ECON 102: Inequality Formulas
In this document, we present the most important formulas used in Module 7.
Income Share
To illustrate, suppose the population
39 104 76 143 103 32
is composed
193 114 121 111
of 20
134 54
individuals with the following incomes:
183 61 75 99 130 117 152 57
The first step is to sort the income:
32 39 54 57 61
75 76 99 103 104
111 114 117
Income Share by Quintile
121 130
134
143 152 183 193
We group the population in five equally populated groups from the poorest to the richest and we compute the total income in each group:
Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5
Total 32 39 54 57 182 61 75 76 99 311 103 104 111 114 432 117 121 130 134 502 143 152 183 193 671
Then, we divide the total of each group by the total income. For example, the income share in the first quintile is
182 × 100 = 8.6749% 182+311+432+502+671
The income shares by quintile in percentage are therefore:
Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5
8.6749 14.8236 20.5910 23.9276 31.9828
Econ 102
Module 7: Formulas
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Cumulative Income Share by Quintile
The cumulative sum of a list of numbers {X1,…,Xn} is the following list:
X1
X1 + X2
X1 +X2 +X3
X1 +X2 +X3 +X4 .
X1 +X2 +X3 +X4 +···+Xn
Apply this definition to the income shares and we obtain the cumulative income shares. The second column is the cumulative population share (20% of population are in each quintile).
Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile 5
Gini Coefficient From Quintiles
Cum. income share 8.7 23.5 44.1 68.0 100.0
Population Share 20.0 40.0 60.0 80.0 100.0
Let hi be the cumulative income share for quintile i not expressed in percentage, then the Gini coeffi- cient is:
G=0.8−0.4×(h1 +h2 +h3 +h4) In this case, it is equal to 0.2228.
Income Share by Decile
We group the population in 10 equally populated groups from the poorest to the richest and we compute the total income in each group:
Decile 1 Decile 2 Decile 3 Decile 4 Decile 5 Decile 6 Decile 7 Decile 8 Decile 9 Decile 10
Then, we divide the total of each group by shares by decile in percentage are therefore:
Total 32 39 71
54 57 111 61 75 136 76 99 175
103 104 207 111 114 225 117 121 238 130 134 264 143 152 295 183 193 376
the
total income as for the quintile case. The income
Econ 102 Module 7: Formulas Page 2 of 3
Decile 1 Decile 2 Decile 3 Decile 4 Decile 5 Decile 6 Decile 7 Decile 8 Decile 9 Decile 10
Cumulative Income Share by Decile
3.3842 5.2908 6.4824 8.3413 9.8665 10.7245 11.3441 12.5834 14.0610 17.9218
We apply the same formula presented earlier for the Cum. income share
Decile 1 Decile 2 Decile 3 Decile 4 Decile 5 Decile 6 Decile 7 Decile 8 Decile 9 Decile 10
Gini Coefficient From Quintiles
3.4
8.7 15.2 23.5 33.4 44.1 55.4 68.0 82.1 100.0
quintiles:
Population Share 10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
100.0
Let hi be the cumulative income share for decile i, then the Gini coefficient is: G=0.9−0.2×(h1 +h2 +h3 +h4 +h5 +h6 +h7 +h8 +h9)
In this case, it is equal to 0.2324.
The Median
The median is the number that separates a sample in two equal groups after sorting the observations.
Odd Number of Observations
If the series is {X1,X2,X3,X4,X5} with X1 < X2 < X3 < X4 < X5, the median is equal to X3. Even Number of Observations
If the series is {X1,X2,X3,X4,X5,X6} with X1 < X2 < X3 < X4 < X5 < X6, the median (one version of it) is equal to (X3 + X4)/2.
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