A farmer is testing two different types of plant food in his garden. He used UltraFeed on five of his plants and Mega Feed on another five. He measured the growth increase of each of the plants and recorded the results, rounded to one decimal place, in the table. Find the mean absolute deviation of the growth of the plants fed UltraFeed. Find the mean absolute deviation of the growth of the plants fed Mega Feed. Which feed produces less variability in the growth?
First, let’s think about what the mean absolute deviation is. Imagine we have a number line. And on it, we plot the mean of our data. And we also plot the other points. When we talk about the absolute deviation, we want to know how far all the data points are from the mean. And the mean absolute deviation is the average of this deviation. To solve for this, we’ll first need to find the mean of both feeds then find the absolute deviation of both feeds and then take the average of that absolute deviation.
We can start by finding the mean of the increase in growth for both types of feed. First, for UltraFeed, we add 1.2 plus 1.4 plus 0.9 plus 1.8 plus 1.7 and then we divide that by five. Once we do that, we find that the average increased growth on UltraFeed is 1.4. Now, for the average of Mega Feed, which will be 1.5 plus 0.6 plus 1.1 plus 1.3 plus 1.0 divided by five. When we do that calculation, we get an average of 1.1.
Now that we have the mean, we can find the absolute deviation for each of the data points. Starting with the UltraFeed and our first data point, to find the absolute deviation, we take the absolute value of the mean 1.4 and subtract 1.2, which is the first data point. That means the first absolute deviation is 0.2. We need to follow the same procedure for the second data point. Since the second data point is exactly the same as the mean, there is no deviation; the deviation is zero. For our third data point, we take the absolute value of 1.4 minus 0.9, which is 0.5. For the fourth point, we need the absolute value of 1.4 minus 1.8, which is 0.4. And for the final point, the absolute value of 1.4 minus 1.7, which is 0.3.
We now need to repeat this process for the Mega Feed. The Mega Feed average is 1.1. To calculate the deviation, we’ll then use the absolute value of 1.1 minus each data point starting with 1.5. 1.1 minus 1.5 is negative 0.4. And we need the absolute value 0.4. For the second point, we have an absolute deviation of 0.5. The third data point is the same as the mean. Therefore, it has a standard deviation of zero. The fourth data point has a deviation of 0.2. And the final data point has an absolute deviation of 0.1.
To have the mean absolute deviation, we need to average all of the absolute deviation data points we found. First, for the mean absolute deviation of UltraFeed, it will be equal to 0.2 plus zero plus 0.5 plus 0.4 plus 0.3 all divided by five. And when we do that calculation, we get a mean absolute deviation for UltraFeed of 0.28. Following the same procedure for Mega Feed, we’ll have 0.4 plus 0.5 plus zero plus 0.2 plus 0.1 all divided by five. And when we do that calculation, we get a mean absolute deviation of Mega Feed as 0.24.
Our final question is asking, which produces less variability? The feed that produces less variability will be the feed whose mean absolute deviation is lower; there is less deviation from the mean. And since 0.24 is less than 0.28, we can say that Mega Feed produces less variability.