Estimating mean and median in data displays | AP Statistics | Khan Academy

We are told researchers scored 31 athletes on an agility test. Here are their scores; it's in this histogram. And what I'm going to ask you is which of these intervals, interval A, B, or C, which one contains the median of the scores and which one, or give an estimate of which one contains the mean of the scores.

Pause this video and see if you can figure that out.

So let's just start with the median. Remember, the median you could view as the middle number, or if you have an even number of data points, it would be the average of the middle two. Here we have an odd number of data points, so it would be the middle number. So what would be the middle number if you were to order them from least to greatest? Well, it would be the one that has 15 on either side, so it would be the 16th data point.

16th data point, and so we could just think about which interval here contains the 16th data point. You could view it for the 16th from the highest or the 16th from the lowest; it is the middle one.

All right, so let's start from the highest. So this interval C contains the 13 highest data points, and then interval B goes from the 14th highest all the way to the 18th highest. So this B contains the median; it contains the 16th highest data point, or if you started from the left, it would also be the 16th lowest data point. So that's where the median is, the median.

Now, what about an estimate for the mean? Well, you have calculated the mean in the past, but when you're looking at a distribution like this, when you're looking at a histogram, one way to think about the mean is it would be the balancing point. If you imagine that this histogram was made out of some material of, let's say, uniform density, where would you put a fulcrum in order to balance it?

If you put the fulcrum right over here, it feels like you would have all—it feels like you would tip over to the left because this is a left-skewed distribution; you have this long tail to the left. If you really wanted to balance it out, it seems like you would have to move your fulcrum in the direction of that left skew, in the direction of the tail.

And so I would estimate to balance it out, it would actually be closer to that, which would be interval A. Interval A would contain the mean. The intention of this type of exercise isn't for you to try to calculate every data point. In fact, they don't give you all the information here and add them all up and then divide by 31.

It's really to estimate and to also get the intuition that when you have a left-skewed distribution like this, you will often see a situation where your mean is to the left of the median. If you have a right-skewed distribution, it would be the other way around.

As we will see, when you see a symmetric distribution, the mean and the median will be awfully close to each other, or when you have a roughly symmetric distribution. If you have a perfectly symmetric distribution, they might be exactly in the same place.

So let's do another example. So here it says we have the ages of 14 co-workers, and what I want you to do is say roughly where is the mean and roughly where is the median? Is it roughly at A, is it roughly at B, or is it roughly at C?

Pause this video and try to figure it out.

So let's first start off with the median. We have 14 data points, so this would be the average of the middle two data points. It would be the average of the seventh and eighth data points. Well, you could say one, two, three, four, five, six, seven, and then the eighth one is here.

So the seventh data point is a 30, and the eighth one is in the 31 bucket, so the average of the two would give you 30.5. Another way that you could think about it is you can just eyeball it and see you have just as many data points below B as you do have above B, and so that also gives you a good indication that B would be where the median is.

So that is where the median is. Now, what about the mean? Well, this is a perfectly symmetric distribution. If I wanted to balance it, I would put the fulcrum right in the middle. So I would say that the mean would also be at B.

And we are done.