<p>Building further on the answer of @1muflon1, the disadvantage of the mean is indeed that it entirely disregards inequality or variation of income. This can be offset (in part) to looking at measures of spread, like the variance or interquartile range.</p>
<p>The economic literature on inequality and welfare measurement contains a large number of concepts to investigate and analyse both inequality and welfare (and their trade-off). One of the most useful concepts (in my opinion) is the generalized Lorenz curve.</p>
<p>Assume that we have a society of <span class="math-container">$N$</span> individuals where individual <span class="math-container">$i$</span> has income <span class="math-container">$y_i$</span>. Let us rank these individuals according to their income, from lowest to highest (breaking ties arbitrarily). Then individual <span class="math-container">$i$</span> is at the <span class="math-container">$\dfrac{i}{N}$</span>th percentile of the income distribution.</p>
<p>The generalized Lorenz curve is a graph with on the <span class="math-container">$x$</span> axis the values of <span class="math-container">$\dfrac{i}{N}$</span> (between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>) and on the <span class="math-container">$y$</span>-axis the value:
<span class="math-container">$$
GL\left(\frac{i}{N}\right) = \dfrac{\sum_{j = 1}^i y_j}{N}.
$$</span>
So it plots the position of individual <span class="math-container">$i$</span> on the versus the total income of all individuals with lower (or equal) income than <span class="math-container">$i$</span> (normalized for the total population). Setting <span class="math-container">$\overline{y}$</span> to be the mean income, we have that:</p>
<p><span class="math-container">$$
GL\left(\frac{i}{N}\right) = \frac{\sum_{j = 1}^i y_j}{N} = \frac{\sum_{j = 1}^i y_j}{\sum_{j = 1}^N y_j}{\overline{y}},\\
 = L\left(\frac{i}{N}\right) \overline{y}.
$$</span></p>
<p>Where <span class="math-container">$L(.)$</span> is the usual Lorenz curve that plots the fraction of income of all individuals poorer than <span class="math-container">$i$</span>. This shows that the generalized Lorenz is simply a rescaling of the Lorenz curve by the average income <span class="math-container">$\overline{y}$</span>.</p>
<p>Notice that <span class="math-container">$GL(1) = \overline{y}$</span> so we can directly see the mean income from the graph. For an example, consider two societies with 4 individuals and with incomes <span class="math-container">$X = (1,2,5,7)$</span>  and <span class="math-container">$Y = (2, 3, 4, 7)$</span> The Lorenz curves are given below (in purple for <span class="math-container">$X$</span> and in red for <span class="math-container">$Y$</span>). We see that the <span class="math-container">$GL$</span> for <span class="math-container">$Y$</span> is entirely above the one of <span class="math-container">$X$</span>.</p>
<p><a href="https://i.sstatic.net/xVyFn.png" rel="noreferrer"><img src="https://i.sstatic.net/xVyFn.png" alt="GL" /></a></p>
<p>One can partially rank different income distributions by seeing if one generalized Lorenz curve dominates another one. This has strong implications in terms of ranking these countries based on some welfare functions.</p>
<p>Let <span class="math-container">$X = (x_1,\ldots, x_N)$</span> and <span class="math-container">$Y = (y_1,\ldots, y_N)$</span> be two income distributions and let <span class="math-container">$W$</span> be some welfare function used to rank distributions, so <span class="math-container">$X$</span> provides more welfare than <span class="math-container">$Y$</span> if <span class="math-container">$W(X) \ge W(Y)$</span>. First, we might impose that <span class="math-container">$W$</span> is increasing (as more income is better) and that it is concave, so we are averse to inequality. For example, we have the following result (I guess this goes back to <a href="http://darp.lse.ac.uk/papersdb/Shorrocks_(Ecca_83).pdf" rel="noreferrer">Shorrocks (1983), Ranking Income Distributions, Economica, 50, p. 3-17</a> )</p>
<p><strong>Theorem</strong>: The generalized Lorenz curve for <span class="math-container">$Y$</span> is above the generalized Lorenz curve for <span class="math-container">$X$</span> if and only if for all concave, increasing social welfare functions <span class="math-container">$W(Y) \ge W(X)$</span>.</p>
<p><em>Proof:</em> Assume the the generalized Lorenz curve of <span class="math-container">$Y$</span> is above the one of <span class="math-container">$X$</span> and that <span class="math-container">$W$</span> is monotone and concave. From concavity we have that:</p>
<p><span class="math-container">$W(X) - W(Y) \le \sum_{j = 1}^N \omega_j (x_j - y_j)$</span></p>
<p>where <span class="math-container">$\omega_j$</span> is the subdifferential of <span class="math-container">$W$</span> at <span class="math-container">$Y$</span> (if <span class="math-container">$W$</span> is differentiable this is just the derivative of <span class="math-container">$W(Y)$</span> with respect to <span class="math-container">$y_j$</span>). From concavity, we see that <span class="math-container">$\omega_j$</span> decreases with <span class="math-container">$j$</span>.</p>
<p>For <span class="math-container">$j &lt; N$</span>, let <span class="math-container">$\Delta_j = \omega_j - \omega_{j+1} \ge 0$</span> and set <span class="math-container">$\Delta_N = \omega_n$</span> then <span class="math-container">$\omega_j = \sum_{i = j}^N \Delta_i$</span>. This gives:
<span class="math-container">$$
\begin{align*}
W(X) - W(Y) &amp;\le \sum_{j = 1}^N \sum_{i = j}^N \Delta_i (x_j - y_j),\\
&amp;= \sum_{i = 1}^N \sum_{j = 1}^i\Delta_i (x_j - y_j),\\
&amp;= \sum_{i = 1}^N \Delta_i \sum_{j = 1}^i (x_j - y_j) \le 0,
\end{align*}
$$</span>
The fact that the last term is negative follows from the assumption that the generalized Lorenz curve of <span class="math-container">$Y$</span> is above the one of <span class="math-container">$X$</span>: <span class="math-container">$\sum_{j = 1}^i x_j \le \sum_{j = 1}^i y_j$</span> for all <span class="math-container">$i$</span>.</p>
<p>For the reverse, let <span class="math-container">$W(Y) \ge W(X)$</span> for all concave, monotone welfare functions. Let <span class="math-container">$\varepsilon &gt; 0$</span>, <span class="math-container">$k \le N$</span>, and define <span class="math-container">$W_{\varepsilon,k} (Y) = \sum_{i = 1}^k y_k + \varepsilon \sum_{i = k + 1}^N y_i$</span>. This is monotone and concave (linear) function. Now if <span class="math-container">$W_{\varepsilon,k}(Y) \ge W_{\varepsilon,k}(X)$</span>, we have:
<span class="math-container">$$
\begin{align*}
&amp;\sum_{i = 1}^k (y_k- x_k) + \varepsilon\left(\sum_{i = k+1}^N (y_k - x_k)\right) \ge 0,\\
&amp;\sum_{i = 1}^k (y_k - x_k) \ge \varepsilon\left(\sum_{i = k+1}^N (x_k - y_k)\right)
\end{align*}
$$</span>
As this holds for all <span class="math-container">$\varepsilon &gt; 0$</span> we get:
<span class="math-container">$$
\sum_{i = 1}^k (y_k - x_k) \ge 0.
$$</span>
This holds for all <span class="math-container">$k$</span>, so we obtain that the generalized Lorenz curve of <span class="math-container">$Y$</span> is above the one of <span class="math-container">$X$</span>. <span class="math-container">$\square$</span></p>
<p>If the two generalized Lorenz curves do cross, we cannot make such comparison. However, there are other &quot;dominance&quot; type results even in these cases (e.g. The GL curve of <span class="math-container">$Y$</span> crosses once the GL curve of <span class="math-container">$X$</span> from above). A good reference is the book of <a href="https://rads.stackoverflow.com/amzn/click/com/0719057329" rel="noreferrer" rel="nofollow noreferrer">Peter J. Lambert, The distribution and redistribution of Income</a>.</p>

CC BY-SA 4.0
2021-07-10T14:45:33.773Z
2021-07-10T14:45:33.773Z
tdm
<ol>
<li>Averages are always skewed by outliers. There is no good solution to that if you <em>insist</em> on using averages.</li>
<li>There is no need for using averages. A common practice when people compare welfare among countries is to use <strong>median</strong> incomes, wages etc. Median is the value for the person who is literally in the middle so in your example above median in group A would be \$10 while median group B would be \$3.</li>
<li>There is no way how one can properly capture inequality using any measure of central tendency. If you want to examine inequality across various metrics you need to examine the distribution and its spread not just its central tendencies. Common way of measuring inequality is to use GINI coefficient, share of income going to top 1% or 10% (or to bottom 99%, 90% or other percentages), or by literally superimposing different income distribution over each other (e.g. plotting cumulative income distribution and seeing if they overlap or one dominates the other, see examples of that in De Kruijk Poverty Dynamics pp 49 &lt;- the book also talks in the same chapter just about poverty not only inequality please do not confuse the two).</li>
</ol>

CC BY-SA 4.0
2021-07-09T12:12:57.757Z
2021-07-09T12:12:57.757Z
1muflon1
<p>I apologize if it is too trivial a question, but I didn't seem to find quick answer and haven't encountered it in my math classes back in high school loads of years ago.</p>
<p>When talking about people's income, as well as many other statistics, people frequently quote averages.</p>
<p>However, averages don't reflect the actual distribution. For example, you can have two groups with the same average amount of money (let's say $10), but divided like this:</p>
<pre><code>Group A:
<span class="math-container">$10  $</span>10  <span class="math-container">$10  $</span>10  $10

Group B:
 <span class="math-container">$1   $</span>2   <span class="math-container">$3   $</span>4  $40
</code></pre>
<p>Now, apparently, there is income inequality in Group B, and it has majority of people poor, with one person having significantly more.</p>
<p>What would be a better way to express the average income that takes into account the income of majority of group's members?</p>

<econometrics>
A better way of expressing income distribution that takes into account the inequality?
Gnudiff
# A better way of expressing income distribution that takes into account the inequality?

(Asked by: username_0 on 2021-07-09)

I apologize if it is too trivial a question, but I didn't seem to find quick answer and haven't encountered it in my math classes back in high school loads of years ago.

When talking about people's income, as well as many other statistics, people frequently quote averages.

However, averages don't reflect the actual distribution. For example, you can have two groups with the same average amount of money (let's say $10), but divided like this:

```
Group A:
$10  $10  $10  $10  $10

Group B:
 $1   $2   $3   $4  $40

```

Now, apparently, there is income inequality in Group B, and it has majority of people poor, with one person having significantly more.

What would be a better way to express the average income that takes into account the income of majority of group's members?

**Tags:** econometrics


---

## 2 answers

### Answer by username_1 on 2021-07-10. Score: 8

Building further on the answer of @username_2, the disadvantage of the mean is indeed that it entirely disregards inequality or variation of income. This can be offset (in part) to looking at measures of spread, like the variance or interquartile range.

The economic literature on inequality and welfare measurement contains a large number of concepts to investigate and analyse both inequality and welfare (and their trade-off). One of the most useful concepts (in my opinion) is the generalized Lorenz curve.

Assume that we have a society of $N$ individuals where individual $i$ has income $y\_i$. Let us rank these individuals according to their income, from lowest to highest (breaking ties arbitrarily). Then individual $i$ is at the $\dfrac{i}{N}$th percentile of the income distribution.

The generalized Lorenz curve is a graph with on the $x$ axis the values of $\dfrac{i}{N}$ (between $0$ and $1$) and on the $y$-axis the value:
$$
GL\left(\frac{i}{N}\right) = \dfrac{\sum\_{j = 1}^i y\_j}{N}.
$$
So it plots the position of individual $i$ on the versus the total income of all individuals with lower (or equal) income than $i$ (normalized for the total population). Setting $\overline{y}$ to be the mean income, we have that:

$$
GL\left(\frac{i}{N}\right) = \frac{\sum\_{j = 1}^i y\_j}{N} = \frac{\sum\_{j = 1}^i y\_j}{\sum\_{j = 1}^N y\_j}{\overline{y}},\\
= L\left(\frac{i}{N}\right) \overline{y}.
$$

Where $L(.)$ is the usual Lorenz curve that plots the fraction of income of all individuals poorer than $i$. This shows that the generalized Lorenz is simply a rescaling of the Lorenz curve by the average income $\overline{y}$.

Notice that $GL(1) = \overline{y}$ so we can directly see the mean income from the graph. For an example, consider two societies with 4 individuals and with incomes $X = (1,2,5,7)$ and $Y = (2, 3, 4, 7)$ The Lorenz curves are given below (in purple for $X$ and in red for $Y$). We see that the $GL$ for $Y$ is entirely above the one of $X$.

[![GL](https://i.sstatic.net/xVyFn.png)](https://i.sstatic.net/xVyFn.png)

One can partially rank different income distributions by seeing if one generalized Lorenz curve dominates another one. This has strong implications in terms of ranking these countries based on some welfare functions.

Let $X = (x\_1,\ldots, x\_N)$ and $Y = (y\_1,\ldots, y\_N)$ be two income distributions and let $W$ be some welfare function used to rank distributions, so $X$ provides more welfare than $Y$ if $W(X) \ge W(Y)$. First, we might impose that $W$ is increasing (as more income is better) and that it is concave, so we are averse to inequality. For example, we have the following result (I guess this goes back to [Shorrocks (1983), Ranking Income Distributions, Economica, 50, p. 3-17](http://darp.lse.ac.uk/papersdb/Shorrocks_(Ecca_83).pdf) )

**Theorem**: The generalized Lorenz curve for $Y$ is above the generalized Lorenz curve for $X$ if and only if for all concave, increasing social welfare functions $W(Y) \ge W(X)$.

*Proof:* Assume the the generalized Lorenz curve of $Y$ is above the one of $X$ and that $W$ is monotone and concave. From concavity we have that:

$W(X) - W(Y) \le \sum\_{j = 1}^N \omega\_j (x\_j - y\_j)$

where $\omega\_j$ is the subdifferential of $W$ at $Y$ (if $W$ is differentiable this is just the derivative of $W(Y)$ with respect to $y\_j$). From concavity, we see that $\omega\_j$ decreases with $j$.

For $j < N$, let $\Delta\_j = \omega\_j - \omega\_{j+1} \ge 0$ and set $\Delta\_N = \omega\_n$ then $\omega\_j = \sum\_{i = j}^N \Delta\_i$. This gives:
$$
\begin{align\*}
W(X) - W(Y) &\le \sum\_{j = 1}^N \sum\_{i = j}^N \Delta\_i (x\_j - y\_j),\\
&= \sum\_{i = 1}^N \sum\_{j = 1}^i\Delta\_i (x\_j - y\_j),\\
&= \sum\_{i = 1}^N \Delta\_i \sum\_{j = 1}^i (x\_j - y\_j) \le 0,
\end{align\*}
$$
The fact that the last term is negative follows from the assumption that the generalized Lorenz curve of $Y$ is above the one of $X$: $\sum\_{j = 1}^i x\_j \le \sum\_{j = 1}^i y\_j$ for all $i$.

For the reverse, let $W(Y) \ge W(X)$ for all concave, monotone welfare functions. Let $\varepsilon > 0$, $k \le N$, and define $W\_{\varepsilon,k} (Y) = \sum\_{i = 1}^k y\_k + \varepsilon \sum\_{i = k + 1}^N y\_i$. This is monotone and concave (linear) function. Now if $W\_{\varepsilon,k}(Y) \ge W\_{\varepsilon,k}(X)$, we have:
$$
\begin{align\*}
&\sum\_{i = 1}^k (y\_k- x\_k) + \varepsilon\left(\sum\_{i = k+1}^N (y\_k - x\_k)\right) \ge 0,\\
&\sum\_{i = 1}^k (y\_k - x\_k) \ge \varepsilon\left(\sum\_{i = k+1}^N (x\_k - y\_k)\right)
\end{align\*}
$$
As this holds for all $\varepsilon > 0$ we get:
$$
\sum\_{i = 1}^k (y\_k - x\_k) \ge 0.
$$
This holds for all $k$, so we obtain that the generalized Lorenz curve of $Y$ is above the one of $X$. $\square$

If the two generalized Lorenz curves do cross, we cannot make such comparison. However, there are other "dominance" type results even in these cases (e.g. The GL curve of $Y$ crosses once the GL curve of $X$ from above). A good reference is the book of [Peter J. Lambert, The distribution and redistribution of Income](https://rads.stackoverflow.com/amzn/click/com/0719057329).


---

### Answer by username_2 on 2021-07-09. Score: 3

1. Averages are always skewed by outliers. There is no good solution to that if you *insist* on using averages.
2. There is no need for using averages. A common practice when people compare welfare among countries is to use **median** incomes, wages etc. Median is the value for the person who is literally in the middle so in your example above median in group A would be \$10 while median group B would be \$3.
3. There is no way how one can properly capture inequality using any measure of central tendency. If you want to examine inequality across various metrics you need to examine the distribution and its spread not just its central tendencies. Common way of measuring inequality is to use GINI coefficient, share of income going to top 1% or 10% (or to bottom 99%, 90% or other percentages), or by literally superimposing different income distribution over each other (e.g. plotting cumulative income distribution and seeing if they overlap or one dominates the other, see examples of that in De Kruijk Poverty Dynamics pp 49 <- the book also talks in the same chapter just about poverty not only inequality please do not confuse the two).


---

