let's put it all together with this example in my open map following data are the distances between 46 retail stores and a large Distribution Center the distances are in miles so we have here the distances from 46 different stores and the central Distribution Center okay now this is a sample of the stores and so we want to find the sample mean of the data and the sample standard deviation of the data here it tells us which makes our life a little bit easier we don't have to decide for ourselves if we're dealing with a sample or a population okay so we're going to start off by finding the sample mean and then we'll find the sample standard deviation okay so I'm going to start by copying this data and moving it to my calculator up here okay so I'll call my list of data X stored in my list let's find out how many we have and store it in variable n we'll use little n since we're dealing with the sample if I evaluate that says we have 46 that matches what the problem tells us so that's good it means we didn't leave any data points out okay and now with my list of data set knowing how many data values we have we're set to find the mean add up all my data values in the list so sum up all the values in list X and divide by n number of data values in the list okay and let's store that in a variable this is the sample mean so we're going to call it X bar click evaluate and it just displays the last line here so it's displaying press X bar so our mean our sample mean is 197.2826 and so I don't forget anything I'm just going to copy that whole number and paste it into the box here you don't want to round so or else you might be off by a little bit right rounding introduces error so don't even mess with it uh just copy and paste the full answer from the calculator okay we have our mean the average distance of the 46 retail stores on average the retail stores are 197.2826 miles from that Central Distribution Center okay all right let's calculate the sample standard deviation now so I'm going to calculate the distance between each data value in my list X and uh the mean so that's x minus X bar that calculates the distance with direction to get rid of the direction we Square I average it so I add it all up and normally I'd divide by n because this is a sample standard deviation we have to divide by n minus one right a little bit different when you're calculating it for a sample and then last step I need to take the square root so this will give me my sample standard deviation we denote our sample standard deviation with the variable s s in um statistics as always means sample standard deviation and so that will let me use it later which I will need to in the next step after this so let's click evaluate let's find our sample standard deviation 32.4884 that tells us how far the data on average deviate from that mean what's the standard amount we deviate from that mean okay all right one more step we need to find our outliers okay so we first need to find the bounds remember most values are within two standard deviations of the mean so we need to find those bounds and then we look for values that are not within two standard deviations for money okay so if I do the mean plus 2 times the standard deviation that's two standard deviations above the mean okay so that's our upper bound 262.2594 okay in X bar minus 2s that goes in the other direction that says that calculates two standard deviations below the mean okay so if I evaluate that 132.3058 we are interested in the Val our data values these data values that are not in between those two numbers the data values not in between these numbers that we just calculated are the outliers okay now unfortunately uh this will only display one at a time right so you could either you know just display it and write it down so 262.2594 and then 132.3058 or if you really want to display more than one thing at a time type print that tells the computer hey I really want to to see this okay now if I evaluate both of them are visible okay you have to be a little bit careful if I say print for this last one and let's see what happens notice I get a third value it prints this value just like I want it prints this value just like I want but remember with this particular version of the calculator it also always displays what's going on on your last line so it displays that lot that last line twice so if you use that print function just be aware of that I usually won't use that print function for the last line just because of that little uh artifact okay so use print function here I see that X bar minus 2 times s that's the last line so that's that um so anything between these two numbers is within two standard deviations of the mean it's not further than two standard deviations of mean so that's kind of the normal range the outliers are the numbers not in between these numbers so looking at our data which are conveniently in order from small so Vegas are there any data values which aren't in between these two numbers so let's look 132 is 132 in between these numbers no it's a little bit smaller than this right just a little but it's smaller so that would be an outlier 132 is an outlier okay now there could be more than one so we have to check 133 would that be an outlier no that's in between those numbers right it's bigger than 132 but smaller than 262. okay and since these numbers are in order I don't have to really check the next number I could go to the other end of the data okay see if this is an outlier okay so looking at 270 277 is that in between these numbers no it's not in between the numbers right so which is an outlier it's more than two standard deviations away from the mean it is unusually big in this case it's bigger than 262 right it's bigger than that upper bound so I'm going to add that number also but I'll do it in a list so separate it with a comma and then add that number in okay so I have at least two outliers let's see if I have any more so I'm going to check up here again 252. is that not with is that within these numbers is that between these two numbers yeah it is it's between those two numbers so it would not be an outlier it's within two standard deviations of the mean okay and it just gets smaller after that so all of these numbers are between 132 and 262. right 132.3058 and 262.2594 132 and 277 those are our outliers because they are not in between these two numbers and so those are the numbers I write here as a list separated with a comma and then I submit and we have it