 Hello and welcome to the session. In this session, we are going to discuss Scholesian theorem. There are procedures to measure the relation between the mean and the standard deviation. The mean is the measure of centrality of a set of observations and the standard deviation is the measure of the spread There are two general rules that establish a relationship between these measures and the set of observations. The first is called the Scholesian theorem and the second is empirical rule. And now we shall study Scholesian theorem in detail. Scholesian theorem applies to any distribution Regardless of shape, it places lower limits and the percentage of observations within a given number of standard deviations from the mean. The Scholesian theorem establish the following rules. One, at least three-quarter of the observations in a set will lie within two standard deviations of the mean. And second, at least eight-ninths of the observations in a set will lie within three standard deviations of the mean. In general, Scholesian theorem states that at least one minus one upon k-square of the elements of any distribution lie within k-standard deviations of the mean where k is a number greater than or equal to one. We can explain this in another way that Scholesian shows that for any number k greater than one the probability that a value of a given random variable will be within k-standard deviations of the mean at least one minus one upon k-square. Here we take an example. We have stock rises that last 35 days. As follows, now we have step-by-step instructions for calculating variance and standard deviation. First, we calculate the mean x-bar. Second, write a table that subtracts from each observed value then list x-observations mean that is x-bar x-x-bar and x-x-bar the whole square. Next, v-square each of the differences that is x-x-bar the whole square. Then we add x-x-bar the whole square column and then divide by n-1 where n is the number of items in the sample. This is the variance. In order to calculate standard deviation take the square root of the variance. Now using the above instructions the sample mean is calculated as that is x-bar is given by 4.4286 that is we have taken the sum of all the divers and divide it by 45 variance is given by 4.1933 and standard deviation is equal to 2.0478 thus we have found we mean that is 4.4286 and the standard deviation that is 2.0478 means that for this study we have 4.4286 minus of 2.0478 which is equal to 2.3808 4.4286 plus 2.0478 which is equal to 6.4764 in this instance the price of the stock ranges from 2.38 to 6.48 therefore based on Shadisha's theorem it indicates that at least three quarters of the observations in a set will lie within the interval mean plus minus of 2 into standard deviation now for the value of k is equal to 2 we have 1 minus 1 upon k square which is equal to 1 minus 1 upon 2 square that is 1 minus 1 upon 4 which is equal to 4 minus 1 by 4 that is 3 by 4 which is equal to 0.75 this is interpreted that at least 0.75 of the observations will fall between minus 2 and plus 2 standard deviations furthermore we calculate 4.4286 plus minus of 2 into 2.0478 which are defined by the points 0.333 and 8.5242 thus we can say that at least 75% of the stock prices fell between 0.33 and 8.5 and a review of the data shows that 1 observation but as 0.333 lies outside this range of values since there are 35 observations in the set 33 out of 35 are within the specified range so we rule that at least 3 quarter will be within the range of satisfied this computation indicates that at least 75% of the data lies within 75% of the mean in the above example we can say that at least 75% of the stock prices are at least within 2 standard deviations of the mean similarly calculation can be made using 3 standard deviations let us take another example suppose we ask 1000 people what their age is if this is a representative sample then there will be very few people of 1 to 2 years old just as there will not be many 95 year olds most will have an age somewhere in their 30s or 40s a list of the number of people of a certain age may look like this next we can turn this list into a data diagram with age on the horizontal axis and the number of people of a certain age on the vertical axis and it will make a bell curved shape a bell curve is perfectly symmetrical with respect to a vertical line through its peak and is sometimes called a gauze curve or a normal curve now let us assume that we have recorded 1000 ages and computed the mean and standard deviation of these ages the mean age amount as 40 years the standard deviation 6 years and the following can be said about the individual data which in this case are the ages at least 75% of all the ages will lie in the range of x by plus minus of 2 into f and s stands for standard deviation in our case it means that at least 75% of the people will have an age in the range of 40 plus minus 2 into 6 which is equal to 40 plus minus 12 which simplifies to a range of 28 to 52 years that is if we put the value of k as 2 in 1 minus 1 upon k square then we get 1 minus 1 upon 2 square which is equal to 0.75 which implies that 75% of the observations will lie between mean minus k into standard deviation to mean plus k into standard deviation so 75% of the observations will lie between mean that is 40 minus k that is 2 into standard deviation which is equal to 6 which gives 28 to 40 plus 2 into 6 which is equal to 52 and then we have at least 88.9% of the ages will lie in the range of mean that is x by plus minus of 3 into standard deviation that is if in our case it means that at least 88.9% of the people will have an age in the range of 30 plus minus 3 into 6 which is equal to 40 plus minus of 18 which simplifies to a range of 22 to 58 years at least 93.75% of all the ages will lie in the range of mean that plus minus of 4 into standard deviation is in our case it means that at least 93.75% of the people will have an age in the range of 30 plus minus of 4 into 6 which is equal to 40 plus minus of 24 which simplifies to a range of 16 to 64 years and similarly we can say that at least 96% of all the ages will be in the range of mean x by plus minus of 5 into x that is the standard deviation and in this case it means that at least 96% of the people will have an age in the range of 40 plus minus of 5 into 6 which is equal to 40 plus minus of 40 which simplifies to a range of 20 to 70 years now we shall learn how to calculate these percentages now to calculate the 75% the 88.9% and the 3.75% etc we look at the number of standard deviations in the respective intervals the 75% goes together with mean plus minus of 2 into standard deviation the 88.9% goes together with mean plus minus of 3 into standard deviation and the 3.75% goes with mean plus minus of 4 into standard deviation so in general we can say that the percentage of people with an age in the range of mean plus minus of k times standard deviation by calculating the value of the quantity 1 minus 1 upon k square by converting that into a percentage summarizing the above we get the following table we should note that we can take any value of k as long as it is larger than 1 for example for k is equal to 2.5 we get the result that 1 minus 1 upon 2.5 divided by square which is equal to 0.84 or 84% in the interval 40 plus minus of 2.5 into 6 which is equal to 40 plus minus of 15 years this completes our session hope you enjoyed this session