<p>No, since you can change the Gini coefficient by altering the distribution of incomes in the top half (keeping the total the same) without changing the mean or median income; you can also change the Gini coefficient by altering the distribution of incomes in the bottom half without changing the mean or median income.</p>

<p>Compare the Gini coefficients of {2,2,2,2,2,6,10,10,10,10,10} and of {1,1,1,1,6,6,6,6,6,6,26}: the former will have a Gini index of about 0.33 while the latter will have a Gini index of about 0.46 though the mean and median do not change (6 each)  </p>

CC BY-SA 3.0
2017-07-04T06:36:38.577Z
Your numerical example does not coincide with the two general examples in your first paragraph. Although the counterexample is enough to prove the "No" of your answer, you have not provided a proof of the first statement.
luchonacho
CC BY-SA 3.0
2017-07-04T07:36:23.627Z
@luchonacho: I regarded my first paragraph as an obviously true statement (you can change the distribution and Gini coefficient without affecting the mean and median), and the answer to the question.  My numerical example changes the distribution both ways: in the top half from  10,10,10,10,10 to 6,6,6,6,6,26 (both adding up to 50) and in the bottom half from 2,2,2,2,2 to 1,1,1,1,6 (both adding up to 10), so the mean is unchanged and the median of 6 is unaffected.  Not surprisingly, each of these changes increases the Gini coefficient as a measure of inequality
Henry
CC BY-SA 3.0
2017-07-04T10:15:16.260Z
My point is that the answer looks misleading, as it seems you are giving an example to support the statement in the previous paragraph. I think a clarification would improve it.
luchonacho
CC BY-SA 3.0
2017-07-04T12:32:18.037Z
@luchonacho Thank you so much for your answers. I am facing this problem since i am testing the median voter theorem, which states that as the gap between mean and median income increases demand for redistribution increases.

If using gini in market incomes to test this as a lot of journals do, they implicitly assume that an increase in this gap is reflected in the gini ? I know that the same gini may cover different distributions but if mean rises realtive to median gini should increase as well no?
Madslassen
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2017-07-04T12:41:51.207Z
As Milanovich states:" 
When individuals are ordered according to their factor or market incomes, the median voter, the individual with the median level of income, will be, in more unequal societies, relatively poorer. His or her income will be lower in relation to mean income"
Madslassen
CC BY-SA 3.0
2017-07-04T12:45:07.143Z
@Madslassen:   It is common for median income or wealth to be below mean income or wealth for many unequal distributions but, as my examples show, this need not be the case.  It may be true sometimes that increased inequality leads to social pressure to reduce it (but consider the last US Presidential election as a possible counter-example).  So there might in some circumstances be a correlation between the mean/median ratio and pressure for reform.  But to argue there is a direct causal relationship would probably be hard.
Henry
CC BY-SA 3.0
2017-07-04T12:51:17.817Z
@Madslassen:  I have given a possible counterexample to your Milanovich quotation, but I do not claim it is realistic.  For example [in the UK in 2015/16](https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/600091/households-below-average-income-1994-1995-2015-2016.pdf), median equivalised net household income per person before housing costs was about £481 per week while the mean was about £593, with a Gini coefficient of about 0.35 and the 97th percentile around £1470.
Henry
CC BY-SA 3.0
2017-07-04T13:09:45.990Z
Thanks Henry. As I understand, it would not be proper to test the Median Voter Theorem by then using the gini, but instead one should use the mean/median ratio or something.

A final thing: The fact that different income distributions can have the same gini I guess is what we illustrate with crossing lorenz curves?
Madslassen
CC BY-SA 3.0
2017-07-04T13:11:54.400Z
Yes - The Gini coefficient is proportional to the area of between the Lorenz curve and the diagonal, and it is quite possible to have different shapes with the same areas, though the curves of the different shapes would then have to cross at least once
Henry
CC BY-SA 3.0
2017-07-04T13:25:54.253Z
@Madslassen I suggest you ask a new question about the median voter issue, as it is not directly related to this one.
luchonacho
CC BY-SA 3.0
2017-07-04T02:03:57.610Z
2017-07-04T02:03:57.610Z
Henry
<p>Simple question: If the GINI coefficient (market incomes) goes up, is it a necessary feature that the mean income rises relative to the median? </p>

<inequality>
Gini coefficients and median / mean income ratio
Madslassen
# Gini coefficients and median / mean income ratio

(Asked by: username_0 on 2017-07-03)

Simple question: If the GINI coefficient (market incomes) goes up, is it a necessary feature that the mean income rises relative to the median?

**Tags:** inequality

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## 1 answers

### Answer by username_1 on 2017-07-04. Score: 1

No, since you can change the Gini coefficient by altering the distribution of incomes in the top half (keeping the total the same) without changing the mean or median income; you can also change the Gini coefficient by altering the distribution of incomes in the bottom half without changing the mean or median income.

Compare the Gini coefficients of {2,2,2,2,2,6,10,10,10,10,10} and of {1,1,1,1,6,6,6,6,6,6,26}: the former will have a Gini index of about 0.33 while the latter will have a Gini index of about 0.46 though the mean and median do not change (6 each)

> **Comments:**
> **username_2**: Your numerical example does not coincide with the two general examples in your first paragraph. Although the counterexample is enough to prove the "No" of your answer, you have not provided a proof of the first statement.  
> **username_1**: @luchonacho: I regarded my first paragraph as an obviously true statement (you can change the distribution and Gini coefficient without affecting the mean and median), and the answer to the question.  My numerical example changes the distribution both ways: in the top half from  10,10,10,10,10 to 6,6,6,6,6,26 (both adding up to 50) and in the bottom half from 2,2,2,2,2 to 1,1,1,1,6 (both adding up to 10), so the mean is unchanged and the median of 6 is unaffected.  Not surprisingly, each of these changes increases the Gini coefficient as a measure of inequality  
> **username_2**: My point is that the answer looks misleading, as it seems you are giving an example to support the statement in the previous paragraph. I think a clarification would improve it.  
> **username_0**: @luchonacho Thank you so much for your answers. I am facing this problem since i am testing the median voter theorem, which states that as the gap between mean and median income increases demand for redistribution increases.  If using gini in market incomes to test this as a lot of journals do, they implicitly assume that an increase in this gap is reflected in the gini ? I know that the same gini may cover different distributions but if mean rises realtive to median gini should increase as well no?  
> **username_0**: As Milanovich states:"  When individuals are ordered according to their factor or market incomes, the median voter, the individual with the median level of income, will be, in more unequal societies, relatively poorer. His or her income will be lower in relation to mean income"  
> **username_1**: @Madslassen:   It is common for median income or wealth to be below mean income or wealth for many unequal distributions but, as my examples show, this need not be the case.  It may be true sometimes that increased inequality leads to social pressure to reduce it (but consider the last US Presidential election as a possible counter-example).  So there might in some circumstances be a correlation between the mean/median ratio and pressure for reform.  But to argue there is a direct causal relationship would probably be hard.  
> **username_1**: @Madslassen:  I have given a possible counterexample to your Milanovich quotation, but I do not claim it is realistic.  For example [in the UK in 2015/16](https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/600091/households-below-average-income-1994-1995-2015-2016.pdf), median equivalised net household income per person before housing costs was about £481 per week while the mean was about £593, with a Gini coefficient of about 0.35 and the 97th percentile around £1470.  
> **username_0**: Thanks Henry. As I understand, it would not be proper to test the Median Voter Theorem by then using the gini, but instead one should use the mean/median ratio or something.  A final thing: The fact that different income distributions can have the same gini I guess is what we illustrate with crossing lorenz curves?  
> **username_1**: Yes - The Gini coefficient is proportional to the area of between the Lorenz curve and the diagonal, and it is quite possible to have different shapes with the same areas, though the curves of the different shapes would then have to cross at least once  
> **username_2**: @Madslassen I suggest you ask a new question about the median voter issue, as it is not directly related to this one.  

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