# The Gini Coefficient of wealth distribution

By Murray Bourne, 24 Feb 2010

In a "perfect" society, everyone would share wealth (or income) evenly. (Don't worry - this is an article about math, not communism!)

The Gini Coefficient is one way to measure how evenly the income (or wealth) is distributed throughout a country.

The Gini Coefficient is calculated as follows. We find out the income of all the people in a country and then express this information as a cumulative percentage of people against the cumulative share of income earned. This gives us a Lorenz Curve which typically looks something like the following.

Image Credit: Wikipedia

In plain English, the graph above indicates the proportion of the income going to the poorest people, middle-income people and richest people.

There will always be rich and poor, but we are interested in how evenly wealth is distributed and most governments put effort into keeping this coefficient as low as possible.

The Gini Coefficient ranges between 0 and 1 (or it can also be expressed as a number from 0 to 100) and is given by the ratio of the areas:

$\mathrm {Gini Coefficient} = \dfrac{A}{A+B}$

If A = 0, it means the Lorenz Curve is actually the Line of Equality. In this case, the Gini Coefficient is 0 and it means there is "perfect" distribution of income (everyone earns the same amount).

If A is a very large area (making B very small), then the Gini Coefficient is large (almost 1) and it means there is very uneven distribution of income. Countries with a high Gini Coefficient are more likely to become unstable, since there is a large mass of poor people who are jealous of the small number of rich people.

## What Does it Mean?

Let's try to understand the above graph.

For example, say we have 10 people in a village and the income for the village is $100 per day. If every person shares this income evenly, they get$10 each per day.

So the income distribution would be as follows. ("Cumulative" just means add up the number you have so far for each step.)

Person Proportion of population (%) Cumulative proportion of population (%) Income (%) Cumulative income (%)
A 10% 10% 10% 10%
B 10% 20% 10% 20%
C 10% 30% 10% 30%
D 10% 40% 10% 40%
E 10% 50% 10% 50%
F 10% 60% 10% 60%
G 10% 70% 10% 70%
H 10% 80% 10% 80%
I 10% 90% 10% 90%
J 10% 100% 10% 100%

So for this society with perfectly-distributed income, we could draw a graph of the cumulative proportuion of population (on the horizontal axis) against the cumulative percentage of income (on the vertical axis) as follows.

In the above case, A = 0 so the Gini Coefficient is 0.

Now, people being people, some of the villagers decide they should be paid more because they work harder, or because they are older, or because they have more children, or whatever. So three of them (persons H, I and J) decide to keep 15% of the income each, and distribute the rest evenly among the others. However, that doesn't work out evenly, so they decide the laziest 3 people in the village (persons A, B and C) should only get 5% of the income. Our table now looks like this:

Person Proportion of population (%) Cumulative proportion of population (%) Income (%) Cumulative income (%)
A 10% 10% 5% 5%
B 10% 20% 5% 10%
C 10% 30% 5% 15%
D 10% 40% 10% 25%
E 10% 50% 10% 35%
F 10% 60% 10% 45%
G 10% 70% 10% 55%
H 10% 80% 15% 70%
I 10% 90% 15% 85%
J 10% 100% 15% 100%

Let's graph it and see what it looks like.

In summary, the bottom 30% of the population earns 15% of the income, while the top 30% earns 45% of the income.

I've shaded 2 regions in the above graph, region A (with light magenta shading) and region B (with light green shading).

Recall the Gini Coefficient is the ratio of the areas:

$\dfrac{A}{A+B}$

Area A = 0.095 (from calculating area B - one triangle and 2 trapezoids - and subtracting it from 0.5)

Area (A + B) = 0.5 (this is half of the rectangle)

So the Gini Coefficient in this case is:

$\dfrac{0.095}{0.5}=0.19$

Let's take it another step. The three richer guys (H, I and J) have a fight and J wins. He demands 50% of the income and leaves it to H and I to distribute the rest.

Then H and I have a fight and I wins. He wants 33% and gives 10% to H and they decide to give what's left (1% or $1 a day) to each of the rest of the village. (Millions of people live on less than$1 per day.)

Person Proportion of population (%) Cumulative proportion of population (%) Income (%) Cumulative income (%)
A 10% 10% 1% 1%
B 10% 20% 1% 2%
C 10% 30% 1% 3%
D 10% 40% 1% 4%
E 10% 50% 1% 5%
F 10% 60% 1% 6%
G 10% 70% 1% 7%
H 10% 80% 10% 17%
I 10% 90% 33% 50%
J 10% 100% 50% 100%

Now we have a very uneven income distribution. The bottom 70% of the population earn only 7% of the income, while the top 30% earn 93% of the income.

Here's the graph.

The Gini Coefficient for this situation is very high:

$\dfrac{0.355}{0.5}=0.71$

Finally, let's take the extreme case, where "Person J" becomes a dictator and decides all the income should go to him and everyone else gets nothing.

The cumulative income is 0% for Persons A to I, then it jumps up to 100% for Person J. Here's the graph.

This time area A is very large and the Gini Coefficient is:

$\dfrac{0.45}{0.5}=0.9$

Why isn't it equal to 1?

The highest possible Gini Coefficient is 1 and this implies 1 person gets all the income.

In our story, we only have 10 people in our example population. If there were, say, 100 million people in the country, and one person had all the income, then the Gini Coefficient would be 0.999999, or very close to 1.

## Using Calculus to find the Gini Coefficient

The above story is simplified and with a large data set, the Lorenz Curve will appear to be a curve, not a series of straight lines.

This time I have modeled the Lorenz curve using:

Cumulative share of income = (cumulative share of people)5

If we use I (for income) and P (for people), this would be written

I = P5

We find the area A using the following:

$\mathrm{area} \hspace {3}A=0.5 - \mathrm{area } \hspace {3}B = 0.5-\int_{0}^{1} P^5 dP$

This gives:

$0.5-\left[ \frac{P^6}{6} \right]_{0}^{1} =0.5-0.16667=0.33333$

So the Gini Coefficient in this case is very high, at:

$\dfrac{0.3333}{0.5}=0.667$

## Gini Coefficients in Various Countries

These are sorted highest (worst equality) to lowest (best equality).

Country UN Gini Coefficient Rank
Namibia 0.743 1
Sierra Leone 0.629 3
Haiti 0.592 7
South Africa 0.578 10
China 0.469 34
Singapore 0.425 51
United States 0.408 58
India 0.368 76
United Kingdom 0.360 83
Australia 0.352 86
Japan 0.249 110
Denmark 0.247 111

China's coefficient is quite high and this is causing a lot of concern. The Eastern provinces are now well-developed and responsible for most of the income growth, whereas the rural west is still quite poor.

You can see the full list here: Gini Coefficient by Country.

## Singapore's Case

Here's the coefficient for Singapore over the last decade. The rapid rise from 2002 and spike in 2007 were due to several factors, including rapid population increases (through immigration) of higher-income people, and a subsequent boost in the overall economy.

The drop in 2008 and 2009 is due to the Global Financial Crisis, where many high-paying jobs either disappeared, or bonuses were slashed.

Information source for graph: Straits Times, Singapore

You may also be interested in: Database of Happiness.

### 34 Comments on “The Gini Coefficient of wealth distribution”

1. Ian MacFarlane says:

The Gini coefficient for the US, of .408, in the table, above, is quite out of date. That number predates the Republican tax cuts that shifted income from the middle class radically to the rich. We now have a Gini coefficient that is closer to .50 and will not recover from this Great Recession until we redress this issue.

2. Murray says:

Thanks, Ian. I meant to mention the age of the data when I wrote the article.

Most Gini Coefficient data is going to be approximate. In countries where barter systems are common, it would be very difficult to come up with a meaningful Gini Coefficient. Same thing for war-ravaged countries.

But your point is certainly true - I never did understand why Bush (and Reagan before him) was hell-bent on taking from the soon-to-be-poor and giving to the already-rich.

3. Steve says:

I'm in a masters program studying post-conflict environments. Would you have any idea what the Gini coefficient is for Iraq? Or how I can estimate?

Thank you for your time. Excellent explanation.

4. Murray says:

Hi Steve. Sounds like a very interesting Masters program.

Now, what is the Gini coefficient for Iraq? This compilation sidesteps the issue with "NA", as it does for several impenetrable countries like North Korea.

Estimating Gini coefficients for any poverty-struck, politically unstable or war-torn country is nigh-impossible.

My only suggestion would be to talk about the pre-2003 invasion period, and to estimate it on "similar" countries in the region (I'm probably going to make wildly inappropriate cultural gaffes here, for which I apologize). Iran arguably is run by a small elite that (no doubt) holds the reins to the bulk of the oil wealth and arguably (again) it was the same situation in Iraq. So given the estimates of around 43 to 44 (by UN/CIA) for Iran, that may serve as a proxy for Iraq's case, pre-war.

But then again, that estimate could be out ±50%!

5. Jose Toral, DBAc says:

Excellet review and explanation of the Gini coefficient.

the gini co-efficient does not consider whether it is a rich or poor country been dealt with.

7. Murray says:

That is correct, Adu. Gini Coefficient is only concerned about the spread of the wealth, not the total wealth.

8. Wendell Verduin says:

I am trying to understand just where South Africa ranks on the Gini coefficient calculation. Usually its numbers are near the top. Why would there be such a difference between the UN and the CIA calculations for South Africa?

9. Murray says:

@Wendell: The Gini at best is an estimate, even for highly organized and controlled countries. (How do "the authorities" know how many people have millions stashed away in off-shore bank accounts?)

South Africa still has millions of people living in poverty and probably there is little record keeping for such cases. I imagine a lot of whites kept their wealth outside of South Africa before and after Mandela came to power.

Hence UN and CIA estimates are going to be wildly out.

10. Rajiv says:

Murray,

There is a big difference between the Gini Coefficient based on wealth compared to that based on income. While today 2011, the gini coefficient based on income for the US is close to 0.5, give or take 0.02, the gini based on wealth distribution (household net worth) is close to 0.85 -- I believe that it has crept up from 0.834 to 0.865 from 2007 to 2009 -- see Wolff - Recent Trends in Household Wealth in the United States

Also read an interview with Wolff -The Wealth Divide The Growing Gap in the United States Between the Rich and the Rest

While income may denote the ability to live day to day in normal times, wealth denotes the ability to survive when the times are bad. Also wealth, as opposed to income is a measure of the potential political power of the individual/household

11. Murray says:

Really interesting articles, Rajiv and thanks for sharing.

It looks to me like the US is headed for some serious re-adjustment. Few societies survive income and wealth disparities like theirs. The anger against the banker bonuses was only the beginning.

12. Shane says:

I see a lot of talk about tax cuts and increased inequality. Government spending is now near 45% and the income gini is in the 0.45-0.5 range. In 1911 government spending at all levels was under 5%, transfers were essentially zero, and the income gini was in the 0.4-0.45 range. Any thoughts on the role of bureaucratic waste, corruption through tax credits and subsidies to the wealthy, regulation biased towards the wealthy, immigration from poor countries, and trade with poor countries? Maybe income gini is not a even useful measure if the question is how to reduce poverty.

13. Peter Fichera says:

Hmmm...Aside from the various ruling elites fudging the numbers--I'll lay odds that any 'official' stats from the old CCCP would give a number close to 0, even though as a matter of practicality, the entire wealth of the country was in the hands of only a very small cadre of Party members!--there is the problem of standards. What is 'too high' a value, and on who's authority? What about too low? If the analogy is to thermal distribution in gasses, where almost all the work is done by a small portion of very energetic molecules, then perhaps what a society needs is a greater inequality of wealth distribution! As an example, it was Paul Allen, Microsoft partner, who funded the first private vehicle to get a man into space and back, not me--as much as I would have liked to!

pgf

14. Murray says:

@Peter: Yes! If it is difficult (impossible?) to get an "accurate" Gini for the advanced countries, how meaningful is it for developing ones, especially those that still hang on to some communist ideology.

I've seen a few articles on the growing Maoist revival in China - now that will be interesting to watch.

15. Allahisamoongod says:

See here for an up-to-date blog entry on the GINI Index:
http://www.oneutah.org/2010/09/the-gini-index/

16. Tony61 says:

The title says "wealth" but the table of Ginis seems to be "income." Confusing.

17. Murray says:

@Tony61: Rajiv mentioned this dichotomy in an earlier comment. Yes, we are assuming income distribution is roughly correlated to wealth distribution, but this does not hold for many societies.

18. John Seater says:

Congratulations to Murray for providing a very clear explanation of the Gini coefficient.

I would like to see the source of the UN Gini numbers. Using US government data on income (not wealth), one gets a Gini coefficient for all income earners of about 0.5 from 1995 to the present with some small fluctuations but no overall trend. If the UN is talking about income, then I don't know where they get their numbers. Their value of 040 is strongly inconsistent with the US government numbers. I don't know the US Gini for wealth, so if the UN is looking at wealth, then maybe that explains the difference.

For those readers interested in relating the changes in the Gini coefficient over time to US politics, there was a slight downward trend from 1994 to 2008, and after that a slight upward trend. I see no obvious connection to any particular tax policies.

19. How to calculate Gini Coefficient from raw data in Python – Plan Space from Outer Nine says:

[...] The Gini Coefficient is a measure of inequality. It’s well described on its wiki page and also with more simple examples here. [...]

20. Amri says:

Thanks Murray... Can the Gini Coeficient and Lorenzt Curve could be applied in Tax or country fiscal policy? How?

Thanks
Amri Wirabumi

21. Murray says:

@Amri: Well, I guess every government tries to keep in mind the current distribution of wealth before coming up with a fair tax system. (And of course, some governments deliberately design an unfair system, and that is very sad.) They know that income equality (or the impression of income equality) is important for social harmony.

22. Inequalities | elifgunaydin says:
23. Glory says:

What if i have to compute gini coefficient without being given any number? Or how can i use my own assumed numbers to compute gini coefficient? Please can i be helped..

24. Murray says:

@Glory: You could follow the same steps I do in the article. Put your values into a table and then find the cumulative frequency.

25. Jack says:

Well, the information on Singapore is totally out as you did not consider the net effect after Taxes and Transfers, a latest article dated Aug 21, 2015 has concluded that many of these countries, such as the United States, Australia, and Germany, actually have greater income inequality than Singapore. However, their taxes and transfers have done a better job of reducing inequality, to the point where they now have a narrower gap than Singapore. Thank you for your special interest on Singapore, that little red dot.

26. Murray says:

@Jack: I mentioned several times in the comments already that any calculation for Gini coefficient is approximate - for advanced or developing countries. Interesting to ponder, but not possible to conclude anything with any great certainty.

27. Joanna says:

Thank you for the insightful information. I have a question? What would happen to the Gini Coefficient if the government gave a one off payment of \$1,500 to all households in a country? Would the income distribution get better or worse?

28. Murray says:

Hello Joanna

The Gini distribution wouldn't change since it takes into account the way the wealth is shared, not the total value of that wealth.

29. John says:

I was going to ask how I could approximate a formula for the Lorenz Curve given the Gini, but I think I figured it out.
If we assume a curve of the form
y=x^n
then n=0 gives the equality line and the higher n, the greater the inequality, larger A, smaller B etc.
Finding the integral of y=x^n, evaluate at x=1, then
B=1/(n+1)
Now G=1-2B, so
G=1-2/(n+1)
Rearranging,
n=2/(1-G)-1
so
y=x^(2/(1-G)-1)
will provide an approximation to y given x and G.
Voila.

30. Murray says:

@John - Interesting, and it does give a range of reasonable-looking curves. The challenge now it to ensure your curve goes through (0,0) and (0,1).

31. John says:

It works because 0^n=0, 1^n=1, and x^0=1.

32. Murray says:

My bad - I looked at it in a rush. Yes - it certainly does work and I like it. Clever!

Here's how it looks in Desmos: Lorenz Curve - Gini Coefficient model

33. John says:

Thanks. The view in Desmos is instructive. It seems like it may be a little flat at the lower end, i.e it would be better if it was a bit more spherical than parabolic. Not sure how to do that. I want to use it as a quick way to approximate the income distribution given the gini for a large number of countries. Any ideas?

34. Murray says:

I believe you mean "circular" rather than "spherical".

Considering the curves given at the bottom of this page, http://www.physics.ucla.edu/~chester/GINI/, I think your parabolic model is actually not bad at all. None of those idealized curves is circular.

By "approximate the income distribution", do you mean something like determining how much of the wealth the top 20% owns?

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