The table shows the lengths of 18 pythons in feet. The outlier measurement of 32 feet is an error. Which of the mean, median, or mode of the listed values would be most affected if the outlier was removed? The values are 18, 18, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23, 23, 23, 23, 24, 24, and 32.
So in order to see which one of our averages would be the most affected, what we’re gonna do is work them out. And we start with mode because this is the most straightforward. And the mode is the most common value. In our case, it would be the most common number.
So if we take a look at our table, we have two 18s, four 20s, two 21s, two 22s, five 23s, two 24s, and then finally one 32. So therefore, if we look at the number that appears the most, it’s gonna be 23. So therefore, the mode is going to be equal to 23. Be careful because a common mistake that students make is that they say it’s five because they write down the number of times that 23 appears. But it wouldn’t be that. The mode would be 23 itself.
Okay, so now, let’s take a look at what the mode would be if we removed the outlier. So once we remove the outlier, our mode would not be affected because there was only one 32. So our mode would still be 23. So therefore, we can say there will be no change in the mode whether the outlier is removed or not. Okay, great, now let’s move on to the median.
So the median is the value that’s in the middle of all of our values when they’re put in numerical or number order. So if we take a look at the table, we can see that this already is in number order because it goes from 18 all the way up to 32. So we have a couple of ways to work out the median. There’s one way that we can use; it just involves crossing off values. And the other way is by using a small formula to help us find what the middle number or middle value is going to be.
So to help us visualize what’s going on, I’ve drawn all our numbers in a row. We could have done it in the table; I’ve just done it like this so it’s nice and easy to see what we’re doing. So the first method I mentioned was crossing off. So by crossing off what I mean is we cross out the lowest value and the highest value and then we work our way up and down until we get to the middle value. So our next values crossed off will be 18 and 24, then 20 and 24, then 20 and 23, 20 and 23 again, 20 and 23 again, 21 and 23, then finally 21 and 23 again.
So then we’re left with two values and we get two values because we’ve got an even number of values. And what we need to do is find the middle of these two values or the average of these two values. Well, to do that, what we do is we add the two values together and divide them by two, so 22 and 22 divided by two which is 44 divided by two which is just 22. But in our scenario, this will be quite obvious anyway because they’re the same value. We can surmise that the middle or the average of them is going to be 22. So therefore, our median is 22.
So the small formula that I mentioned before is this one here, which is 𝑛 plus one over two. And this is what we’re gonna use for this other method. And what this tells us is which value is going to be the median or middle value. And this is 𝑛 plus one over two, where 𝑛 is the number of values that there are. So in our scenario, we’re gonna have 18 plus one because there’s 18 values divided by two. So that’s 19 divided by two. So a half of 19 is 9.5. So therefore, we can say that the median value is gonna be the 9.5th value.
What that means in real terms is it’s gonna be halfway between the ninth and the 10th values. So if we count up our values, we can see that the ninth and 10th values are gonna be the same as we had previously. So therefore, we’ve got 22 and 22. So that’s our ninth and 10th values. So halfway between them is gonna be the same as we found out earlier, 22. So we get the same median which is 22.
So now what we need to do is see how our median will be affected if we remove the outlier. Well, the outlier was 32. So I’ve now removed the 32. So what we’re gonna do now is use the crossing out method to start with to see if the median has changed. So when I do that and I use crossing out method, again I’m left with a middle value of 22. This time, I’m only left with one middle value. And that’s because we have an odd number of values cause we got 17 values. So because it’s 22, that means our median again is gonna be 22.
And then if we use the other method just to double-check, we’re gonna have 17 plus one. It’s 17 because we removed one of our values; we removed 32. So 17 plus one over two is 18 divided by two which is nine. So we take a look at our ninth value; it’s 22. So we get the same median. So yet again, we’ve got no change. So our median and our mode have no change if the outlier is removed. So therefore, it’s looking like it’s gonna be the mean. But now, we’re gonna work that out.
So to calculate the mean, what we do is we add together all of our values and then divide by however many there are. So what we get is 397. And that’s because if we add all our values together, we get 397. Then we divide this by 18. And then, if we work this out on the calculator, we get 22.05 recurring. So now, what we can do is calculate the mean after we’ve removed the outlier, which is the 32. So then what we’re gonna get is 365. And that’s because that’s what we have. We added together all the remaining values. However, we wouldn’t have to add them all together. And that’s because what we could do is just remove or subtract 32 from 397. And that’s cause we know that we’ve removed the outlier which is 32. And then, this is divided by the 17 values that we have left. And when we do that, we get 21.47 et cetera.
So therefore, we can clearly see there’s a decrease in the mean from before the outlier is removed to after the outlier was removed. So therefore, we can say that an answer to the question “of the averages, the mean would be the most affected if the outlier was removed.”