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### Course: Digital SAT Math > Unit 7

Lesson 4: Center, spread, and shape of distributions: medium# Center, spread, and shape of distributions — Basic example

Watch Sal work through a basic Center, spread, and shape of distributions problem.

## Want to join the conversation?

- how does Sal teach both math and English ; he is very good at both(45 votes)
- Teaching is an interleaving skill that allows you to hone your own skills(5 votes)

- my brain stopped braining (36 votes)
- Instead of complaining, ask questions or tips for u to figure out this lesson. this info won't help you or us but doubt u! so What do u need help with?(0 votes)

- the average marks of boys is 68 and that of the girls is 89.if average marks of all is 80,find the ratio of no.of boys to no.of girls(3 votes)
- The answer is 3:4. The explanation is a bit long, but I hope will help you answer similar questions in the future.

We must know that if the number of boys in the class is equal to the number of girls, then the average mark is the average of 68 and 89, which is 78-79 (the computed average is 78.5, but such a number cannot represent students (humans.)) As we see, the average mark in the question is not 78-79 but 80. It means that the number of boys and girls is not the same. Though it's impossible to find the exact number of students in the class based on the information given, we can determine the ratio of the number of boys to girls by finding the difference between 80 and the two average marks, separately. Since 89-80= 9 and 80-68= 12, the ratio is9:12or 3:4. The question arisen is if this ratio is of boys to girls or of girls to boys. To answer this question, you should simply see to which average mark 80 is closer- to this of the boys or of the girls. 80 is closer to 89 than 68, which means there are more girls in the class, so the ratio of girls to boys is 12:9. The question asks for the ratio of boys to girls which is9:12, or 3:4 if simplified. If you want to think about this question in another way, think about a line segment that contain the point A,B and C when B is between A and C. Now imagine that A is placed further from B that C is. It looks like this: .......A..............B.....C....... This exactly the same case in the question you have written.

Hope it helps! Good luck!(47 votes)

- What class was this learned in? I have gone through HS Pre-Calc and have NEVER gone over this in school. IIRC the SAT covers Alg I, Geometry, and Alg II, not AP Stats.(9 votes)
- Basic Casework Solves The Problem. Mean, Median and Mode should have been taught in your high school's Finite(3 votes)

- are the videos changed? coz the comments and video are completely mismatched(9 votes)
- Khan Academy does sometimes change the video for a skill if they make an updated version that is more intuitive, and this keeps the old comments the same. However, as far as I know this isn't one of those times, at least in the past 3 years. A lot of comments and questions will talk about their own practice problems that are relevant to the topic in the video, and you might have been confused by those.(6 votes)

- how to you find the center of a distribution?(4 votes)
- You find the middle.

1, 2, 3, 4, 5, 6, 7. The middle is 4.

1, 2, 3, 4, 5, 6, 7, 8. (4+5)÷2=4.5. Median is 4.5.(10 votes)

- anyone taking sat in october ?(11 votes)
- Is there going to be questions involving Standard Deviation in new SAT?(6 votes)
- what happens to the range(2 votes)
- We get the range from doing the largest value minus the smallest. If both of these values increase by 10 (the same amount), then the distance between them doesn't change. So the range doesn't change.(9 votes)

- how to calculate median of a random data(2 votes)
- Order the numbers in the data set from smallest to largest and the middle number is the median. If the data set even then divide the 2 middle numbers to find the median.(7 votes)

## Video transcript

- [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.