[MUSIC] Now, central tendency is all very well. But rarely does it tell us the whole story. Consider the following image, which represents two histograms. So here, all I've done is shown the outline of these two histograms, namely the red and the black ones, such that they can be super imposed onto the same set of axis. Now these are being constructed such that both of these sample distributions have a sample mean of zero. So to perhaps put some real world perspective to this, consider these as perhaps the daily returns of two shares. The red share and the black share. Now, although they have the same measure of central tendency, the same mean, and you can sort of see from the symmetry here that this would also tend to affect the median of these distributions as well. Clearly these two distributions are very different from each other, mainly they share a different amount of spread or dispersion. So as far as our descriptive or summary statistics are concerned, yes, measures of central tendency such as the mean, median, and mode are useful. But we'd also like to consider another attribute of these distributions, mainly how dispersed or spread out they are. So here we're going to introduce some common are common and very widely used measures of dispersion or a spread, you can use those terms interchangeably. Now before we do this formally, if we consider this as the daily returns on to stocks whereby we collected some data maybe over a few months or a few years. And these are the histograms which are being generated, now which stock would you choose to invest in? Well clearly, that's going to depend on your personal appetite for risk. Now if we consider the black stock, we see here this has a much smaller dispersion than the red one. So if you were more of a risk-averse investor, by definition it means you're averse to, you dislike risk. Then clearly the black stock would be preferable to you over the red one. Because if you choose the black one, well think about it in terms of costs and benefits, the trade offs involved. By choosing the black stock where there is little variability in the returns, then of course there's a very small chance that you would ever lose a large amount of money. And clearly those risk-averse investors are indeed scared of losing large sums of money. But of course, on the flip side, you're are really restricting your potential upside by owning the black stock. Because there it's very unlikely that you'd ever make a very large return. In contrast those risk loving investors will tend to pick the red stock over the black one. Why, because of course the potential now for achieving a very high return is more achievable by holding the red stock than the black one. One could see, potentially, you might make as much as maybe 7.5% return on a single day's trading. In contrast, by holding that red stock, just as you are exposing yourself to the possibility of making a large sum of money, the flip side is you run the risk of losing a large sum of money as well. Because of course that return, as much as it can be a large positive value, it can be a large negative value as well. So we're not saying one stock is better or worse than the other. It will depend upon the individual investor, and his or her attitude to at risk. But nonetheless, we can see that we can have two distributions which share perhaps the same central tendency, but clearly differ a great deal in terms of their dispersion or spread. So what are our key statistical measures to try and quantify this amount of spread or dispersion? We'll begin with the so-called sample variance which we will denote by S squared. Now, think of the sample variance like an average. So when we considered measures of central tendency, our average or mean X bar was the sum all of the observations. So the sum of the XIs divided by the number of observations. So in this sense, the sample variance is also an average, but it's not an average just of the X observations, as we saw with X bar. Rather, it's a average of the squared deviations about the mean. So, let's consider an extreme example. Imagine your dataset consisted all of the entirely same values. Let's suppose each time you observed a value, it was equal to 5. So the first observation, X1 = 5. The second observation X2 = 5, so on and so forth until the nth observation, also equal to 5. Now trivially, the average, the sample mean of those values would be 5. So if we now consider the deviations from the mean for each of these observations, each of those deviations is going to be equal to 0. x1- xbar = 5- 5, the difference between those is 0, and that would extend across all of those n observations. Now we noted previously that the mean was a good measure of central tendency, in that it would always be the case that some of the deviations about the mean will add up to 0. But there, in our earlier example, we considered some positive and negative deviations from the mean existing within our dataset. Now because those positive and negative deviations cancel each other out, well we really care about any deviation from the mean, regardless of sine. So what we tend to consider are not the deviations from the mean but the squared deviations from the mean. Now, returning to our example of a full set of fives in our dataset, then of course each deviation from the mean is 0, and of course each squared deviation from the mean would be 0 squared and hence still 0. So in this extreme case all of our squared deviations from the mean are 0, and hence the average of those is also 0. Which is perhaps as we may wish it to be for any measure of variability, because if there is no variability within our dataset, we would want to assign a value for the variance of 0. So more generally, looking at situations where there is some variation across our observations, how would we calculate the sample variance? What formula should we use? Well, I mentioned the sample variance was a kind of average, and indeed it is. But rather than averaging the individual observations, as we do to work out the sample mean x bar. Here we're going to take an average of all of the squared deviations about the mean. So as the deviations from the mean tend to become greater and greater due to greater variability within our dataset, then of course upon squaring, those squared deviations tend to become a very large value. Now, for reasons I don't want to get into because it's rather technical and we'll need some more advanced statistical theory to address this. When we take an average of these squared deviations from the mean, we don't actually divide by n. We divide by n-1. Now for the purposes of this recording and this course, just trust me on this fact. But if you decide to take your studies of statistics to a much greater level Will discover the reason for why. But for practical purpose if n is very large, whether one divides by n or n or n minus 1 makes very little practical difference. So in summary the sample variance is an average, the average squared deviation about the mean, just with this divisor of n minus 1 rather than n. So as a dataset has more variability within it, if we return to those share returns, the red stock had much greater variation than the black stock. Then the average of those square deviations about the mean, ie the sample variance, would be much greater for the red stock over the black one. So, we have the sample variance S squared. Now, if our variable was measured in particular units, then as we have these squared deviations about the mean in our variance calculation, then the actual units of measurement of the variance would always be the square of the original units. Now frequently, the square of the original units may have no practical meaning. So if we're looking at share returns, let's say as percentage returns, the percentage on a given day. Then the units of the variants would be percent squared. Now that's a concept many of us can't easily relate to. Of course, sometimes the square of a unit may have some real world meaning, if we were looking at height, for example, measured in metres. Then the variance of height would be measured in metres squared, well, you can also roughly visualise what a square metre looks like. But in general, the squared unit has little real world applicability. Which is why in practice we tend not to focus so much on the sample variance s squared, rather its positive square root, the sample standard deviation. Because if we square root the variance, that means we are square rooting those squared units of measurement, and hence we return to the original units of measurement. So, it's very common to deal with the standard deviations over the variances, just to ease the interpretation of the result. So for any of you, perhaps they're interested in finance, and you'd like to know what are the key measures of risks or volatility in the financial markets. Be it related to the returns on an individual stock or maybe movements in an exchange rate. Well, the standard deviation is a very widely used measure in finance. The statistical measure to simply try and quantify the risk associated with any particular asset. So if we sort of bring together our measures of central tendency, and combine it with measures of dispersion, we can now focus on two key features of attributes of a distribution. Some sense about where it's centered, what is the typical perhaps return, the average return say on a stock. Typically denoted by our sample mean x bar, though conscious of its sensitivity to any outliers in our dataset. But also some measure of the dispersion and in terms of financial assets. The risk of holding these assets, which we could quantify either through all the sample variance. But to ease the interpretability, will tend to focus on the sample standard deviation instead. So these are some very widely used and very important descriptive statistics, so do make sure you're comfortable with these, and look out for these as when they're mentioned in the media. [MUSIC]
