What is economic inequality?
Wikipedia says: “Economists generally think of three metrics of economic disparity: wealth, income, and consumption”. Let’s leave aside how much people freely choose to consume, and look only at their underlying wealth and income.
As Wikipedia says, the balance between income and wealth changes with career progress. For most people, the majority of their wealth is not inherited, and is thus a product of household income. Income thus appears to be the main driver of inequality.
Note: breakup of households (due to increasing divorce rates) is also believed to have a significant effect on household inequality, as you might expect.
Income can be measured at the level of the individual, or (more meaningfully) at the level of the household. The household is the same level at which the ONS measure income inequality.
Issues remain about how to compare full-time workers with part-time, and comparing self-employed with employees. Using household income (partly) overcomes these issues.
Inequality can be compared before or after taxation, and can opt to take into account public services. These would make figures even harder to understand.
So, IMHO concern should be focussed on net income inequality at the household level. Ideally this should include income from capital, asset transfer (eg inheritance) and savings.
How do you measure economic inequality?
… There are various numerical indices for measuring economic inequality. A widely used one is the Gini coefficient https://en.wikipedia.org/wiki/Gini_coefficient , but there are also many other methods https://en.wikipedia.org/wiki/Income_inequality_metrics. The Gini measure has a number of shortcomings. It is not intuitive, it does not indicate the income level at which the inequality exists, it gives peculiar values under certain demographic conditions, and negative income/wealth values also generate strange values. However it does have general acceptance and it takes account of all income inequality.
Note: attempting to understand the integral calculus involved in the Gini coefficient can cause headaches – stick to the graphical description!