At an art gallery, Jill hung 14 paintings, Kevin hung six paintings, Laurie hung nine paintings, Meg hung eight paintings, and Matt hung nine paintings. Find the mean absolute deviation of the number of paintings hung.
Well first of all, let’s think about how to calculate the mean absolute deviation. This is sometimes abbreviated to MAD, and it’s a measure of how spread out the data is, by how much does it vary from the mean on average. Well, there are really four steps to finding the mean absolute deviation of a set of data. First of all, we have to find the mean of the data, and let’s call that 𝑥 bar. Next for each piece of data, we have to calculate the absolute value of the difference of that piece of data from the mean. Then, we add up all those absolute differences. And lastly, we divide that sum by the number of data elements that we’ve got.
Right, let’s work through those steps with our data. First, how many pieces of data have we got? One, two, three, four, five. So, 𝑛, the number of data elements, is five. Now, we need to work out the mean number of paintings that each person had hung in the art gallery, and 𝑥 bar, the mean, is the sum of all those data elements divided by how many there are. So we’re gonna count up all the paintings that have been hung in; that’s 14 plus six plus nine plus eight plus nine, which makes a total of 46 and then divide that by five. So on average, they hung 9.2 paintings per person in the gallery. So that’s step one complete.
Now we’ve got to calculate the absolute values of the differences from the mean for each person. So Jill hung 14 paintings. The mean was 9.2; the difference between those two is 4.8. In fact, the mean is 4.8 lower than the number of paintings that Jill hung. Now we have to take the absolute value of that, so we’re going to ignore the negative sign. So the number of paintings that Jill Hung varies from the mean number of paintings per person by 4.8. And for Kevin, the difference between the number of paintings he hung in the gallery, which was six, and the mean, which was 9.2, is 3.2. In fact, Kevin hung 3.2 fewer than the mean number of paintings. Jill hung 4.8 more than the mean number of paintings.
And you can see that this absolute value process is ignoring whether it’s under or over; it is just by how much did they differ from the mean number of paintings. Laurie’s nine paintings was a difference of 0.2 from the mean, Meg’s eight paintings was a difference of 1.2 from the mean, and Matt’s nine paintings was again a difference of 0.2 from the mean. So now, we’ve calculated the absolute values of the differences from the mean number of paintings for each person. Now in step three, we’ve got to add all these values up. So that is 4.8 plus 3.2 plus 0.2 plus 1.2 plus 0.2, which gives us a total of 9.6. So that step three done. And all that remains to calculate the MAD is to divide that sum by the number of data elements. So that’s 9.6 divided by five, which gives us 1.92. So that’s all four steps complete. So our answer is that the mean absolute deviation is 1.92.
Now just before we go, let’s try to visualise what that actually means. The mean number of paintings hung by each person in the gallery was 9.2. Jill hung 4.8 more than that, Kevin hung 3.2 fewer than that, Laurie and Matt hung 0.2 fewer than that each, and Meg hung 1.2 fewer than that. Now if I just try to work out the mean of those deviations, I’d work out 4.8 plus negative 3.2 plus negative 0.2 plus negative 1.2 plus negative 0.2 and then divide that by five, the number of pieces of data I had. But the problem with that is if I added all those numbers up, I get an answer of zero divided by five, which will be zero. So on average, the deviation from that mean would be zero.
The positives are balancing the negatives; that’s not really giving us any extra information. So by taking the absolute values of those deviations, I can calculate the average deviation from that mean without worrying about whether they’re above or below. And when I do that, I can see that on average, each person’s number of paintings varies from the mean by about 1.92. Some are above and some are below, but that’s not really the issue. So there we have it, mean absolute deviation.