Talk:Lorenz curve

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I agree, I had trouble learning about the Lorenz curve from this page because the cdf is different. The cdf plots the cumulative % of the variable against the category. The Lorenz plots the cumulative % of the variable against the cumulative % of population (calculated from the categories). The first sentence and the graph labeling should be revised. —Preceding unsigned comment added by Intlthahc (talk • contribs) 20:37, 12 January 2008 (UTC)

Is it really a "cumulative" dist. fn. ?? The the y-axis gives the sum as a PERCENTAGE of the total, whereas in the cdf, the y-axis will give the un-normalized sum. This will produce a logrithmic-shaped graph rather than an exponential one... Nigel, May 7, 2006

Nigel, after sorting individuals from poorest to richest, the point (x,y), tells us the cumulative number of individuals (on the x-axis) and the cumulative wealth of those individuals (y-axis). A cumulative distribution function would put wealth (not cumulative) on the x-axis and number of people (cumulative) on the y-axis.

Polymer scientists approach the distribution of polymer molecular weights in a polymer samples (same concept as distribution of wealth among individuals) in yet another way. We talk about number distributions and weight distributions (neither one cumulative), often use logarithmic scales for the molecular weight, and use the ratio of the weight average molecular weight to the number average as a measure of dispersity. I would not be surprised to hear of other fields that treat mathematically equivalent problems in still other ways. I suspect that "least confusing" depends on precisely what one's question is and/or what one is already familiar with -- Emil M Friedman — Preceding unsigned comment added by Emilfriedman (talk • contribs) 18:11, 23 September 2014 (UTC)

Convex function?

The curve is apparently always convex. Is there any corner case that it's not convex? — Preceding unsigned comment added by 74.117.104.162 (talk) 12:30, 23 October 2019 (UTC)

• is a convex function.

If the variable being measured can take negative values but has a positive mean, then the Lorenz curve will sink below the line of perfect inequality and is a convex function.

If the variable being measured can take negative values and has a negative mean, then the Lorenz curve will be above the line of perfect equality, except at the end points, and is a concave function.

^ I've edited this out. This convex statement is BS. It can easily be non convex and still be monotonic increasing. The graph pictured is misleading. Jasmine85 (talk) 10:07, 3 September 2009 (UTC)

The Lorenz curve is convexe and not concave as is said in the text, see the graph below.

Can we get a real life example for the illustration from some place? Paranoid 20:10, 6 Jan 2005 (UTC)

The Lorenz Curve is used in geography as well to represent unequal distribution of the world's population over area...please add that in

'... we call this line the line of perfect equality or the 45° line.' It's only 45° when both axes have equal scales. Holy Cow 20:55, 18 March 2006 (UTC)

Discrete Probability Functions

What does a Lorenz curve look like for a discrete probability function? What are the formulas for calculating the curve in such a case? DCary 04:19, 26 May 2006 (UTC)

x(F)=F^-1(f(x))?

Is x(F), the inverse of F(x), equal F^-1(f(x))? --Kwj2772 (talk) 13:21, 17 July 2009 (UTC)

Income or wealth?

The article waffles back and forth on whether this curve typically represents income or wealth, which are completely distinct measures. Which one is kind of important. --75.94.164.123 (talk) 14:11, 28 June 2011 (UTC)

Needs examples of both! JdelaF (talk) 07:10, 12 March 2023 (UTC)

convex and increasing

would like an explanation (it may have been buried in the math, didn't look) as to why the curve cannot hump above the perfect equality line when non-negative variables are used. It would seem that a decreasing increase is possible. — Preceding unsigned comment added by Barnsward (talk • contribs) 19:42, 15 August 2011 (UTC) never mind, i see that if all are lined up in ascending order on the x axis there has to be an increasing increase. — Preceding unsigned comment added by 69.123.186.12 (talk) 23:17, 15 August 2011 (UTC)

Mean Deviation Proof

Is there a proof of the below, can some one add citation, or simply prove it.

$F(\mu )-L(F(\mu ))={\frac {\text{mean deviation}}{2\,\mu }}$

Definition and calculation

In "Definition and calculation" section: Please define Fi, Si, and Li; or at least F, S, and L. I couldn't find any "...where S equals..." Thank you 71.139.166.86 (talk) 23:12, 22 February 2014 (UTC)

Better graph suggestion

I think the image found here is very helpful. It shouldn't be difficult for someone with basic skills in graph making (which I lack) to recreate it in a free version. --_{Piotr Konieczny aka Prokonsul Piotrus| reply here} 09:08, 29 April 2014 (UTC)

PS. Another very useful graph to add would be a practical example of a lorenz curve. For example, for the world ([1]). Or comparing some major countries, like in OECD. We have the File:Lorenz curve of Denmark, Hungary, and Namibia.png, which I'll add, but the countries on it are "meh" (who cares about them, with all due respect...). --_{Piotr Konieczny aka Prokonsul Piotrus| reply here} 09:24, 29 April 2014 (UTC)

My image-making skills aren't great, but I'll try...

What's wrong with Denmark, Hungary, and Namibia? :-) We're a little too USA-centric around here anyway. Personally, I like the idea of showing a curve shifting over the course of a few decades - it's not too hard to get time-series data for most interesting subjects. Anyway. How about we pick a couple of different countries that make good examples (I don't want the graph to get too crowded), and from there it's easy to plug the data into a graph. bobrayner (talk) 18:34, 1 May 2014 (UTC)

@Bobrayner: Were you able to work on the image? I agree with trying to avoid US-centrism; at the same time we have to consider the usefulness of such graphs. Most people will want to learn about US than Nambia or Hungary. Ideally we would have graphs for all countries, of course... --_{Piotr Konieczny aka Prokonsul Piotrus| reply here} 07:44, 6 May 2014 (UTC)

OK: Arbitrary choice time. How about America, Angola, and Argentina? Gives us some contrasting economies whilst still including the USA.

There's no point in me building a graph til we've decided what data to put in it. :-) bobrayner (talk) 14:44, 7 May 2014 (UTC)

The World Bank has income quintiles (plus top and bottom decile) which would give a basic graph for each country, but the datasets are surprisingly incomplete - they have recent numbers for Argentina but not the USA or Angola, using the examples above. In fact, they don't have post-1999 income quintiles for any of the big rich anglophone countries. I'm wary of stitching together stats from each country's statistical authorities into a single graph. Suggestions? bobrayner (talk) 15:42, 9 May 2014 (UTC)

Re: helpful images.

Why isn't there a simple graph illustrating the 80:20 rule?

… and extending it to (64:4), (51.2:0.8), etc. Also GINI/Lorenz curves (p-p plot) should exhibit the Pareto. Also why no mention of the form of the eq for the GINI/Pareto of the simple form Y = 1-(1-x)^[log.8/log.2]? BTW, imho Lorenz curves using the GINI format should be the most standard way of illustrating income &/or wealth distributions! I don't know why the US Census or FedReserve folks don't use them. JdelaF (talk) 07:28, 12 March 2023 (UTC)

The real world sample graphic isn't showing Denmark at all, contrary to its caption. Lewis Goudy (talk) 16:00, 25 March 2017 (UTC)

Who is on the bottom?

In the case of perfectly equal wealth distribution, why is the lorenz curve considered to be a straight line? For example, in that case, how would you decided which 10% of the population is the "bottom" 10% ? As a mathematical technicality, don't we have to assume that individuals with equal wealth will be ranked in some arbitrary order?

Tashiro~enwiki (talk) 15:34, 9 October 2016 (UTC)

The independent variable is not the set of individuals but rather the size of that set as a fraction of the size of the total population. We are modeling a discrete situation using continuous mathematics, so we represent the cumulative presentation as a piecewise linear function rather than a step function. In the equal wealth case that function turns out to be a straight line, which is what we would get in the limit of large population if we had used a step function. Lewis Goudy (talk) 16:14, 25 March 2017 (UTC)