Transcript
1. The Normal Distribution
00:00 - 05:17
A statistical distribution is a way to represent the possible values and their respective probabilities for a random variable. It describes the likelihood of observing different outcomes or values of a variable in a population or a sample. We say that data like stock returns are normally distributed when more values are found closer to the mean, less values are found further away from the mean, and half the values are below the mean and half the values are above the mean. This gives us the well known bell shaped curve that we can see here. The standard deviation is used to measure how spread out the data is. The more spread out the data, the wider the bell shaped curve and the larger the standard deviation. A really useful property of the normal distribution is that 68% of the values in the sample or population are within one standard deviation of the mean. This means 34% lie one standard deviation below the mean and 34% lie one standard deviation above the mean. Going a bit further, 95% of the values lie within two standard deviations of the mean and 99.7% of the values lie within three standard deviations of the mean. Normal distributions tables can be used to show this information, although Excel has some really useful normal distribution functions to make our lives a bit easier. So here we are back in our Statistics for Finance template workbook and I've scrolled down to the normal distribution section. Now I want to introduce a function in Excel and it's called the NORMSDIST function, and it's written like this, NORMSDIST function. And what this does is it calculates the area under the curve, and remember the area under the curve is the same as the probability, the area under the curve from the far left hand side of the normal distribution up to a certain number of standard deviations from the mean. So if I start in the middle of my table here where I've got zero standard deviations from the mean, zero standard deviations from the mean is the mean, it's right in the middle of our normal distribution. So if I use the NORMSDIST function and if I take Z as my number of standard deviations from the mean which is zero, and if I say true for cumulative distribution and hit the enter once I close the bracket, I get 50%. And that's because the area under the curve from the far left hand corner of the normal distribution up to the mean is 50%. Now remember, if I move one standard deviation above the mean, that's between the mean and that point is 34%. And so if I go one standard deviation above the mean, the area under the curve from the far left hand corner up to that point will be the 50% up to the mean plus another 34% which means we have 84.13%. And if we go all the way to four standard deviations above the mean, we get to 100% because in theory, 100% of the values for a normal distribution lie within four standard deviations of the mean. So if I just have a look at this point here, two standard deviations above the mean, I have 97.22. And we can see that in the diagram and it's represented by the double asterisks. So if I just do the double asterisks, you can see 97.72 is the area under the curve from the far left hand corner up to that point. The remaining area, the area under the curve to the right of two standard deviations above the mean must be one minus the 97.72% which is 2.28%. And so we can use the NORMSDIST function to find the area under the curve. Now if I start from the far left hand corner, four standard deviations below the mean, the area under the curve, do you know what it would be? Have a guess while I type out my NORMSDIST. Zero or minus four standard deviations from the mean and we want the cumulative true. Now what do you think it is? Did you say 0%? Now if you did, well done. Because the area under the curve, if we start on the far left hand corner of the curve, the far left hand corner is four standard deviations below the mean so there's no area under the curve as yet. But as we get closer to the mean, the area under the curve gets bigger and bigger. And so, if I have a look at this 15.87, one standard deviation below the mean, I've represented that on my diagram with a single asterisk. We can see there from the far left hand corner up to one standard deviation below the mean, the area under the curve is 15.87%. So the NORMSDIST function is a great function to use in Excel to find the area under the curve. But remember, it works in a certain way. It finds the area under the curve from the far left hand corner up to a certain number of standard deviations away from the mean.


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