Hoover index

Key takeaways

Hoover index belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hoover index to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hoover index from memory before moving on to harder problems.

Reference excerpt

The Hoover index, also known as the Robin Hood index or the Schutz index, is a measure of income inequality. It is equal to the percentage of the total population's income that would have to be redistributed to make all the incomes equal. i.e. The Hoover is the total amount (as a percentage of the national-income) by which people have less than their equal income-share. The Hoover Index can be calculated by the following subtraction: The percentage of the people getting less than their equal-share (i.e. less than the national mean income), minus their percentage of the national income. It can be graphically represented as the longest vertical distance between the Lorenz curve (which graphs cumulative income vs cumulative population (income-ordered population-percentile)and the 45 degree line representing perfect equality). It would be informative to express the Hoover in terms of its average cost to individuals who get less than their equal-share: If the Hoover is divided by the percentage of the population whose income is less than their equal-share (i.e. the mean income), that gives the average cost of that Hoover-value, per person whose income is less than their equal-share. ...that cost being expressed in terms of the national mean income. If, instead, the Hoover is divided by the percentage of the total national income received by the people getting less than their equal-share (i.e. less than the mean income), then that gives the percentage by which those people, as a group, would get more than they currently do, if income were equal. ...in other words the cost, to them, of that Hoover value, expressed in terms of their actual current income. That latter cost can also be gotten by dividing, instead of subtracting, the two numbers that were subtracted to get the Hoover. ...i.e. dividing the percentage of the population whose income is less than the mean by their percentage of the national income. ...and then subtracting 1. The Hoover index is typically used in applications related to socio-economic class (SES) and health. It is conceptually one of the simplest inequality indices used in econometrics. A more frequently encountered inequality measure is the Gini coefficient which is based on the summation, over all income-ordered population-percentiles, of the cumulative income up to each percentile. That sum is divided by the maximum value that it could have (its value with complete equality), to express it as a percentage of its maximum-possible value. The result is subtracted from one, to get a measure of inequality. A report from the National Library of Medicine, of the National Institute of Health, described a statistical study that compared how the Robin Hood and the Gini are correlated with mortality:

Results: The Robin Hood index was positively correlated with total mortality adjusted for age (r = 0.54; P < 0.05). This association remained after adjustment for poverty (P < 0.007), where each percentage increase in the index was associated with' an increase in the total mortality of 21.68 deaths per 100,000. Effects of the index were also found for infant mortality (P = 0.013); coronary heart disease (P = 0.004); malignant neoplasms (P = 0.023); and homicide (P < 0.001). Strong associations were also found between the index and causes of death amenable to medical intervention. The Gini coefficient showed very little correlation with any of the causes of death. The Gini, like the Theil (below), is an impartial measure of inequality over the entire population. That can be of interest and use, but the Robin Hood differs, as a not-impartial examination of the total amount by which members of the population get less than their equal-share.

Computation Let x i {\displaystyle x_{i}} be the income of the i {\displaystyle i} -th person and x ¯ {\displaystyle {\bar {x}}} be the mean income. Then the Hoover index H {\displaystyle H} is:

H = 1 2 ∑ i | x i − x ¯ | ∑ i x i . {\displaystyle H={\frac {1}{2}}{\frac {\sum _{i}|x_{i}-{\bar {x}}|}{\sum _{i}x_{i}}}.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Hoover index

Start with the simplest possible case. Write down what Hoover index claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hoover index before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hoover index ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hoover index

In research: Hoover index appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hoover index in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hoover index is common in secondary-school and first-year university syllabi. It links to neighbouring topics Income inequality metrics, Welfare economics, so understanding it makes those chapters shorter.
In everyday life: Look for Hoover index outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Hoover index in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hoover index means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hoover index out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hoover index in simple terms?

The Hoover index, also known as the Robin Hood index or the Schutz index, is a measure of income inequality. It is equal to the percentage of the total population's income that would have to be redistributed to make all the incomes equal. i.e.

Why does Hoover index matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hoover index?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hoover index.