What the hell is a median, and why should I care?

Often times, as I peruse the various blogs around real estate, particularly ones that are focused on the Bay Area, I will see many misconceptions about what exactly the definition of "median" is. And this misconception isn't limited to amateurs and newcomers, as even professional Realtors (such as here, and here) are prone to confusion about the term.

It's a fairly common mistake to confuse median with mean (more commonly known as "average").

In simple terms, the median is literally the number in the middle.
So in a three number set such as {1, 9, 10}, the median is 9, while the mean (or average) would be 6.66 (1+9+10=20, 20 divided by 3 = 6.66).

If there are an even amount of numbers in a set, the median is taken by dividing the two in the middle by 2. So in a four number set {1,8,9,10} the median is 8.5 (8+9=17 divided by 2 = 8.5). While the average in this case would be 7.

So how does this apply to San Francisco real estate, and why should we care? Well because San Francisco has a mix of properties selling at very different price tiers. And the result, as I am about to show, is that trying to determine short term market trends by looking at median prices can give you some very misleading answers.

Let's imagine 3 different scenarios:

Scenario 1
If you have 1,000 properties sell for $1,000, then the median is $1,000 and the average is $1,000.
Scenario 2
If you have 999 properties sell for $1,000, and one property sell for $2,000,000 then the median is still $1,000, but the average is now $2,999.
Scenario 3 (i.e. the "mix" scenario)
If you have 250 properties sell for $500, 250 properties for sell $1000, 250 properties sell for $2000, and 250 properties sell for $1 million, then your median is $1500, and your average is $250,875.

Obviously no one is buying a house for $250k in scenario 3, they are either paying a lot less, or a lot more, and that's why medians are used more often then averages in RE, because it tells us what people are actually paying, and it's a lot less susceptible to being manipulated by the top 2%.

Now, using these same scenarios, let's say all properties go down in price by 20% and also the bottom portion of the market doesn't sell at all due to a 25% reduction in sales volume.

What does that do to our medians and averages?

Scenario 1
Median and average both drop to $800.
Scenario 2
Median drops to $800, average drops to $2932
Scenario 3 (i.e. the "mix" scenario)
Despite a catastrophic bust in real estate, the median has risen to $2000, and the average has also risen to a whopping $534,000. Weird isn't it? Prices are down 20% and sales volume is down 25%, but the median went up 33%, and the average went up 112%!

When you have a relatively small sample size like we have in SF, and wide variances in pricing tiers based on micro-neighborhoods, it is really difficult to extract anything useful out of a city wide median or average, and that's assuming all things are equal. Add in the value of property improvements and it's just plain impossible. Prices could be going up city wide, or just in nice neighborhoods, the lower priced property could just not be selling at all, quality could be improving, or it could be a mixture of any or all of these. In short: there's no way to tell what the market is doing just by looking at median prices.