The example I have used here is not real but relates to a real problem that is too complex to explain on this site.
Within a large shopping facility, we distribute staff across the various areas according to a prioritised plan. As staff reduce through breaks, dealing with specific customer issues etc, we move the free staff into the busier areas. This helps us provide a quicker response to our customers. There are 5 departments and key areas for staff to be in each. The key areas are organised by their importance. For example:
The electrical department has the following key areas: Sales Counter 1, Sales Counter 2, Demonstration stand, Roaming staff member, Sales Counter 3, Returns.
Each department has a different number of key areas. It is the responsibility of the managers to ensure that each department has sufficient staff. So if the electrical department had all their key areas covered, but the other departments had no staff at all, this would be a problem and staff need to be moved.
Therefore, I would like to put a figure on how even the distribution of staff is. This is not about actual numbers of staff in each department, but solely about the key areas being covered given some departments have a lot more key areas than others. I've started with percentage coverage of key areas in each department, for example:
Electrical Department: 25% Baby & Child: 50% Home & Garden: 75% Clothing: 10% Sports & Leisure: 0%
Looking at that, you can see staff are not evenly distributed and managers should move staff about to make it:
Electrical Department: 25% Baby & Child: 25% Home & Garden: 25% Clothing: 25% Sports & Leisure: 25%
That would be evenly distributed.
So my question is: How can I take those 5 percentage figures to form a sole figure that represents a meaningful value about how well we are distributing our staff?
Hoping that I understood what you meant (that you want a single number, based on those five, that would reflect how equally values are distributed).
Denote your input as $p_1, ..., p_n$. Then define $$h = \max_{i} \frac{|p_i - \bar p|}{\bar p}$$ where $\bar p = \frac1n \sum_{i=1}^n p_i$.
Then $h$ would be a positive number that would measure deviation from equal distribution. $h$ close to $0$ would signify, that all the percentages are approximately equally distributed. A big $h$ would mean that some department is either under- or over- staffed.