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# Comparing means of distributions

## Video transcript

Kenny interviewed freshmen and seniors at his high school asking them how many pieces of fruit they eat thee they eat each day the results are shown in the two plots below and so the first the first statement that we have to complete is the mean number of fruits is greater for and let actually let me go to the actual screen is greater for we have to pick between freshmen and seniors and then they say the mean is a good measure for the center of distribution of and we pick either freshmen or seniors so let me go back to my scratch pad here and let's think about this so let's first think about the first part so let's let's just calculate the mean for each of these distributions and I encourage you to to pause the video and try to calculate it out on your own so let's first think about the mean number of food for freshmen so essentially we're just going to take each of these data points add them all together and then divide by the number of data points that we have so we have one data point at 0 so we have one data point at 0 so I'll write 0 and then we have two data points at one so we could say plus two times one and then we have two data points at - right plus two times two and then let's see we have a bunch of data we have four data points at three so you could say we have four threes so let me circle that so we have we have four three plus four times three and then we have we have three fours so plus three times four and then we have a 5 so plus five and then we have a six I'm going to do this in a color that you can see and then we have a 6 right over here plus 6 and how many total points did we have well we had one two three four five six seven eight nine 10 11 12 13 14 Oh actually we carefully had 15 points and I didn't put that one in there so actually let me just so we have 15 points and I can't forget this one over here so plus my pin is acting a little funny right now but we'll power through that plus 19 plus 19 so what is this going to be so this is just going to be zero this is going to be two this is going to be four this is going to be twelve my pin is really acting up it's almost like it's running out of digital ink or something and this is going to be another twelve and then we have five six and 19 so what is this going to be 2 plus 4 is 6 plus 24 is 30 plus plus 11 is 41 plus 19 gets us to 60 60 divided by 15 is 4 so the mean number of fruit per day for the freshman is for pieces of fruit per day so this right over here that right over there is our mean four they let me put that in a color that you can actually see now let's do the same calculation for the seniors so we have one data point where they didn't need any fruit at all each day not not too healthy then you have one one so I'll just write that is weak I could write that as one times one but I'll just write that as one then we have two twos so plus two times two then we have one two three four five threes five threes so plus 5 times 3 and then we have we have three three fours so plus 3 times 4 and then we have two 5s plus two times 5 and then we have a 6 we have a 6 plus 6 and we have a 7 so I'm going to eat 7 pieces of fruit each day a lot of fiber plus 7 and now how many data points did we have well we have one two three four five six seven eight nine 10 11 12 13 14 15 16 data points so we're going to divide this by 16 so what is this going to be this is just 0 let's see this is just right over that's zero this is four this is 15 this is 12 this is 10 so we have 1 plus 4 is 5 plus 15 is 20 plus 12 is 32 plus 10 is 42 42 plus 6 is 48 48 am i doing 42 plus 6 is 48 plus 7 48 plus 7 is 55 did I do that right let me do it one more time 1 plus 4 is 5 plus 15 is 20 32 42 42 plus 13 is 55 so this is equal to this is equal to 55 over 16 which is the same thing as let's see that's the same thing as 3 & 3 that 3 times 16 is 48 so 3 and 7/16 so the mean for the seniors the mean for the seniors 3 and 7/16 as right around let's see this is 3 that's 4 so 7/16 it's a little less than a little less than 1/2 it's right right around right around there so the mean number of fruits is definitely greater for the freshmen they have 4 on their mean number of fruit eaten per day is 4 vs. 3 and 7/16 the mean is a good measure for the center of the distribution of so when we think about whether it's freshmen or seniors so the mean because is is fairly sensitive to when you have outliers here for example someone here was eating 19 pieces of fruit per day that's that's an enormous amount of fruit they must be only eating food you could imagine if it was even a bigger outlet if someone's eating 20 or 30 pieces of fruit just that one data point will skew the entire mean upwards that wouldn't be the effect on the mode because the mode is the middle number if even if you change this one point all the way out here it's not going to change what the middle number is so the mean is more sensitive to these to these outliers so these really these points that are really really high really really low so and because because the seniors don't seem to have any outliers like that I would say that the mean is a good measure for the center of distribution for the seniors or a better measure for the center of distribution for the seniors so let's fill both of those out so the mean number of fruits is greater for the freshmen and the mean is a good measure for the center of distribution for for the seniors and you actually even see it here we saw that the mean number for freshmen was at 4 but if you if you just ignore this person right over here just you kind of thought about the bulk of this distribution right over here for really doesn't look like the center of it the center of it looks closer to 3 here and what happened is is this one person eating 19 pieces of fruit per day skewed the mean upwards while here that 3 and 7/16 really did look closer to the actual distribution closer to the actually I shouldn't say I mean both in both times we actually did calculate the mean of the actual distribution but here since there's no outliers it does seem the mean seemed much closer to I guess you could say the middle of the middle of this pile right over here let's check our answer and we got it right