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Comment by frogpelt | Hacker News Reader

Is it meaningful to compare the median annual income of software engineers to the average annual income of the general population?

I'm legitimately curious, what is the statistical significance of an average versus a median?

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It's statistics 101 to not compare median to average, simply because an average is more impacted by outlier values and so a member of the general population making $15m a year would lift the "average" income far more than it would lift the "median" income. The numbers set "1, 2, 3, 4, 1000" has an average of 202 but a median of 3, and 3 is clearly a better representation of those numbers as a group.

Also we've got multiple data sets at play and so we shouldn't be using a "ratio" but rather a z-score.

A z-score is a measure to normalize all the different distributions and see how many standard deviations away from the average each value is. It's the accepted practice for comparing across two different data sets.

Udacity covers it in their data science course: https://www.udacity.com/course/viewer#!/c-st095/l-81689336/m...

It's not necessarily wrong to compare the median to the average once a sufficiently large dataset is reached given LLN, but I'd argue that a z-score is still far too basic to capture the nuances of such a varying dataset with so many points. Some type of hierarchical model would probably be best for this dataset.

Fair point on the z-score still to simple but better than a ratio.

As far as the LLN, also right but this assumes that PayScale has anywhere close to enough datapoints for that number to be reached which I would question. Also not sure if we can consider the LLN in the same way if you are comparing a total population with a sample here?

That I guess is the other compounding factor actually, the population of Software Engineers is actually just a sample of the overall population of "the general public's income" and so that does impact this as well. We're not comparing group A to group B but rather group A to group ABC...N.

"average income" as reported is usually the median.

Edit: I'm not sure why I'm getting downvotes.

Anywhere you read about average incomes from a serious source it'll be the median, no one uses "mean income" for average when dealing with incomes, since the distribution is so skewed.

If you even just google "average income" it'll report you the median. Median is so completely standard when reporting on income data that it gets short-handed to "average".

Anywhere you read about average incomes from a serious source it'll be the median, no one uses "mean income" for average when dealing with incomes

This is demonstrably untrue. A search for "average income" returns articles referencing both median and mean incomes (and usually clearly distinguishing between the two).

We're on the internet, you can link to examples if it is true.

I didn't downvote you but people probably did because, if all someone says is "average," most people will assume they're talking about the mean absent any other context. Yes, it's a bad measure to use for certain things but lots of bad measures are used in lots of places.

(Now, if a piece talks exclusively and explicitly about medians and then uses a term such as "this average" that's probably OK because they've already stated that median is the only average they're discussing.)

All Google is doing BTW is basically "deciding" that although you asked for average income, what you really wanted is median income because that's what people are looking for. So it basically reworded your query for you.

Average in this article is GDP per capita, which is the mean, not the median.

I think you are probably getting downvoted because the article explicitly says that they are using the mean for the general population:

"Moreover, we compare median annual values for software engineers with mean annual values for the general population. Considering how large the income share of top earners is in several countries, annual median values for the general population might well show a different picture."

It's a mistake to compare the two -- statistics 101.

So, I guess the point of the notebook was probably not the statistics but the scraping and plotting of the data.

And furthermore, the author just used the data that Bloomberg used here: http://www.bloomberg.com/visual-data/best-and-worst/highest-...

Both arithmetic mean and median are averages, so unless they specify what average they used for the annual income, you may only assume what the author meant.

EDIT: Never mind, the author also says:

>Moreover, we compare median annual values for software engineers with mean annual values for the general population

I'd appreciate a link to anyone, anywhere, at any time, using the word average to refer to the median. The median is the middle, not the average, and often nowhere near the average.

http://www.theguardian.com/money/2014/mar/25/uk-incomes-how-...

This is standard statistical terminology, where "average" is a general term used to describe a central tendency of data. Canonical example averages are arithmetic mean, median, mode.

In popular usage, this has become muddled because the arithmetic mean is also an estimate of the peak/center of a Gaussian distribution from a sample distribution.

Note that "arithmetic mean" is what you colloquially call "average" and think of as "the mean" but it is not in any way the only "mean", either.

See e.g. https://en.wikipedia.org/wiki/Average

To understand the significance, lets start by considering that the top 1% earn about 20% of the total income, and 60% of the lowest earners only have about 4.2% of the total income.

If you remove both those extremes, then average and median becomes somewhat closer to each other.

If you removee the bottom 60% of people you aren't representing what is happening very well.

And removing the bottom 60% will most likely remove the median.

More than "likely", by definition, if you remove the bottom 60%, you are removing the median (which is where the bottom 50% and the top 50% meet)

It certainly distorts results where the median and average in the general population differ by a lot. Where I live the average is almost 50% higher than the median.

It's not very meaningful but in general thanks to outliers at the upper income level the average is higher than the median income which means you are underestimating the ratio of median software engineer pay to average income which means that the multiplier could be much higher than the data tells you. If you're a software engineer this is good news for you but bad news if you are an average worker.

You can compare them because they have the same units. Whether or not it makes sense to make statistical conclusions from the comparison is open for debate.

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