Last time I set you a puzzle—how could an organisation improve satisfaction with each of its products, but have lower satisfaction overall? This is one form of a statistical anomaly known as Simpson's paradox which deserves to be better recognised than it is.
Simpson's paradox in worth reading about in detail, but the short version is that overall data shows the opposite effect to that found in subgroups. It happens when there is a hidden weighting variable, which acts as a confounder when we draw conclusions.
The hidden variable in this instance is the change in sample size for each product. The overall score is lower because it is based on more large player customers (who are less satisfied) and fewer tiny player customers.
| 2006 | 2007 |
| Sample size | Satisfaction | Sample size | Satisfaction |
| Overall |
600 |
81 |
600 |
78 |
| Tiny player |
300 |
85 |
100 |
86 |
| Small player |
200 |
78 |
200 |
79 |
| Large player |
100 |
73 |
300 |
74 |
We'll assume that this sample is perfectly representative of the population—in other words this company really has sold far more large players this year. If not, then the problem could well be with the sampling, and ensuring representativeness would be the answer
So, is satisfaction going up? In a word…yes and no. How's that for fence-sitting? Interpreting this data is not straightforward. Satisfaction is improving for each product, but we need to think about how to interpret the change in proportion of sales for each one. Our customer base, overall, is less satisfied than last year.
Strategically, this shift towards products where we perform less well could be very significant and worrying. This wouldn't be the first company to slide into oblivion by making great products that no one buys (even the mighty Steve Jobs fell foul of this).
Conclusion
- Be aware that changes to the make up of your sample can affect the overall result
- Make sure you sample properly
- Consider weighting to adjust for problems
- Look out for instances of Simpson's paradox
- Don't interpret relative proportions causally
- Read about the Berkeley sex bias case for a striking real example
- Think about interpretation when it does happen
- Performance is improving with each product
- But is there a deeper problem?