How parameters change as data is shifted and scaled

so I had some data here in a spreadsheet you could use Microsoft Excel or you could use Google Spreadsheets and we're going to use the spreadsheet to quickly calculate some parameters let's say this is the population let's say this is we're looking at a population of students and we want to calculate some parameters and this is their ages and we want to calculate some parameters on that and so first I'm going to calculate it using the spreadsheet and then we're to think about how those parameters change as we do things to the data if we were to shift the data up or down or if we were to multiply all the points by some value what does that do to the actual parameters so the first parameter I'm going to calculate is the mean then I'm going to calculate the standard deviation then I want to calculate the median and then I want to calculate let's say the interquartile range enter I'll call it I Q R so let's do this let's first look at the measures of central tendency so the mean the function on most spreadsheets is the average function and then I could use my mouse and select all of these or I could press shift with my arrow button and select all of those okay that's the mean of that data now let's think about what happens if I take all of that data and if I were to add a fixed amount to it so if I took all the data and if I were to add 5 to it so an easy way to do that in a spreadsheet is you select that you add 5 and then I can scroll down and notice for every data point I had before I now have 5 more than that so this is my new data set or so I'm calling data plus 5 and let's see what the mean of that is so the mean of that notice is exactly 5 more and the same would have been true if I added or subtracted any number it the mean would change by the amount that I add or subtract and so and that shouldn't surprise you because when you're calculating the mean you're adding all the numbers up and you're dividing by the numbers you have and so if all the numbers are 5 more you're going to add 5 in this case how many numbers are there 1 2 3 4 5 6 7 8 9 10 11 12 you're going to add 12 more 5s and then your divided by 12 and so it makes sense that your mean goes up by let's talk about how the mean changes if you multiply so if you take your data and if I were to multiply it times five what happens so this equals this times five so now L all the data points are five times more now what happens to my mean notice my mean is now five times as much so the measures of central tendency if I add or subtract well I'm going to add or subtract the mean by that amount and if I scale it up by five or if I scaled it down by five well my mean would scale up or down by the same amount and if you numerically looked at how you calculate a mean it would make sense that this is happening mathematically now I'm going to look at let's look at the other measure the other typical measure of central tendency and that is the median to see if that has the same properties so let's calculate the median here so once again you want to order these numbers and just find the middle number which is it too hard but a computer can do it awfully fast so that's the median for that data set what do you think the median is going to be if you take all the data plus five well the middle number if you ordered all of these numbers and made them all five more the orders you could think of it being the same order but now the one in the middle is going to be five more so this should be ten point five and yes it is indeed ten point five and what would happen if you multiply everything by five well once again you still have the same ordering and so it should just multiply that by five yep the middle number is not going to be five times larger so both of these measures of central tendency if you've shift the all the data points or if you scale them up you're going to similarly shift or scale up these measures of central tendency now let's think about these measures of spread see if that's the same with these measures of spread so standard deviation so stdev I'm going to take the population standard deviation I'm assuming this is my entire population so let me know why is it so I make sure I'm doing so standard deviation of all of this is going to be 2.99 let's see what happens when I shift everything by five actually positive what do you think is going to happen this is a measure of spread so if you shift I'll tell you what I think if I shift everything by the same amount the mean shifts but the distance of everything from the mean should not change so the standard deviation should not change I don't think in this example and indeed it does not change so if we shift the data sets in this case we shifted it up by five or if we shift it down by one your measure of spread in this case standard deviation should not change or at least the standard deviation measure of spread does not change but if we scale it well I think it should change because you can imagine a very simple data set the things that were a certain amount of distance from the mean are now going to be five times further from the mean so I think this actually should we should multiply by five here and it does look like that is the case when I multiplied this by five so scaling the data set will scale the standard deviation in a similar way what about interquartile range where it's essentially we're taking the third quartile and subtracting from that the first quartile to figure out kind of the range of the middle 50% and so let's do that we can have the quartile function equals quartile and then we want to look at our data and we want the third quartile so that's going to calculate the third quartile minus quartile same data set so now we want to select it again so same data set but we this is not going to be the first quartile so this is going to give us our interquartile range this is the calculate the third quartile on the data set and this calculates the first quartile on that data set and we get 2.75 now let's think about what the inter court whether the interquartile range should change and I don't think it will because remember everything shifts and even though the first quartile is going to be five more but the third quartile is going to be five more as well so the difference shouldn't change and indeed look the distance does not change or the difference does not change but similarly if we scale everything up if we were to scale up the first quartile in the third quartile by five well then their difference should scale by five and we see that right over there so the big takeaway here I just use the example of shifting up by five and scaling up by five but you could subtract by any number and you could divide by number as well you're the typical measures of central tendency mean and median they both shift and scale as you shift and scale the data but your your typical measures of spread standard deviation and interquartile range they don't change if you shift the data but they do change and they scale as you scale the data