Numerically Summarizing Data (Part 2)

Ajouté : 2026-05-17

[00:00:00]okay so the instructions on how to find mean and median on your calculator are down here okay so this is what we used okay so let's go back to number three compute the mean of the new data set so the new data set has eight employees remember okay how does the new employees travel time affect the mean how about median okay so I want you guys to find the mean of the new data set okay so you can pause this video and you can do this on your own and then come back and check your answer right now you can either use your calculator you can use the one bar stats function that we just learned okay all you have to do is really go back to the list l1 right and then you can add 180 that's the eighth person the new employees data value so just add 180 to the list and then do one bar stats right that's it or if you want to do it manually then just add all the eight values and then divide the total by eight all right so go ahead and pause the video find the mean and then come back to check your answer okay all right so let's see here what I got was I got mu equals forty four point twenty five minutes so this is the mean of the new data set after the new employee joined the company all right is that what you got all right now we want to compare this mean with the old mean right so mu before before the new employee okay so that close it again it was 29 no 24.9 it was twenty four point nine minutes before I said this was only for about seven employees okay so notice that since the new persons travel time was very very large like 180 minutes right it affected the mean a lot look at this me was twenty four point nine minutes before but now is forty four point 25 right okay so we can say that the new employees travel time increased increased the value of the mean okay all right on how about the median right how about the median so median for the new data set after a new person joined the company and if we found that to be twenty five point five minutes this is from the previous page okay now how about what was the median before the new person it was twenty five twenty five right yeah just twenty five okay so twenty five minutes okay so the median values did not really change much look at that twenty five minutes and twenty five point five minutes right so even though the new person's travel time was very large that did not really affect the median so much okay so we can write that new employees time did not affect the value of the median so much okay so that brings us to the next very important definition resistance by a numerical summary of data is said to be resistant if extreme values so very large or very small values relative to the data do not affect its value substantially so as we just discussed the new employees travel time that's 180 minutes right that's a very large value relative to the data and that did not affect the value of media much so the median is called a resistant measure of center okay so the median is resistant okay so when it comes to median we are just looking at the physical middle so extreme values will not affect the value of it an extreme value like 180 will just sit at the end and it will not affect the medium so much if the value was 1800 minutes instead of 180 minutes right it does not change the median at all since the median is the center by location right now well extreme values affect the mean yes the new employees time 180 minutes brought the mean from 25 well almost 25 right well 24 point nine minutes dude 44 point 25 minutes 180 minutes was so large that it threw off the mean so the mean is not resistant so mean is not resistant okay so because the median is not affected by extreme values median is a better measure of central tendency if the data contain extreme values so if the distribution is skewed right or skewed left extreme values usually sit where the longer tails are so it's cute right and skew left right here all right so this is where extreme values are okay right so you have a skewed left or skewed right distribution then use median as the measure of center and use the mean if the distribution is roughly symmetric so let's look at three common distribution shapes with mean and median so let's start with the mean okay so the mean is the balancing point of the distribution so if we want to balance something symmetrical like a bell shape let's start with this one right you want to balance that where does that balance well right in the middle right so meaning is here right in the middle that's where the mean is okay of a bell-shaped or distribution now if we want the balance something like skewed right skewed right right where does that balance well most of the data are right here on this side side right but there are values over here and these values hold a lot of weight because these values are extreme values right so while the majority of the data are on the left side somewhere here of the distribution these extreme values are going to pull the mean in that direction right so the mean is going to be somewhere here right closer to the tail and we just saw that in action in the last example right when we added the value of 180 the mean went up to 44 point 25 right so the value of a 180 so that'll be likely maybe eating somewhere it appeared 180 okay pull the mean to the right okay towards the tail because the mean is not resistant it's very sensitive to extreme values and likewise in skewed left distribution the extreme values are over here to the left so it is going to pull the mean towards left so it mean is somewhere here towards the tail the mean is not resistant it is very sensitive to extreme values okay so now let's consider the median it's the physical middle so 50% of data values are going to be less than the median and a 50% of the data values are going to be greater than the median so it's so if you have a bell-shaped distribution median is going to be right in the middle just like mean so 50% of data or below the median 50% of data or above the median so for bell-shaped distribution the mean and the median are equal or very similar how about skewed right well the median is going to be hanging out with the majority of the data so somewhere here okay so again 50% of data values are on the left side of the median and then 50% of data values are on the other side of the median okay and for skewed right the value of mean is greater than median okay mean is closer to the tail right okay so mean is greater than the value of median actually we can use these two values to explain the relationship right forty four point twenty five twenty five point five these two values are from the new data set that has the extreme value of 180 minutes right so that's the skewed right distribution so 180 is like up here so mu is forty four point 25 that's like means right here right oops forty four point 25 and the media is twenty five point five twenty five point five look at that median is less than the mean right mean is greater than the median right so there you go okay and how about the skewed left well the median is going to be somewhere down here I'm sorry somewhere here up here where the the majority of the data are right so again the median is the physical middle so fifty percent of data values are on the right side of the median and fifty percent of data values are on the left side of the median right so the value of mean this time is less than median okay because mean is going to be towards the tail right where the extreme values are okay so the mean is the center of gravity the median is Center by location all right okay so our last measure of central tendency is the mode okay and it's the most frequent observation of the variable that occurs in this data set okay so we have a lot of possibilities we can have one mode we can have two modes more than two modes and we can even have no mode by zero it depends on the data so let's take a look at this example all right so we have how many values here 1 2 3 4 5 6 7 8 9 values 15 14 32 15 7 14 8 14 and 12 right okay so what is the mode okay so looks like we have one two three 14 right okay how about 15 15 repeats as well right we have 1 - 15 right but mode is 14 because this is the value that repeats the most right so since the data and have only one mode the data is called unimodal okay unimodal and then the mode is 14 okay the next one and take a look at this example all right we have a b c d ba e if h okay which letter repeats the most well actually a and b tie for the greatest frequency right because we have two A's and also we have two B's right so there are both modes okay since this dataset has two modes we call that data set by model right and the modes are a and B both alright now if you have more than two modes right then the data set is called multimodal and when no data value is repeated then we say there is no mode alright okay so here is a summary of what we just learned in this section right okay so mean median your mean median and mode and here's how to find each one computation right okay here is a formula for population mean and sample mean okay and also how to interpret each one and when to use each one right so use the mean if data are quantitative and the frequency distribution is roughly symmetric and use median when data are quantitative and the frequency distribution is skewed left or skewed right then you use mode when the most frequent observation is the desired measure of central tendency or the data are qualitative oh okay so this part is kind of important so let's go down here so the mean and median can be found only with oh not both only with quantitative data but the mode can be found with both quantitative and qualitative data okay well why can't we find the mean and median for qualitative data well when we learned qualitative variable in the first chapter we learned that arithmetic operations such as addition and subtraction with qualitative data do not give any meaningful results for example student ID numbers these are qualitative data because we don't add and subtract ID numbers to get anything meaningful right so finding a numerical summaries like mean and median for student ID numbers does not really make sense okay so that's why we cannot find the mean and median for qualitative data okay is for quantitative data only okay and again rounding rules for measures of a center so for the mean and median we're gonna carry one more decimal place than it is present in the original set of values for the mode we can go ahead and leave the value as is without rounding okay so that's the rounding rules okay final example is an application so we have that Miss Forbes finds the mean height of a random sample of a 14 students in her statistics class to be 68 inches just as she finishes explaining how to get the mean Daniela walks in late Danielle is 65 inches tall what is the mean height of the 15 students okay so 14 students and plus Danielle okay they'll be 15 students okay all right so I am going to write down my information so I'm just gonna start writing some tidbits from this problem so first of all the mean height of 14 students right so the mean height of 14 students is 68 inches okay and oh by the way I'm using X bar because we are talking about a sample of 14 students right okay so the sample mean symbol is X bar okay so that's why I'm using X part instead of MU okay then Danielle walks in late so she is the 15th person all right so her height is 65 inches okay there you go okay notice that I'm not using X bar here right I'm just using X okay because we're talking about just Danielle's height just one person's height instead of the mean of 14 students okay great all right oh we're gonna worry about the unit's at the end ok so right now you don't have to write down the unit's all right so here's the problem what is the mean height of the 15 students right so do you want to know what the mean of 15 students so that's what we don't know all right so there you go now let's write down other things we know right so we know that miss Forbes found that the mean height of 14 students by adding up 14 students heights and dividing by 14 and that is 68 inches right so we know that this is the sum of 14 Heights divided by 14 and we know that that's 60 eight okay now we also know that if we wanted to find X bar with 15 students so we want to find X bar with 15 students we need to do some 14 of sorry 15 Heights and if we divide that by 15 then we're gonna get the mean right and that's what we are looking for okay all right so what do we do now well we can actually figure out this part right here the sum of 14 Heights right we can kind of think of this as an algebra problem this whole numerator is unknown so how could we solve for the numerator well multiply both sides by 14 right so sum of 14 Heights divided by 14 equals 68 so we can multiply both sides by 14 okay so we know that the sum of 14 right is 68 times 14 okay so 68 times 14 is 952 all right so this means that if we add all 14 students Heights right the total is 952 inches it does not mean that every single student is 68 inches some are short some are tall right but when we add them all up we get 952 inches right okay so let's go to this equation here right how can we get the sum of fifteen students heights well we just need to combine the sum of 14 students heights and Daniele's height right okay so you know what let me just come down here okay so this will be sum of 14 Heights fool us Danielle's height okay that will give us the sum of 15 Heights and we divide that total by 15 okay so we're almost there so some 14 Heights that's 950 to pull us Danielle's height which is 65 okay and we divide that by 15 okay so 952 plus 65 that's 1017 so we divide that by 15 and that's sixty seven point eight inches okay so that's the new mean okay so I mean let me just write down here so the you mean height of the 15 students is sixty seven point eight is I and that's that alright so that's it for this video I will see you in the next one bye guys