All right, guys, it's time. Welcome to another webinar session. Today we're gonna talk about statistics. I hope you are doing well and it is your gift on the world. Please check that one and share your ideas with us. Let me start timer. One person says C. Okay. C got one vote so far. What about others? Okay, another friend says, answer should be C. Okay guys, just a reminder, please, share your answer with Soham. I will check the answers too, but I might not check as fast as him, so. We got two more votes for C. Gotcha. Okay, it has been 47 seconds. Maybe we can just wait one more friend to share their answer with us. So, do we get just one more answer or not? Not yet, I guess. Okay, we got one more for C. Okay, we got one more, cool. Alright, guys. So, the arithmetic mean of nine consecutive whole numbers equal to 2006. The largest of these nine numbers is... First of all, you gotta remember how to find arithmetic mean. Okay? Also, you need to remember how does consecutive numbers work. Finding arithmetic mean, guys, you gotta add all of the numbers, like let's say you have five numbers, you will add five of them and divide by five. In that case, since we have nine numbers, we gotta add nine numbers then divide by nine. And the result is 2006. But here is the deal, guys. If we say we just get... more friends coming, I cannot admit them for some reason, I believe we are good. Okay. So, if we say, guys, we have all of those different variables represents nine different numbers, it might be a little difficult. You need to work with what does consecutive numbers mean. Well, for those case, guys, if you know that, like for example, five and six and seven are consecutive numbers. But it's not about giving us example, it's about how can you rewrite the numbers in terms of others. Like let's say we gotta find the largest of these nine numbers here. We can say we have a smaller number, middle number, and the biggest number. If we compare only three different numbers, if we say smaller number here is x, the middle number should be x plus one. There is only one difference because they are consecutive, and the biggest number should be x plus two. And we can work from there. So, if you wanna say, guys, since we have nine numbers, let's say smallest number is x, then we can go from there. Let's see what we're gonna have. So, the first number, second, third, fourth, fifth, sixth number, seventh number, eighth number, hope it's not the comma, and ninth, ninth number. We have all of those numbers. Then we gotta divide this term by nine, then you gotta get 2006. But I feel like some of your friends are saying, Sir, no, what are you doing? Look, yes, we always use some of the easiest or shortcut ways. Even that seems algebraic way, guys, I wouldn't use that. What I mean, I can still use setting up equations, but not necessarily this way. I believe we have been discussing this classroom. You need to use some shortcut ways. Why? Look, if we have nine numbers, guys, and we know that nine is an odd number, that means we know some of the number, whole number is exactly in the middle. Let's just say, in the middle number is x. Why not? If you know the rules, you can play however you want. This is how we do algebra in math. The middle number is x, guys. Then starting x, I am going to work with one of the next number and one of the previous numbers. We should get x plus one and also x minus one. Interesting, keep going. We can say x plus two, then x minus two. So far, we have five numbers. We just add two and two each side. We have x plus three and also x minus three. We have x plus four. Also, we have x minus four, guys. At the end, you add all of them and divide by nine to get two thousand six. This is how it works, you know. As you see, I am not going to continue with the top one because then you just keep adding extra numbers. When you do more and more steps, more operations, it's going to confuse you. Why do I choose these numbers? Because, guys, if you pay attention here carefully, we can cancel some of these terms. Minus four and plus one. God, we are good. Minus three, plus three. Why not? Minus two, plus two. Yep. Minus one, plus one. Exactly. We have only nine x here at the top. You can count. And we have nine x divided by nine is equal to two thousand six. I believe you all know how to simplify this. It literally gives us x is two thousand six. You see, without doing too much effort, we find x, which is the number in the middle. Oh, it's here. My bad. It should be here, this one. The number in the middle. So, here is the thing, guys. You know, some questions are a little tricky. Even if it's number six, you may just say, oh, I found two thousand six. It's not in answer choice, but sometimes they just give you two thousand six and you jump in. Don't do that. So, we are going to find x plus four, guys, which is the largest. As you know, largest means we got to do plus four, whatever number you found. And, of course, two thousand six plus four should be two thousand ten. Any question for me? For this one? Or you can ask Suhan. I mean, not necessarily ask. Just type in the chat. Then we can, you know, guide you. Okay. I believe we are good. Then we can keep moving on. This is the layout. Today we're going to check statistics. So, I really believe that you know so far how to solve and how to use this problem solving strategies, guys. I am not going to, you know, review again and again. Please check these numbers really quick. I mean, it has been like five, six times already. And if you have any questions, just ask us. All right. The basic of statistics. We're going to talk about mean today. The arithmetic mean, basically, not the social mean, you know. Average mean is average, or how to find it, sum of all of the numbers divided by number of terms you have. Then the mean of the data set is given. It also tells the total. Many problems can be easily solved by considering the total or comparing the totals when some numbers are added or removed. We have questions like that. I'm going to show you. Median means the middle number, when the data is arranged from least to greatest values. What if there is an even number of numbers? We're going to check this answer on the next page. You can check it in the middle to not confuse you. In that case, guys, you got to just add those middle two numbers and divide by two, basically. You're going to get average of two numbers, you know. Another question. Also, the mode represents the number repeats the most. Could we have more than one mode? Yes. Could we have no mode? Yes. Let's check really quick. Yes. For example, in that data set, 1, 1, 2, 2, 3, we have two of the numbers are repeating. You see, 1 and 2. They repeat twice. And sometimes, for some of the number sets, we have no mode. You see 1, 3, 6, 9, 10. There is nothing, no mode in this data set, because we have all numbers on the first one. OK? Range means, basically, you got to do largest number minus smallest number. OK? I'm not going to give you example about this, because we got to work with it anyway. Did you know that, guys, the mean and median of an arithmetic sequence is the number in the middle? For example, we have given the sequence 4, 1, 4, 7, 10, 13, 16, and 19. We got to notice that both the mean and median are 10, guys. Because remember, when you put them in order, arithmetic sequence is supposed to be in order from smallest to biggest anyway. Basically, the middle number is the median. Also, mean, not necessarily like regular mean, just arithmetic mean. When I say mean, I meant arithmetic mean, guys, not to be mean someone, whatever. So arithmetic mean, means here, you got to add all of them and divide by 1, 2, 3, 7. Then you're going to find 10 again. OK? The sum should be 7. You can check if you want. Any questions so far? That's good. That means no. All right, guys. First question comes from 2003. And it was number 9. There were five parrots in pet store. The average price of each of them was $600. Let's just say $600. One day, the most beautiful parrot was sold. The average price of each of the remaining four birds was $500. What was the price of the parrot that was sold? Let me start your timer. And let's see who says which one should we answer. Thank you. Bye-bye. Thank you very much. okay so the votes are we've got six for e and one for c all right six for e versus one for c interesting maybe that friend who says c is gonna change their mind to make e or someone for me changed their mind to make other okay guys we have a lot of votes and it has been one minute and 30 seconds let's just start them all right guys as you remember how do we find average is add all of the number of items then divide by number of values we have number of values here is five we have five items five parents whatever an average price of each of them was 600 so to be able to do this total you know like if we have five you know first one plus second one third one fourth one and fifth one those letters represents their price you know divided by five should be equal to 600 we can find their total easily by solving this i believe you all remember that when there is division you gotta do multiplication first we will cancel that division so the total price for five parrots were basically 600 times five which is three thousand i will just write it like that three thousand then they say one day the most beautiful one was sold the average price of each of the remaining four birds was 500 well in that case let's say they sold e a plus b plus c plus d divided by four remember i am only dividing by how many items do we have and the average was 500 in that case guys it's a little ugly 500 we can say guys we are able to find a plus b plus c plus d basically the total price for the first four well we gotta cancel for y multiplying so we should just make it a plus b plus c plus d is equal to 2000 because you know we had four birds and the average was 500 four times 500 2000 so we gotta find e i believe you all figured out that when we just compare uh compare those two equations by subtracting you know at the top we have e left over and three thousand two thousand is one thousand answer should be e guys whoever thinks it was e that was correct any question for this one or any part you feel not good okay gotcha let me see you can move on all right the next one it is from 2011 numbers a b and c have the following properties the arithmetic mean of a and b is equal to 17 and arithmetic mean of a b and c is equal to 15 the number c is equal to each other let's start your timer and right away two three people answer c okay we get c three votes for c so we got three more for c oh okay three friends says c three friends says d okay oh no three no three it's six total for c oh i thought you said three more for d my bad man i couldn't hear you gotcha okay okay 45 seconds so far let's make it one minute then we can start guys All right, time is up guys. So they say if the properties of A, Dartmouth mean of A and B is 17, that means I can say A plus B divided by 2 is equal to 17 guys. Well, we can find A plus B easily by canceling that division of 2 by multiplying both sides by 2. So you should get A plus B is equal to 34. This is the first thing you will get. Then we have also the mean of arithmetic mean of A, B and C is equal to 15. So A plus B plus C divided by 3 is equal to 15 guys. That means A plus B plus C is equal to 15 times 3 is 45. Then we get to find the number C. Well, we can still do some comparison or we can just multiply the first equation by negative 1 and just then combine them. That becomes negative A minus negative B is equal to negative 34. Those A and B are gone. We have C is equal to 45 minus 34, which is 11 guys. Any question for this one? I believe we are good. Okay. And the next one. I don't see anyone texting, messaging. Okay. Number three from 2020. Anna wants to walk 5 kilometers on average each day in March. At bedtime on March 16th, she realized that she had walked 95 kilometers so far. What distance does she need to walk on average for the remaining days of the month to achieve her target? Let's start the timer. Right away, someone answers C. Look at that. Two more answers C. All right. C is ahead of others. What about other answers and other friends? all right guys we have only three answers we can do more i know if i can get two more answers it has been one and a half minutes so far everyone is answering c oh everyone literally yeah okay gotcha then if it is not c it's gonna be interesting to me all right i hope it's c guys let's see all right anna wants to walk five kilometer on average each day in march first thing first we gotta find her goal five kilometer on average each day so you need to think about how many days are there in if you count you know january has 31 february depends and march has 31 so we get 30 when you multiply 31 by 5 guys it should be 155 i believe yeah okay this is her target guys she she plans to do like that amount of kilometer total in march but as you know she realized that she had walked only 95 so far that's not too much so she's kind of panicked so 155 minus 95 says she needs to complete 60 more guys in almost next half of the month as you know we know that at that time on march 16th she realized well 16th day is also gone when we do 31 minus 16 guys she has only 15 days left well we know that the total and total amount of kilometers she needs to take and we know that the total number of days she has we divide them basically we can find the average average distance basically so 60 divided by 15 gives you four guys i'm not gonna ask if there is any question because you all says answer was four i am proud of you class all right we can move on to next one number four three sisters whose average average is 10 all have different ages remember guys we get all different ages when they get together in pairs the average age of two such pairs are 11 and 12 it can be any of those two we don't know that yet what is the age of the eldest one and your time starts now okay someone answers d okay okay one vote for d what about others and one answers e all right d versus e let's see another person answers e 16 okay one more vote for 16 okay and we got one more vote for two more votes for 16 okay guys it is one minute and almost 30 seconds i will start shortly what to get here first let's say we have those sisters guys a b and c you know if the average is 10 we can say a plus b plus c divided by 3 is 10 that gives us a plus b plus c is gonna be 30 well we gotta focus with middle sense as well when they get together in pairs the average age of two such pairs are 11 and 10 we can say for example it doesn't matter which one we choose a plus b divided by 2 gives us 20 i mean 11 furthermore a plus b is equal to 22 also we can say guys let's say b plus c over 2 is 12 furthermore b plus c is gonna be 24 well we have three different equations here let's just check let's see if you add second and third one guys you can get a plus 2b plus c is equal to what 46 and if we take if you know multiply negative one the first one and subtract negative a minus b minus c is equal to minus 30 when you combine them you should get 2 minus b is b b is 16 well we don't know if b like 16 is the answer of alice or not but guys we can at least plug in b here to check like what's going on here if a plus b is 22 and if b is 16 guys we should get let me see a should be 6 and if you plug here again 16 and c is gonna be 8 then you can check the conditions guys if it's gonna work for you know all of the situations here the average is 10 yeah we know the totals they all have different ages we have 6 8 and 16 that should work then we can just you know say the eldest one has what is 60 16 years old any question for this part please yes that is very good okay i got you guys and the next one number five the mean score of the students who took a math test was six exactly 60 percent of the students passed the test the mean score of the students who passed the test was eight what was the mean score of the students who failed the test let's see what our friends think about that one let me start timer someone answers c okay another person answers c two more say c and go one more for c gotcha and go one more for c and that's a majority okay here you go so six total for c gotcha c is the only answer i got oh okay i thought someone says d my bad no okay right it has been almost one minute and 30 seconds anyway i believe we are good so the mean score of students who took a math test was six okay guys do we know how many students are there no okay since they gave us percent as well it's a exactly 60 percent of them look i recommend you guys choose number of students kinda either 10 or 100 or something we can work with person that you know so let's say at the beginning we get 10 10 kilos in that classroom 10 students okay if the mean score of the students who took the test was six we can find the total scores by those you know kiddos because i can say the total scores divided by 10 since there are 10 students is equal to 6 in that case i am going to get the total score is 60 guys from that classroom then they say exactly 60 percent of the students pass the test that means six of them because we both know that 60 percent of 10 is six students the mean score of the students who passed the test was eight so we know that we know the total scores for those total score for students who pass the test divided by i am gonna say six because there were six kiddos and the mean was eight that means guys total scores for students who pass the test was six times eight is 48 all right then they say what was the mean score of students who failed the test well we know the total score they got as a class which was 60 and we know that the total score who get as a you know passing kiddos so if we do the difference between those values we get 12 that 12 guys represents total score by the kiddos who fail the test we do not know their mean score but we know how many of them failed test at the beginning you see we said let's say 10 kiddos in that classroom 10 minus 6 is gonna be four students in that classroom well when you do 12 the total point divided by number of kiddos is going to give you mean score guys which is three since i mean even majority of france says three and they get the right answer still if you get any question just ask us or i'm going to keep going yes okay i believe you are good okay and meanwhile we can still check if someone has questions all right check number six please guys which two numbers can be removed from the set 17 13 5 10 14 9 12 and 16 so that the arithmetic mean remains unchanged remember we gotta remove two numbers all right okay one person says okay you get one more two e Okay, one more says E. All right. Bye-bye. Bye-bye. It has been 40 seconds guys. Okay, six more people vote for E. All right. Oh, six more, that's blocked. That means we should start, guys. Look like Magic Love Class thinks it's E anyway. Let's see if it was E or not. So, each two numbers can be removed from that set. So that arithmetic mean remains unchanged. To be able to see if the arithmetic mean remains unchanged, guys, we gotta find the arithmetic mean of those numbers, you know. So, let's see when we add those. 17 plus 13 gives us 30. Plus 10, 40. That means 54. Plus 9, 63. Plus 12, 75. Plus 5 should be 80. So we have 1, 2, 3, 4, 5, 6, 7, 8. Wait, did I add 8? I didn't add 16, yes? It should be 9, 6 then. We get 9, 6. Then we had 8 numbers, guys. When you... 1, 2, 3, 4, 5, 6, 7, 8. 8 numbers. When you divide them, you should get 12. If I'm not mistaken. If I'm mistaken, just let me know via chat. Anyway. So, you get 12 here. That means arithmetic mean of those... The set of numbers were 12. You gotta remove 2 numbers, guys. There is a rule of those type of questions. If the arithmetic mean of those 2 numbers is also same as the arithmetic mean of the whole group, that means arithmetic mean is not gonna change, guys. Well, we found 12 as an arithmetic mean. But, since we are talking about 2 numbers, I gotta say 12 times 2 should be 24. No matter what are the numbers, but you gotta add those 2 numbers to get 24. Because when you divide that total by 2, you should get 12. Well, we are basically looking at some of the numbers which gives us 24. If you check this one, it should be 29. Nope. This one, 22. Nope. That one, 25. Nope. 22. Nope. But if you check this one, guys, 14 plus 10 is 24. And that's it. I believe our friends are good. I'm gonna move on. We are fast today. That's good. Okay. And number 7, guys. Vilo is practicing the long jump. The average distance she has jumped so far today is 3.8 meters. 3.8 or 3.8, they are saying, you know, meters. On her next jump, she jumped 3.99 meters. And her average increased to 3.81 meters. What distance must she jump with her next jump to increase her average to 3.82 meters? This one from 2018. And your time starts, guys. This is our answer for 4.03. Okay. So, you are saying O versus D. Am I right? Yes. Gotcha. Now, we have one answer for 4.01. Oh, okay. Okay. One more answer for 4.03. And another person answers 403. Okay. One more person answers 403. Okay, number 29, I'm gonna give you a little more time. I got you. Another person answers D as well. Oh, okay. Okay, one more person answers D. Okay. So far, seven friends answer. Imagine none of you get the right answer. That would be tough, man. Look, if I am saying that, there is a reason actually. Not necessarily none of you get the right answer, but I'm just saying. Please check your answers carefully. I even give you extra time. I said, look, this is number 29. Please be careful. One of the hardest questions. Not necessarily hard, but they might be tricky, you know. Because we have all of those decimals. You may do mistakes. It's normal. So, two of the D people changed to C. I think you said two of our friends make it C. It's getting interesting. So far C has three versus D has four votes, guys. What's going on with D or C? Or maybe it's others. Who knows? Okay, a new person answers C. All right, right now we have C and D get the equal amount of votes. Let's see what was the answer, guys. So here, guys, the average distance she has jumped so far today was 3.8 meters. We do not know like how many times she, number of jumps she has. Like let's say she had X number of jumps, you know. If the average distance she has jumped so far is 3.8, can I just make it 3.8 times X, guys? Because, I mean, I'm going to say like basically number of jumps resulted, number of jumps X and that's number of jumps resulted 3.8 as an average. That means I'm going to have like 3.8 X. Okay, after the next jump, she got 3.99. I'm going to add that. And they say the average increased to 3.81. Well, the first time I had X amount of jumps, the second part since we had one more jump, it becomes X plus one times 3.81, guys. I hope that piece makes sense, you know. Okay, here is the deal. When you work with those type of equations, guys, when you see there are a lot of decimals, I recommend you to get rid of them. You can make it 3.8 or 3.80. They are, excuse me, they are literally the same thing anyway. But the first question you will ask yourself here is, guys, after decimal point, there are two digits. I would multiply each term by a hundred to be able to cancel those decimals, you know. Well, when you do that, you get, instead of 3.80, you get 38 X plus 3.99. I think it's 380 X. Oh, my bad, you are right. I just, yeah. I didn't pay attention to that zero because 3.8 or 3.80 is the same thing, you know. I just confused myself. 380 X, thanks, buddy, plus 3.99 is equal to, so we have here, guys, X plus one times 3.81. Well, you know, we are not going to multiply a hundred twice. We just make that number 3.81 becomes 3.81. Then we got to do X plus one times 3.81, guys. After that, you all notice, I believe, you got to do just distribution. So 380 X plus 3.99 is equal to, when you just distribute that one and that one value here, you get 3.81 X plus 3.81. Well, after that, we got to cancel the smaller X value from both sides. At the end, we get 3.99 is equal to 3.81 minus 3.80 is just one X or X plus 3.81, guys. Well, since we have X number next to X, we got to cancel that by subtracting. It was plus. That gives us, guys, X is going to be 9.81, should be 18. OK, after we found X, guys, what else we can do? Think about that. After we found X, K is going to be the distance of the 20th jump. OK, so the next jump, basically. And to get average of 3.82, because we're going to need 3.82 as an answer. Oh, I said 20th jump because it was 18, he, I mean, she started 18th jump, then next jump was 19th jump, and jump, and 20th jump is going to be the next one, which the average is going to give us 3.82 meters. So the total is going to be, guys, 3.82. No, 3. We have all the 3.81, yes? Oh, we have average, is it? Times 20. What was that, pardon? It will be 3.82 times 20. That's because the average of the 20 jumps is 3.82. And then on the other side, 3.81 times 19. Then we have plus, should be just K or whatever, because we got to find it, because this is what's going to be, let's just say K. Then we got to solve this one, guys. All right. So 3.81, I didn't put 3.81, but if I can just make it 3.81 and 3.82. I got to use calculator, guys, I don't know what's the answer. Let's just do that. Actually, so there's actually a shortcut that you can do with this problem. So when you move 3.81 times 19 to the right-hand side of this equation, you can write 3.82 times 20 as 3.82 times 19 plus 3.82. And then 3.82 minus 3.81 is just 0.01. And that gets multiplied by 19 to get 0.19. And then finally, you add your 0.19 to your 3.82 to get the answer, 4.01 meters. Good, but would you like to kind of annotate here? Because I get lost after a couple of steps. Yeah, I don't have annotation permissions. You don't have it? You should have. Let me see what's going on. Okay, now I can see it. Okay. So the way you do this, so you have K is equal to, so you have, you write 3.82 times 20 as 3.82 times 19 plus 3.82 minus 3.81. Minus 19. Gotcha. So this term right here and this term right here, those two combine to give you your 0.19. Yeah, yep. And then 3.82 and answer is 4.01. All right. I can say that this was a quite a tricky problem. Most of you answered D on your first attempt. So only two people got, only three people got C on their first attempt. Yeah. That was the reason I was just keep saying, please be careful with your answers and calculations. Any question for this one so far? Okay, gotcha. We should be good then. All right. Hopefully we can solve one more. We should have time. All right. The next one, guys. If the average of 10 different positive integers is 10, the greatest of these numbers can be, look, this is also tricky question, please be careful. This one was number 29. Don't tell me you just find the answer right away. It's not that simple. I am just warning you. Hey, one vote for A, two votes for B. Right. We got one more for B. Gotcha. Okay, we got one more for B and one more for A. All right, interesting. Bye-bye. Okay, one more vote for B. I think we can start now. So you realize that the average of 10 numbers is 10. So the sum of the 10 numbers is, well, 10 times 10 is 100. And that is the sum of your 10 numbers. So for friends who found 9 to 1, I believe you didn't check this part, 10 different numbers. I believe this is to be the case. So if you want to maximize one of your 10 numbers, what you have to do to the other 9 numbers is make them as little as possible. So the 9 smallest numbers that you can have, you do 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9. Those are your 9 smallest numbers. Those 9 numbers sum to 45, and then to make the sum equal to 100, you make the last number 55. And this is your answer. Any questions, guys, you got to figure out how to add those quickly or in a shortcut way to find 45. I would just make 9 plus 1, 10, 8 plus, and so forth. It's totally up to you. Any question for this one, guys? Yeah. Sorry, Swami. You can just show us with the arrows, yeah? Then it's going to be 45, guys. We have less than one minute left. So I am not going to start the next one. It says we have seconds. Any question we have so far with either this one or other questions, guys? I don't see any questions in the chat. There is no chat, yeah. And it is time, actually, guys. All right. I enjoyed your session. Even the topic says statistics, it was most likely more to me. I mean, introduction of statistics, guys, it's not actual statistics. You are going to see actual statistics when you go to high school and college. So I hope you enjoyed it as well. And I am going to see you next week. Take care. Bye.

webinar

arithmetic means

consecutive numbers

problem-solving

interactive learning

algebraic methods

mathematical concepts

logic-based questions

shortcut strategies

algebraic thinking