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let's say that we run a car dealership and we have 11 salespeople and we record the number of cars each of the 11 salespeople sold in the past week so this is you know that's one of them 2 3 4 5 6 7 8 9 10 11 so we record how much each of them sold so maybe one person sold maybe they sold five cars maybe the next person sold seven cars maybe the next person sold 10 cars and so that's how we would do it so we're going to record all of that but then weird we're told that the median car sold is six so if we ordered all of these numbers so let's just assume that we ordered them all from least to greatest so maybe and I'm just making up numbers here maybe this is for this is for maybe this is 5 5 5 6 7 7 8 9 10 the median car sold me that the middle number here is going to be 6 now we don't know whether all of these other numbers are the actual numbers I just made those up but we know that the middle number needs to be 6 so let me just fill out only the middle number so the middle number if you have 11 data points the middle number is going to have 5 on either side so that's that one right over there 1 2 3 4 5 1 2 3 4 5 so we know that the median we know the median actually let me do that purple color that median is going to be 6 now the other thing we know is that the range of cars sold is 4 and let's remind ourselves the range is the maximum number of cars sold minus the minimum number of cars sold and if we have sorted all of these out this if we take if this is our max right over here because we've sorted them all out and this is our min right over here our range is going to be our max minus - our min so let me write that down so you take your men and your max you have your range range is equal to the maximum minus the minimum and they're telling us that that is 4 this range is 4 so the maximum minus the minimum the difference between the the number of cars that the the most productive salesperson sold and the number of cars at the least productive salesperson sold that difference is going to be four so given all of those assumptions giving all of that information I'm now going to give you a statement and our challenge is if we assume everything I just said is true is this statement going to be true or is this statement going to be false or do we not know so let me write this down so is this going to be true false false or do we not know do we not have enough information to say it's for sure going to be true for sure going to be false or for sure we don't know and so I encourage you to pause the video now and try it out alright let's so let's work through it together this is kind of a fun puzzle here where we've been given some clues and then we have a statement we forget can we can we make this statement can we say it's true or can we say for sure that it's false or do we just not know so a couple of ways a couple of ways to to tackle it one way to tackle it is well look one of the salespeople must have sold six cars if the median sold is six and there's eleven salespeople here the middle number here that that if we order it one of them must have sold six cars if we had an even number here then that would be the average of the two middles but if we have an odd number here that is literally the middle number of cars that somebody the number of the middle number of cars sold so someone sold six cars so if we assume if we assume it we're trying to find a world where someone sold more than ten cars so if someone sold six cars is there a way with a range of four that someone sold more than ten well just think about it let's just assume that the mint is six we don't know if that the mint is six but let's just try it out if the minimum is six if the minimum is six what's the maximum going to be well remember the range the range is equal to we know that the range is four the range is equal to the maximum minus the minimum maximum mine the minimum and so the maximum in this case would be 10 would be 10 max would be equal to would be equal to 10 if the min is equal to 6 and instead of colons I'm going to write it equal sign if the min is equal to 6 then the max at most can be equal to 10 and we can't take a higher min because we know that the 6 6 is going to be one of the values we could try a lower minimum value could say the minimum value is 5 or 4 or 3 or 2 or one but then the maximum would go down even more because remember the maximum is no more than 4 larger than the minimum so if we assume 6 and we know that one of the salespeople sold 6 cars then the maximum that any of the sales people can sold is 10 and so the statement at least one of the salespeople sold more than 10 cars that's got to be false that's got to be false now there's another way that you could think about it you could assume that someone sold more than 10 you could assume in the last example or the last way of thinking about it we assumed that the min was 6 but now let's just assume let's just assume that someone sold that let's just let's just try it out let's see if it's possible it's just assume that the max is 11 that someone sold more than 10 cars if the max is equal to 11 what's the min going to need to be well we just have to remind ourselves range is equal to max minus min so 4 is equal to the maximum 11 minus the minimum so 11 minus what is equal to 4 well 11 minus 7 is equal to 4 so if we assume that the maximum is 11 then the minimum is going to have to be the minimum is going to have to be equal to 7 now can the minimum be 7 and is when your median is 6 no if your median is 6 that means you have if you that means if you have an odd number of data points that means one of the data points is 6 and if you had an even number of data points that means that the middle two are going to average out to be 6 which means you even have data points that are less than 6 so your minimum can't be 7 you're going to have a value that is at least as low least as low as six so your assumption can't be true can't be true so once again the assumption based on the statement can't be true this statement is going to be false