Problem solving and data analysis
Center, spread, and shape of distributions — Basic example
- [Instructor] Mr. Jadav raised all of his students' scores on a recent exam by 10 points. What effect did this have on the mean and median of the scores? Now, there's two ways we could approach this. One is we could come up with a simple example that meets the constraints, a plausible combination of scores, and then see what is going to be true, which of these statements get contradicted or don't get contradicted. That's one technique, and that might actually be a simpler one if you're under time pressure, under, in something like the SAT. And another technique is to do it a little bit more rigorously. So let's do the simple way first. So when you say, okay, you know, his students, you imagine a classroom with 20 or 30 students, but they don't say it's 20 or 30 students. You could imagine that it could be three students. And if it could be three students, then these statements need to be true for any of the plausible combinations of scores. So let's just think a simple one. Let's just imagine that the three students all got 80s. And I just randomly picked those numbers because it's very easy to calculate both the median and the mean here. The mean here is 80, and the middle score here, the median, is also 80. So both the median, median is equal to 80 here, and the mean is equal to 80. Now, if you add a 10 to all of these, if you add a 10 to all of these, then it becomes a 90, a 90, and a 90. And then your median and mean are both going to become 90. Median is going to be equal to the mean, which is going to be equal to 90. So at least for this case, which is plausible, it's not necessarily the case that they're talking about, but for this case, when you increased by 10, but when you increased all the scores by 10, both the median and the mean increased by 10. Now, let's see what these statements say. The mean increased by 10 points, but the median remained the same. Well, this combination, we just said this was plausible. This could've been Mr. Jadav's students' test scores, but it contradicts this statement. The median didn't remain the same in this case right over here, so we could cross that out. The median increased by 10 points, but the mean remained the same. Well, once again, this little case that we came up with, it's plausible that that was the scores of his students, but it contradicts this statement. So this statement is definitely not true for all the possible combinations of his students' test scores. The mean increased by 10 points, and the median increased by 10 points. Now, this combination we picked, this particular case we picked, this doesn't prove that this is always going to be true, but at least it doesn't contradict it. So we can't cross it out just yet. The mean and the median remained the same. Well, we were able to come up with this case, which is plausible. This could have been his students' scores, and it contradicts this. In order for us to be able to select a statement, we have to feel good that it would be true for any combination of scores that his students had. So we could cross that out as well. So if I'm under time pressure, I'm taking the SAT, I would definitely do this choice and then move on. Now, I'm sure a lot of you probably want a little bit more of a rigorous proof that we could say, hey, like, for any combination of scores, the mean would increase by 10 points, and the median would increase by 10 points. And for that, we could do a little bit of a justification, borderline proof right over here. So let's just imagine, you know, this is the score one, and then this right over here is the median score. And then we keep going, and then this is the nth. That is the nth score right over there. Now, if we added 10 to everything, the order isn't gonna change. So all of these scores, these n scores, are just gonna go up by 10. So this is going to be S one plus 10. And then you're gonna have S two plus 10. And then you're gonna have your former median plus 10. But now this thing's going to be the middle value, and then you're gonna have your highest value. I'm assuming that I ordered these from lowest to highest. And then you're going to have your highest score plus 10. So if all the scores go up by 10, whatever was the, kind of, that median, that's still gonna be in the middle, but now it's gonna be 10 higher. So your new median is going to be 10 higher. So hopefully this justification shows you that, for whatever combination of scores, if you order 'em in this way, and you need to order 'em to figure out the middle value, the median value, that the median indeed would go up by 10. Also, feel good that the mean would go up by 10. So how would you calculate the mean? So the mean is going to be the first score plus the second score, all the way to the nth score, and you're going to divide it by n. That is what, that is going to be equal to the mean. Now, if you added 10 to all of these, if it was S one plus 10 and plus S two plus 10 and we went all the way to Sn plus 10 divided by n, well, what's that gonna be? What's that gonna be? Well, if we take all the 10s out, we're adding 10 n times. So we could rewrite this as S one plus S two plus Sn. And then we have 10 plus 10 n times. So we could write plus 10n and then divide that whole thing, divide that whole thing by n. I just rewrote this. I just, instead of adding the 10 n times, I just wrote 10 times n, or n times 10. Well, this right over here, I could instead write this like this. This is going to be that divided by n plus that divided by n. And what is this value right over here? This is your mean, your old mean. And 10n divided by n, well, it's gonna be plus 10. So your new mean, your new mean is gonna be the old mean plus 10. So hopefully that gives you justification, not just using a special case to rule out that contradicts these other cases, but a justification why this would be true for all combinations of scores. We could feel very good about this.