How to identify an incorrectly calculated median gender pay gap

I have a spotted an incorrect median gender pay gap published by a well known name in a certain industry. They shall remain nameless for now since I am trying to get them to accept their error and publish a new gender pay gap report on their website. I know they have made an error because their published data violates the laws of mathematics as I will explain in this blog. All it takes to spot such an error is a simple calculation you can do in your head and an understanding what the median measures.

I should state straightaway that Cleveland Police & other employers mentioned in this post are not the nameless employer I am referring to!

Are Cleveland Police correct to claim they have no gender pay gap?

Here are the vital statistics for the Cleveland Police force in 2018 created from my free downloadable tool which has data for over 11,700 employers. They state that for every £1 earned by the median man, the median woman also earns £1. They also state that just over 20% of the two highest paying income quarters and just under a half of the two lowest paying income quarters are women. Is it mathematically possible to have data like this?

5 Percentiles in brief

Let’s remind ourselves of the definition of the median, quartiles and quarters. In the graphic below, I have an employer with 7 men and 5 women who are standing in a line in order of their hourly earnings. The line has been split into four equal sized groups of employees with 3 employees in each group which are called Income Quarters.

Median, Quartiles and Quarters are all subsets of a wider group of statistics known as Percentiles. From left to right in the graphic above a number of percentiles are given their own name and the following 5 percentiles are the boundaries that surround and define the 4 Income Quarters.

0th percentile is the Minimum = £10 per hour
25th percentile is the Lower Quartile = £10 per hour
50th percentile is the Middle Quartile better known as the Median = £17.50 per hour = average of the two men in the middle earning £15 & £20 per hour.
75th percentile is the Upper Quartile = £30 per hour = average of the two men earning £25 & £35 hour.
100th percentile is the Maximum = £50 per hour

Every employee can therefore be compared to these 5 boundaries to decide which income quarter they are in. So a man earning £25 an hour lies between the median and the upper quartile and is in the upper middle income quarter.

How the laws of Mathematics work

In the bottom two quarters (the 6 employees earning less than the median), there are 3 men and 3 women i.e. the gender ratio is 50:50 men:women. The alternative method getting to this ratio is to observe that 1/3 of staff in the lower income quarter are women and 2/3 of staff in the lower middle income quarter are women. The average of these two numbers (1/3 & 2/3) is 1/2 hence 50% of staff in the lower half of the hourly earnings line are women. We can take such an average since there are equal number of staff in each quarter.

In the top two quarters (the employees earning more than the median), there are 4 men and 2 women i.e. the gender ratio is 67:33 men:women. The alternative method of getting to this ratio is to observe that 1/3 of staff in the upper income quarter are women and 1/3 of staff in the upper middle income quarter are women. The average of these two numbers (1/3 & 1/3) is 1/3 hence 33% of staff in the upper half of the hourly earnings line are women. We can take such an average since there are equal number of staff in each quarter.

I define the Gender Ratio Differential (GRaD) to be the difference between these two ratios i.e.

GRaD = Gender Ratio of Top Half – Gender Ratio of Bottom Half = 67:33 – 50:50 = +17 : -17

Can you see why if the GRaD for women is negative (-17 in this instance) then the median woman must be earning less than the median man? Conversely if the GRaD for women had been positive instead, the median woman would be earning more than the median man?

The median woman is the woman standing in the middle of the female only line (5 in all) when sorted by hourly earnings. She is highlighted in the graphic above and is earning £15 per hour. A negative GRaD for women tells us that a majority of women are in the bottom half of the overall income line. That majority must include the median woman since by definition of the median, there are equal numbers of women on either side of her in the line. To make up a majority, the median woman has to be part of that majority and consequently she ends up in the bottom half of the overall hourly earnings line.

The median man is the man standing in the middle of the male only line (7 in all) when sorted by hourly earnings. He is highlighted in the graphic above and is earning £20 per hour. A positive GRaD for men tells us that a majority of men are in the top half of the overall hourly earnings line. That majority must include the median man since by definition of the median there are equal numbers of men on either side of him in the line. To make up a majority, the median man has to be part of that majority and consequently he ends up in the top half of the overall hourly earnings line.

Thus the median woman earns less than the median man because the GRaD for women is negative. If the GRaD for women had been positive, the median woman would earn more than the median man.

Cleveland Police’s GRaD is negative for women

Cleveland Police’s GRaD can be calculated as follows –

Cleveland Police GRaD = 78.5 : 21.5 – 52 : 48 = +26.5 : -26.5

You should be able to work out that the average % of women in the top two quarters is 21.5% (= average of 21% & 22%) and in the bottom two quarters is 48% (= average of 41% & 55%).

Since the GRaD for women is negative, the median woman at Cleveland Police must be earning less than the median man and hence their reported gender pay gap is incorrect.

There is another logic that can be used to explain why this is so. The median woman is the woman standing in the middle of a female only line when the women are standing in order of their hourly earnings. If we assume that the %s in the Cleveland Police chart represent actual numbers of employees of each gender in each quarter, then they would have 261 men (=79+78+59+45) and 139 women (=21+22+41+55). The man in the middle of the male line is the 131st man and the woman in the middle of the female line is the 70th woman. Counting from the lower income quarter, the 131st man must be in the upper middle income quarter since there are 104 (=59+45) in the bottom two quarters. Likewise, the 70th woman must be in the lower middle income quarter since there are 55 women in the lower income quarter.

The advantage of the alternative logic is that it tells you which quarter the median man and woman are in where as the GRaD calculation simply tells you which half they are in.

The exception to the rule

The GRaD calculation is a very easy one to do in your head whenever you are reading a gender pay gap report for an employer. However, there is an exception to the rule that you have to be aware of and if you’ve understood the concept of the Gender Ratio Differential, you may have spotted it already.

Compare these two graphics below. The first is the same 12 employee example I used earlier, the second also has 12 employees who line up in the same order but have different hourly earnings.

For the new graphic, laws of maths tell you that the median woman is again in the lower middle income quarter and the median man is in the upper middle income quarter. This time though both the median man and median woman are earning the same £10 per hour i.e. there is no gender pay gap.

The reason is obvious. 9 out of the 12 employees earn exactly the same i.e. £10 per hour. This includes 4 of the 5 women and 5 of the 7 men. Basically when a majority of staff are earning exactly the same and the effect is seen in both men and women, then by definition of the median, the median man must be earning the same as the median woman hence the gender pay gap is the same even though the GRad is not zero for both genders.

What kind of industries see lots of staff earning the same pay rate? Lees Cleaning with a GRaD of +13.5 : -13.5 shown here are an example and that shouldn’t surprise us since the cleaners are very likely to be earning the minimum wage. Hospitality and Care sectors also pay a lot of their staff a basic wage and it so happens that these are the 3 industries where you are most likely to find employers with no median gender pay gap. They can still have a mean gender pay gap which will be driven by the salaries of the managers and directors but the median is often zero.

This exception means I need to modify my GRaD rule

If the GRaD is 0:0 then the median gender pay gap must be zero.
If the median gender pay gap is zero then the GRaD does not have to be zero if the employer is likely to have a majority of staff on the same pay scale.
If the employer has a pay system with a lot of pay scales then for the median gender pay gap to be zero, the GRaD must be zero as well.

Do we know if Cleveland Police come under rule 2 as above. The answer is no they don’t since when we read their gender pay gap narrative, I find this paragraph that explains how they did their calculation at the bottom of page 1.

“The calculation of the median average involves listing all the hourly rates in ascending order and picking the middle rate if the i.e.: 45 hourly rates in the list the 23 hourly rate in the median. If there are 46 in the list the median would be the mean of the hourly rates 23 and 24.”

This clearly refers to 40 odd pay rates so rule 2 doesn’t apply. It also explains why they made a mistake. They were taking the median pay scale not the median woman. Before you say “how stupid” I had already warned the EHRC (Equality & Human Rights Commission) in point 2 of this post “12 ways to improve gender pay gap reporting” that the statutory guidance was very poorly written and easily misunderstood by those who do not have statistical skills.

GRaD can identify other errors.

Take a look at HairCare Ltd here. Have they made an error? If so, what did they do wrong?

Their GRaD is -17.5 : +17.5 i.e. positive for women. By the rules I’ve set out here, the median woman should be earning more than the median man. Yet they are claiming that the median woman earns 41 pence in the pound less than the median man. That can’t be right.

In case you’re thinking, this might be a majority minimum wage payer, the exception is not relevant here because they are not claiming their median gender pay gap is zero. The exception is only relevant to employers claiming no gender pay gap.

What could they have done wrong here? Unfortunately they do not provide a written narrative but two possibilities suggest themselves to me.

They inadvertently entered 98% for women in the upper income quarter when they submitted their data to the government’s gender pay gap portal when they intended to enter 98% for men and 2% for women.
They really do have a gender pay gap in favour of women i.e. the median woman earns more than the median man as indicated by their GRaD. But when entering such a pay gap on the government portal, you have to enter it as -41% if the pay gap favours women. If they entered +41% instead, it will be counted as a pay gap favouring men.

I don’t which is correct but clearly they need to take a look at their submission again.

– Want to know more about spotting errors? –

I have written a number of articles about errors made in gender pay gap calculations.

Finally visit my Twitter thread to see my comments on gender pay gaps in the media. Some notable ones are here.

– Need help with interpreting your pay gaps? –

I offer the following services.

Analytics – I can dig deep into your data to identify the key drivers of your pay gaps. I can build a model using a large number of variables such as pay band, seniority, job function, location, etc and use this to identify the priority areas for closing your gaps.
Training – I run training courses in basic statistics which are designed for non-statisticians such as people working in HR. The courses will show you how to perform the relevant calculations in Microsoft Excel, how to interpret what they mean for you and how to incorporate these in an action plan to close your gaps.
Expert Witness – Has your gender pay gap data uncovered an issue resulting in legal action? Need an expert independent statistician who can testify whether the data supports or contradicts a claim of discrimination? I have experience of acting as an expert witness for either plaintiff or defendant and I know how to testify and explain complex data in simple language that can be easily understood by non-statisticians as can be seen from my testimony to the Treasury Select Committee.

If you would like to have a no-obligation discussion about how I can help you, please do contact me.

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