the first descriptive statistic we often run into and in our chapters on descriptive statistics is the mean the arithmetic mean of the average value across a sample of data which we denote typically by x-bar and it's basically the sum of all the X values divided by the number of data values in the data set the next level to which we're in student script statistics tends to be the variance of our data set so how much does our data vary around the mean so the formula for that is really quite simple but here we're gonna enter into an entry where I'm indicating the deviation value a deviation is how far does a data value fall from the mean and that's gonna be a very very important value throughout statistics we're often in our applications looking to minimize the overall deviation and a data set so typically we've got outliers data points that are not at our mean plus or minus some value we want to know how much they tend to vary around it cuz two data sets with the same mean could still have very widely different distributions in their data set as we go by squaring the deviation deviation being the difference between the x value and how far it is from the mean by squaring that value we tend to exaggerate the values that are farther from the mean so that a data set that's very very narrow will have a lower variance than a data set that's very very wide even if the means of the two data sets are the same so we square those values as we add up deviations that we get that exaggeration of the outliers notice that compared to the mean we're dividing this value by n minus 1 that's very very important because that indicates the degree of freedom in the data when we calculated the mean there were n data values and we divided by n to get our values we didn't know what the mean was yet but notice that in the variance formula we already have x-bar as one of our elements in the data value so basically the sum of the data it has less freedom than it did up here in the mean an example being if I tell you that two numbers have an average of five and one of them is a six what is the other one you know the average the other value has to be a four because that's the only way you can get an average of five so when I already know the mean I have less freedom the last data value in my data set is something that has to be what it is in order for that to be the mean so when we're looking at the divisions here we divide by n to get the mean but we divide by n minus 1 to get our variance we don't have quite as much freedom in the data and that's a concept you'll see throughout statistics that confuses some people but it's actually quite simple what is the degree of freedom of the data so if I'm looking at the variance of a 10 element data set and I already know the mean of the 10 elements I really only have 9 degrees of freedom going in is that 10th value is already set or I couldn't have known my mean in the first place so the variance is a measure of how wide our data is where I'm exaggerating the outliers because the farther the farther I am from the mean the larger that deviation the more the squared value tends to exaggerate the value beyond the variance I can start to look at not just how wide is the data set but how symmetric is it is it skewed one way or the other and the skewness of your data set is gotten by again further exaggerating the outliers that the farther you are from the mean the more you exaggerate as you go through so this this formula looks fairly complicated but it's not as complicated as it looks because the basis of the formula is the deviation X minus x-bar just like in the other formulas we're cubing that value as we sum them to really exaggerate those outliers and one of the big advantages to cubing a deviation is that the result could end up being negative or positive the variance can only be positive so that we're squaring those deviations but here we're cubing them so we could end up with a negative number or we could end up with a positive number and that's exactly what we're looking for is the data set skewed to the left or to the right is it to them to the minus side or to the plus side and we'll be able to know that as things come through this formula includes SS is the standard deviation the standard deviation of a data set is the square root of the variance s squared is considered the variance so we're dividing here by s to the third standard deviation cubed so notice we have X bar built into the formula we have X bar is actually built into the definition of variance as we go through getting up to Rs so in terms of our degree of freedom not only we have to divide by n minus 1 but is an element of n minus 2 in essence as we work our way down the page here we're giving up a certain amount of freedom by building the results of earlier results the mean and the variance into the formula that we're using so the more we do that the less freedom we have for all the data values that would be involved which is why you see the bottom of this equation getting more complex once I know whether the data is skewed left or right what I'd really like to know is how tall is it or low is it if you imagine a bell curve what someone would call the normal curve that we'll talk about a lot in this class there's a classic shape that it tends to take on but some bell curves are narrower and taller than others and some are flatter and wider than others and kurtosis is a measure of how much that is so again if you're getting used to the pattern here you can see how the equation works because now we're gonna take that deviation as we add it up and we're gonna raise it to the fourth power to really exaggerate the farthest outliers so that big outliers that are far from the mean will tend to pull that curve down on average where most of most of the outliers are near the mean the good curve will tend to be very tall and near it so a measure of kurtosis is a measure of how tall or short the basic curve is and again because we're again taking into account more and more of the data that brought us to this point we're losing degrees of freedom as we go so if you think of this that the variance was the deviation squared the skewness is the deviation of the third power the kurtosis is the deviation to the fourth power it's not surprising that we're we had and first back here on the mean we had an minded and with n minus 1 on the variance we end up with basically N squared n minus 1 times n minus 2 is largely N squared so we end up with N squared here on the bottom and down here we end up with n cubed here on the bottom so you know an N squared and cubed you can imagine where this would go if you went into in higher moments of the mean and these are referred to as the moments of the mean and these are the four we basically need to do is to descriptive statistics the first two most of the time ninety ninety-five percent of the time if you know the mean and the variance of your data set you understand what that data set looks like in terms of a distribution if you really want to know the finer details of that you want to know if that distribution is skewed to the left or the right of the mean that's the skewness variable and you want to know how if it's taller or shorter than the bell curve would typically be and that's thicker ptosis those are the four moments of the mean that we care about in this class when you hear me talk about it I'm less concerned with the equations you can always look them up but the important element for this discussion is that we're looking at higher and higher moments of the mean looking at that first power second power of third power fourth power and the higher we raise those exponents the more we take advantage of that variation and the deviation that we have to understand the full picture of the data set so just by knowing these four numbers we basically know exactly what that curve is going to look like which is important in descriptive statistics because if we know what a curve is going to look like we'll be able to apply a whole bunch of rules to the probability functions under that curve to be able to make inferences or predictions but what should happen in the real world based on that data and that's the real statistical bridge between descriptive statistics and inferential statistics