So, let's talk about numerical summary measures. If we want to summarize data using just one number, then, we use the mean or the median. So, everybody knows the mean which is simply the average. The median is the number where half the data are larger and half are smaller. So, it is in some way, a midpoint. Let's look at that histogram which shows annual household incomes in the United States in 2010. Remember that in the histogram, area corresponds to percentage. So, we can find the median by looking at the point where half of the area is below, and half of the area is above. So, here it is marked that the median is at about $50,000. And we can see that roughly half the area of the histogram is below $50,000, and half the area of the histogram is above $50,000. So, now we know that both the mean and the median summarize the data, and the question is which one should we use? It turns out that if the histogram is symmetric, then the mean and the median are actually the same. This histogram shows 100 of the first measurements that were made of the speed of light. The numbers on the horizontal axis show the speed measured after subtracting of 299,000 kilometers per second. We see that this histogram is roughly symmetric around this axis. Other data that result in symmetric histograms would be, for example, heights and weights of people. But when we look at the histogram of annual household incomes, we see that the histogram is far from symmetric. A histogram like this where the right side is much longer than the left side is called skewed to the right. In that case, the mean can be much larger than the median. In fact, for the annual household incomes, the mean household income is about $67,000. Whereas, we saw that the median is $50,000. For such a skewed histogram, it is better to use the median. Let's look at this hypothetical example, where we look at 10 homes and we know that the median sales price is $1 million. Then, we know that five homes sold for $1 million or more, and five homes sold for less than $1 million. On the other hand, if we are told that the average sales price is $1 million, then, it's not really clear what that means. If the average is $1 million, then, we know that the sum of the 10 sales prices is $10 million. Now, let's assume that one house sold for $8 million. Then, we know that the total sales prices for the other nine houses must be $2 million. The average sales price of the other 9 homes is those $2 million divided by 9, and that's about $200,000. So, if we think that the average sales price of $1 million means that most houses sold for roughly $1 million, then we are very wrong in this case. So, the lesson here is that if the histogram is very skewed, as is the case for incomes and house prices, then, knowing the average doesn't really tell you much. It's much better to use the median in those cases. Let's go back to the histogram of annual household incomes. It says that the top 10% reported an income of $135,000, that means that 90% of households had incomes of less than 135,000, and 10% had incomes above $135,000. This point is called a 90th percentile. Furthermore, it says that the top 25% had household incomes above $85,000. This would be the 75th percentile because 75% of households have incomes below that. The 75th percentile is also called the third quartile, that comes from the fact that if one divides the histogram in four equal parts, then one gets the 25th percentile, the 50th percentile, and the 75th percentile. So, the 50th percentile is then the second quartile, and we already learned another name for that which is the median, and the 25th percentile is the first quartile.
