let's get into our fifth section of unit zero for AP biology in this section we're going to go over something called standard deviation which deals a lot with what we talked about in the last section which is variation and to start out this section I always like to show this graph this graph shows data on human height for both men and women on the bottom here you can see the height in inches and on the side here you can see the frequency of the population now at first this graph might be misleading because it kind of looks like women here are taller than men but remember on the side here the Y AIS this is the frequency not the height the height is down here in the x-axis this shows us that males are taller on average than females what we typically see is there is variation within the men you can see this variation in red and the average male is around 70 71 in tall however there's less variation within females and they have a shorter average height which is around 65 in so what we're looking looking at in these two different data samples is variation within and variation between the two different groups and that's important because we're going to look at something called the bell curve or something called standard deviation this specifically deals with something called normal distribution this is the probability distribution where the values of a random variable are distributed symmetrically or otherwise known as the bell curve these graphs typically look like a bell basically what we're looking at here is how do data samples sets vary within their data and that's specifically what standard deviation measures it's the measure of how the data sets vary or deviates from the mean data sample sets can have less variation or more variation what we typically see is the mean being the middle which is the average and within each of these data sets is going to be variation you can see there is less variation within this data set and more variation within this data set now typically within this bell curve we have something called the standard deviations and standard deviations just tell us how much data is within that group for example we have our mean or our average here in the middle what we typically see is 68% of variation is going to fall between negative one standard deviation and positive one standard deviation that means if we take a data set from some variable whatever the variable is 68% of the data should fall between these two standard deviations for that same data set if we look at two standard deviations 95% of the data should fall between those two points so again if we subtract one standard deviation and we add one standard deviation 95% of the data should fall between these two points now at this point you might be wondering how do we know how much each percent Falls within each standard deviation and that kind of math is way above my head and way above the material for an AP Biology class but this is just backed with with data and math and science but I encourage you if you're interested look this up and do a little more research now again we can go one step further we can look at three standard deviations and within three standard deviations of the mean there's going to be 99% of the data so just as again as a wrap up we have our average or the mean we have our first standard deviation that encapsulates 68% of the data our second standard deviation which encapsulates 95% of the data and our third standard deviation which encloses 99 % of all the data now how do we figure out what these standard deviation points are all right like I warned you there is some math involved here this is the equation for standard deviation but don't freak out yet it's really not that bad I'm going to show you the steps of how to work through one of these standard deviation problems I'm going to go through an example problem and then I'm going to show you a shortcut of how to figure this out a lot easier before we get into how to work through one of these problems go over each of these variables this x with a line above it is called xar or it's the mean or average for the data each X here represents a data point this Big E is I like to call it this is the summation sign this is just adding up all the values Within These parentheses and N is the number of data points for a data set here is a list of the steps of how to work through a standard deviation problem first you find the mean of the data set which is xar you then find the difference of how each data point differs from the mean so you take the data data point and subtract the mean you square that difference and you find the sum of all of these differences for each one of the data points you then take that sum and divide it by the number of data points minus one that whole number you take the square root of it and that is your standard deviation now again there was a lot that I breeed through there let's go over a practice problem because it makes a lot more sense when you actually give data points all right so here's our data points 1 2 3 4 5 and if we calculate the average or the mean we find that the average is three while the number of data points is five now what we have to do is we have to take each data point which is X and subtract the mean from it so for the first data point you can see it's 1 - 3 which is -2 -2 2ar is 4 for the second data point it's 2 - 3 which is - 1 - 1^ 2ar is 1 our third data point is 3 3 - 3 is 0 0 squ is 0 our fourth data point is 4 4 - 3 is 1 1 2qu is 1 and our fifth data point is 5 5 - 3 is 2 2^ 2 is 4 so what we have to do now is take the sum of all of these numbers which is 10 so again we have our data points we have the number of data points which is five we have the average which is three the sum of all of the variation is 10 so we plug it in here up top so the standard deviation is the sare < TK of 10 / n minus one or the number of data points minus one so standard deviation is going to be the square < TK of 10 5 would be the number of data points of 5 - 1 is 4 so it's going to be standard deviation is the < TK 10 / 4 so standard deviation is the square root of 2.5 because 10 divid 4 is 2.5 so standard deviation is 1.58 so what do we do with this standard deviation which is 1.58 what does that actually mean once you've calculated standard deviation what you're going to do is you're going to place the average in the middle of this spell curve and what you're going to do is you're going to add and subtract that standard deviation to figure out where those points are on this graph so again in our bell curve we have the average which is in the middle which is three remember our standard deviation is 1.58 so we add 1.58 and we subtract 1.58 from the average so 68% of our data should fall between 1.42 and 4.58 now we do the same thing for two standard deviations we add another 1.58 and we subtract another 1.58 so for this data set 95% % of the data should fall between 0.16 and 6.16 and typically we stop here at two standard deviations for AP biology but you could go on to three standard deviations you would just add and subtract another 1.58 after I show students how to calculate standard deviation through the math problem like to show them you can do this easily on Google Sheets so as you can see in my Google spreadsheet I place the data here without any units just the numbers below here I place equals stde which stands for standard deviation I select my my data and put it within the parenthesis and I click enter and Google Sheets will spit out the answer for me which is 1.58 for my class I have my students go through one or two of these practice problems just as practice to understand the method behind calculating standard deviation however as we go through the class I really don't care and I encourage them to use these programs to calculate standard deviation a little bit quicker because in my mind I think the students should know there is math Behind these types of Concepts within statistics and biology but as as long as they know that concept they don't have to calculate that standard deviation every single time there are programs there are online resources there are websites that do this for you as long you as you can kind of describe the concept that's all I'm really worried about and it's really important to remember that not having a standard deviation is just mean get it because use the average to mean okay bad joke I know I'm sorry