins Craft Bakery every year and are replaced by 8 men and 32 women so they always have 80 men and 320 women. However, what changes each year is from which pay quarters do the men and women leave from.

If 2020 is year 0, let’s look at what will happen in year 1 in each pay quarter. Note I need to allow fractional employees to exist i.e. 1.5 men and 2.8 women, etc, in order to demonstrate the mathematics here.

**Upper Pay Quarter**– 10% of men and 10% of women will leave as per assumption 2 i.e.**4.1 men and 5.9 women leave**. They are replaced by 2 men and 8 women as per assumption 4. This means at the end of year 1, there will be 2.1 fewer men and 2.1 more women in this pay quarter i.e. there will be 38.9 men and 61.1 women.**Upper Middle Pay Quarter**– 10% of men and 10% of women will leave as per assumption 2 i.e.**1.4 men and 8.6 women leave**. They are replaced by 2 men and 8 women as per assumption 4. This means at the end of year 1, there will be 0.6 more men and 0.6 fewer women in this pay quarter i.e. there will be 14.6 men and 85.4 women.**Lower Middle Pay Quarter**– 10% of men and 10% of women will leave as per assumption 2 i.e.**0.6 men and 9.4 women leave**. They are replaced by 2 men and 8 women as per assumption 4. This means at the end of year 1, there will be 1.4 more men and 1.4 fewer women in this pay quarter i.e. there will be 7.4 men and 92.6 women.**Lower Pay Quarter**– 10% of men and 10% of women will leave as per assumption 2 i.e.**1.7 men and 8.3 women leave**. They are replaced by 2 men and 8 women as per assumption 4. This means at the end of year 1, there will be 0.3 more men and 0.3 fewer women in this pay quarter i.e. there will be 17.3 men and 82.7 women.

Note how the total number of men leaving is 8 and the total number of women leaving is 32 but the breakdown by pay quarter differs.

Now let’s recalculate the **GSN1k** for year 1.

**M(lh)**= 24.7 = 17.3+7.4,**F(lh)**= 175.3 = 82.7+92.6,**M(uh)**= 53.5 = 14.6+38.9,**F(uh)**= 146.5 = 85.4+61.1**MSN**= (24.7-53.5)/2 = -14.4,**FSN**= (175.3-146.5)/2 = +14.4**GSN**= (14.4 – (-14.4) )/2 = +14.4**GSN1K**= +14.4*1000/400 = +36

So the GSN1k for Martin’s Craft Bakery has narrowed from +40 to +36 i.e. a fall of 10% which happens to be annual employee turnover we assumed. If you now start from year 1 and calculate year 2 using the 5 assumptions, the year 2 GSN1K will be +32.4, a fall of 3.6 which is again 10% of the year 1 GSN1K of +36. In other words, if the 5 assumptions hold, the GSN1K falls by the same percentage as the employee turnover each year.

That makes for a very simple mathematical formula which is –

**GSN1k**[**y**] =**GSN1k**[**0**] * (1 –**t**)^**y**, where**y**= number of**y**ears since year 0 i.e. today.**t**= annual employer**t**urnover as a decimal e.g. 10% is entered as 0.1- The formula works just as well if you want to use the actual GSN rather than GSN1k

You can now extrapolate this formula forward into the future for as many years as you want which I have done in the table to the right. Three columns are shown for annual employee turnover of 5%, 10% and 20% where the GSN1k starts at +40 in year 0. Notice this formula will never reduce the GSN1k to exactly zero so you need to choose a target value that more or less represents zero for your employer. Since I am assuming that Martins Craft Bakery has 400 employees, then 1 employee equals 2.5 employees per 1,000 employees hence why I set the target for closing the median gender pay gap for Martins to be a GSN1k of less than 2.5. As you can see, it will take 27 years if turnover annual employee turnover is 10% and 13 years if annual employee turnover is 20%.

When turnover is 5%, the table doesn’t tell us how many years it takes for GSN1k to fall below 2.5 since it is greater than 30 years. I could have easily extrapolated it further into the future but there is a direct formula that tells you how many years it will take GSN1k to reach a target value if the 5 assumptions hold. This is –

**G**= target value for**GSN1k**.**t**= annual employer**t**urnover as a decimal e.g. 10% is entered as 0.1**Y**= number of years for GSN1k to fall below G.**Y**= [ LN(**G**) – LN(**GSN1k**[**0**] ) ] / LN( 1 –**t**)- where
**LN()**is the natural logarithm formula in Microsoft Excel. - Again you can use GSN rather than GSN1k if you prefer.

So if **t**=0.05, **G**=+2.5, **GSN1k[0]**=+40 then **Y** = (LN(2.5)-LN(40))/LN(1-0.05) = 54 years.

Of course, the 5 assumptions are not going to hold for every year even if on average they happen to be true. To see how such variation can affect Y, please read my article “*Eliminate your pay gap by playing Blackjack*” which uses computer simulation to explore this. That article includes a spreadsheet you can download to carry out your own simulation. My simulations result in the same conclusion that should be obvious from the table above, namely that in years when you have high employee turnover, that is the time to go all out on efforts to significantly reduce the underrepresentation of the relevant employee category in the relevant pay quarter. In years of low employee turnover, you won’t be able to do much.

The simulations address another issue with the formulas I’ve given above which is that it is not possible with the 5 assumptions for a GSN to change from positive to negative and vice versa. In reality this will happen due to random chance when the GSN is close to zero.

**Will every employer with GSN1k of 40 take 27 years to reduce that to 2.5?**

As I said before, the formula for estimating the number of years to close a pay gap for a certain level of employee turnover is only true if 5 assumptions are made. Thus the estimate is simply an average and it is up to the employer to work out by how much they are likely to vary from those assumptions. As my Blackjack article noted, growth provides the best opportunity to change a pay gap since you are increasing the total number of employees rather than keeping it constant and thus the new employees where you hope to address historical under-representations will make up a bigger part of your workforce.

What about the 8 employers we started out with? Even though on average, they will take 27 years to close the pay gap with 10% annual employee turnover, are there other factors that might lead us to say this employer will be faster and that employer will be slower?

Here are my thoughts. Feel free to take a different view since I am not an expert in these industries!

**Rayleigh Schools Trust**– the upper pay quarter which presumably includes the senior teachers and heads is more male than the other 3 quarters but even then it is still 3/4 female. The real issue is increasing the number of men in the lower pay quarters. If this is a primary school trust, I cannot see many men wanting to work in these roles. I think this will take longer than 27 years due to social expectations of men and women beyond the capacity of this employer to resolve alone.**HSBC Global Asset Management**– HSBC is a massive global organisation and this is only a small branch. So the pay gap could disappear tomorrow through restructuring of subsidiaries alone, something I highlighted in my article “*the good, the bad and the Unilever*“. Of course, the finance industry is known for very high salaries being predominantly earned by men so progress is not going to be quick but I do think they can make quicker progress than a primary school.**Nestle Waters UK**– Like HSBC, this is again a subsidiary of a much larger organisation and restructuring can close it overnight. I do hope Nestle are not falling into the trap of thinking a pay gap that favours women is a good thing which I have seen some organisations do like Novartis UK. They have a large pay gap and it should be addressed with the same vigour as one that favours men. Unlike HSBC and the finance industry, Nestle and indeed the food industry at large (which I used to work in) has no shortage of women in high paid roles and I think they will be one of the quicker employers to close their pay gap.**Warner Bros Entertainment**– Another small subsidiary of a much larger company. Again I think this gap can be closed quicker than 27 years. At the moment, conglomerates are not required to report for the whole group and I explained in my article “*10 quick and easy ways to close your pay gap*” why this is not desirable. Recommendation E of my 7+5 recommendations for amending UK pay gap legislation calls for this to be changed.**Martins Craft Bakery**– I do not think they are part of a conglomerate so restructuring is out here. What is notable from the pay quarters (or fingerprints as I like to call them) is that men are skewed towards the top pay quarter in what is a female dominant employer. Given my earlier comments about the food industry generally being a good place for women to progress their careers, I think looks like an issue specific to this employer which needs addressing. I suspect rapid progress could be made here if they choose to put their mind to it.**NE Derbyshire Council**– This is a 50:50 M:F employer with women making up the majority of the lower pay half and men making up the majority of the upper pay half. There is no opportunity to restructure as part of a conglomerate, they are likely to be on public sector union negotiated pay scales which are unlikely to change so any change has to come from more women applying for higher paid roles and more men for lower paid roles. The latter may be the harder bit and so I can easily see them taking 27 years to close this pay gap.**Wateraid**– As a charity, they are unlikely to be paying high salaries. I would not be surprised if a large number of employees are earning the same. However, I can see this employer being the fastest to close their pay gap and that is because of the lower middle pay quarter today. This is 75% women compared to 55% women in the upper pay half. That suggests that if a promotion drive aimed at women in this pay quarter was launched which moves them up to the upper middle pay quarter, they could make rapid progress. They still need to address the lower pay quarter which is predominantly women but the quarter above offers an opportunity to take a big step at the beginning.**Europa Warehouse**– Despite having one of the smallest actual pay gaps (median women earns 98p for every £1 earned by the median man), I can see this employer struggling to close the last mile. First it must said that the reason why the pay gap is so small is that the majority of employees are probably on the same pay scale with only a few managers earning a lot more. But this employer is 75% female in the lower pay quarter whilst nearly 50:50 elsewhere. That suggests a set of roles, maybe admin, that are overwhelmingly female and paid slightly less than other roles. If so, making those roles less female may take a long long time.

Of course this where the employer can own their narrative when they submit their gender pay gap data. Rather than letting me try to explain why it will take 27 years, they can instead say why that is a load of rubbish because of what they are planning to do. That is why I would like to add to recommendation 5 of my 7+5 recommendations for amending UK pay gap legislation a call for the GEO to publish the estimated number of years it will take the employer to close the pay gap. That could prompt employers to say “*hang on, that’s not right and here’s why*“. I see too many employers just submitting numbers when in fact they should owning their pay gap report and swap numbers & the associated estimate of Y could be the stick that get them moving.

**Is the Maximum GSN1K useful?**

I’d like to finish on a related point which is an optional extra but some people may find it useful.

The 8 employers I’ve used here all have the same gender swap number but they have different maximum GSN1k’s. By maximum, I mean given the gender ratio the employer has today, what is the worse possible swap number they could have? For example, NE Derbyshire is 50:50 M:F and the worst possible outcome for the swap number would be if men made up 100% of the upper pay half and women 100% of the lower pay half. Then the GSN1k would be +250 so placed against that, NE Derbys current +40 looks a long way from worst case scenario. That differs from Rayleigh Schools Trust where its worse case scenario is +87 so the current +40 is almost half way to worst case.

The calculation of the maximum GSN1k for an employer is simple to do. All you need is the swap number calculation for the smaller category of employees and then imagine all of that category being put in the either the upper or lower pay half. So in the case of Nestle & HSBC, women are the minority category so we use the FSN calculation. The same goes for NE Derbys since it doesn’t matter which one we use since it’s 50:50 there. For the other 5 employers, men are the minority and so we use the MSN calculation. It turns out a simple formula is the following –

**M**= number of employees in the minor category**T**= total number of employees**Max GSN1k**=**M**/2*1000/**T**

Note employers submit pay quarter data to 1 decimal place hence why rounding error means NE Derbys actual maximum GSN1k is +248 not +250.

As I said, I see the Max GSN1k as an optional extra but it is another way of picking up employers who appear to have a relatively low swap number but when you calculate it as a percentage of a maximum, you realise they are close to a worse case scenario. It is the gender dominant (male or female) employers like the bottom two above that will get picked up by this metric but note they are still facing the same amount of time as other employers on average when it comes to closing their pay gap.

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