Yes! You CAN Average Averages

So, you’ve heard that you should never average averages. 

The person who told you this “rule” was probably well-intentioned; someone who didn’t want you to make a mistake by taking an average of averages without understanding the implications of doing so. Great! But there’s one problem: this rigid rule means you’re probably missing out on how useful averages of averages can be for equity in your data projects. 

Let’s see how the differences between these types of averages play out in an example. 

We run a long-term care facility that houses 184 senior citizens. We’ve just put in a swimming pool and we want to know if our residents are using it. We’re hoping that, on average, people are using it 3 times a week. From talking with the residents, we suspect that male and female residents use the pool at different rates. We also want to make sure we take into account the perspective of both our high-mobility (can walk unassisted) and low-mobility (need assistance walking) residents. 

To start, we asked residents how many times a week they used the new pool and the results look like this:

So, to find out how often the “average resident” uses the pool, we can add all the responses and divide them by the number of responses. This gives us a simple mean (as opposed to mode or median), average in which every resident is equal. 

This pool-use average is: 2.777 times per week, which we’ll round up to 3. 

However, we may want to know more about the subgroups we considered before: gender and mobility. 

Here’s a simple, mean average of the responses from each group:

Now, time for some rulebreaking: what if we take the average between each of those four groups?

If we add the averages of men with high mobility, the women with high mobility, the men with low mobility, and the women with low mobility, and then divide by 4 (the number of subgroups) we get a different average:

( 3.71  +  3.93  +  0.50  +  1.21 )  /  4   =  2.33.  We round down to 2. 

Hang on – what?!

We started by deciding that an average use of 3 times per week would be a good indicator of whether the pool was working well. But our average just shrunk from 3—meaning success—to 2—a failure. What’s going on?

In the first type of average, we give equal weight to each resident. In the second type of average, an average of the subgroup averages, we give equal weight to each type of resident instead of equal weight to each resident. This can help to reduce power imbalances for minority groups in our sample. 

Neither of these averages is “wrong”. They’re just measuring different things. 

From an equity standpoint, our facility might decide to take an average from the subgroups (the dreaded “average of averages”) on purpose in order to give equal say to each type of resident, regardless of how many of them there are. Doing averages like this can help to reduce power imbalances for minority groups in our sample. The women with low mobility report barely using the pool at all, but in the simple average their experience is overwhelmed because there are only 36 of them and there are 121 residents with high mobility. 

 If on the other hand, we decided it was most important that our average weights individuals equally instead of groups, then we might go with the simple mean. It gives us the average pool use across all residents, but doesn’t offer any balance between groups. The weight of each group will be determined by how many people are in it. This type of method has the advantage of being what most people think of when you say “average”, the average of averages will most likely need some explaining. 

 Do you want to weight each type of person—each sub-group—equally, or do you want to weight individuals equally? 

At We All Count, we encourage you not to underestimate your data audience. Both methods described above should have a place in your equity toolbox. Both averages are useful. And your big takeaway should be: don’t be bullied into never using an average of averages!