 As Salaamu Alaikum, welcome to lecture number 14 of the course on statistics and probability. Students you will recall that in the last lecture I discussed with you in detail the box and whisker plot and towards the end of the lecture I discussed the Pearson's coefficient of skewness. As you will remember the Pearson's coefficient of skewness is defined as mean minus mode over the standard deviation and in those situations where the mode is not very easily found we can apply the empirical relation between the mean, median and the mode and in that situation the formula for the Pearson's coefficient becomes 3 times mean minus median over the standard deviation. . . . . . . . . . . . The answer that you will get will be in the same units as the units of the data itself, but when we divide it by the standard deviation we get a pure number and now we are in a a position to compare the skewness of any one distribution with the skewness of another distribution. A pure number can be compared with a pure number, but for example, if there is a distribution of a pound or a second distribution of inches, then the pounds cannot be compared with inches in a very proper way. So as you noticed, the Pearson's coefficient of skewness requires the computation of the mean as well as the standard deviation. Now if you want to measure the skewness of a distribution without having to compute the mean and the standard deviation, there is another formula and that is the one given by Baale. This formula students is based on the median and the quartiles and these are of course measures of partition and these also enable us to measure the skewness in a very interesting way. For an absolutely symmetric distribution, the distance between the median and the first quartile is exactly the same as the distance between the median and the third quartile. But in case of a positively skewed distribution, the distance between the median and the third quartile is longer than the distance between the median and the first quartile as you now see on the screen. So if I subtract x tilde minus q 1 from q 3 minus x tilde students, I will obtain a positive answer as you now see on the screen. q 3 minus the median minus median minus q 1 is the same thing as q 1 plus q 3 minus 2 times the median and as I just explained, in case of a positively skewed distribution, this quantity will be greater than 0. Only the opposite situation prevails in case of a negatively skewed distribution. In this case, the distance between q 1 and x tilde is more than the distance between x tilde and q 3 and if you subtract x tilde minus q 1 from q 3 minus x tilde, you will obtain a negative answer. q 3 minus the median minus median minus q 1 is again the same thing as q 1 plus q 3 minus 2 times the median and because of the fact that in this case, median minus q 1 is greater than q 3 minus median, hence the quantity q 1 plus q 3 minus 2 times the median comes out to be less than 0. So, this is the way you have seen that quartiles and the median also provide a way of determining the skewness of your distribution q 1 plus q 3 minus 2 times x tilde compute. If it is a positive quantity, it indicates that the distribution is positively skewed and vice versa, but students, the Baoules coefficient is given by as you now see on the screen q 1 plus q 3 minus 2 x tilde divided by q 3 minus q 1. The reason is that when we divide the numerator by q 3 minus q 1, once again we obtain a number which is unit less a pure number and hence we are able to compare the skewness of one distribution with another. Another important point is that it has been mathematically shown that this coefficient that I have just presented to you lies between minus 1 and plus 1. So, when you compute the closer it is to minus 1, the greater will be the skewness in the negative direction and the closer it is to plus 1, the greater will be the skewness in the positive direction or agar aapka answer 0 k bahut nisdeek, positive ya negative ho to is ka matlab ye hai, k your distribution is approximately symmetric. Let us apply this concept to the example of the ages of the children of the manual and non-manual workers that we discussed in the last lecture. As you will recall, we had data regarding the age of onset of nervous asthma in children and we had this information for the children of manual workers and the children of non-manual workers. Also, we had computed various sample statistics pertaining to the ages of these children and we had the figures that you now see on the screen. First, let us concentrate on the distribution of the ages of the children of the manual workers. Now, because q 1 is equal to 6.0, x tilde is equal to 8.5 and q 3 is equal to 11.0, you will realize that the distance between q 1 and x tilde is exactly the same as the distance between x tilde and q 3 and this leads to the symmetric picture that you now see on the screen. On the other hand, the situation of the ages of the children of the non-manual workers is quite different. q 1 is equal to 5.5, x tilde is equal to 9.2 and q 3 is equal to 10.8 and this implies that the distance between q 1 and x tilde is much greater than the distance between x tilde and q 3 and this leads to a negatively skewed picture that you now see on the screen. So, the distribution of the children of the manual workers is symmetric and the distribution of the ages of the children of the non-manual workers is negatively skewed. If we, for both data sets, compute the coefficient of skewness, as you see on the screen, the coefficient comes out to be 0 for the children of the manual workers and for the children of the non-manual workers it comes out to be minus 0.37. And the negative answer indicates that the distribution of the children of the non-manual workers is indeed negatively skewed. Students, the next concept that I am going to discuss with you is the concept of kurtosis. This term was introduced by Karl Pearson and it literally means the amount of hump in your distribution. In other words, this is the concept of the peakedness or the flatness of your distribution. Students, when the values in your data set are closely bunched around the mode, we say that the distribution is leptocurtic and it is a relatively high peaked distribution, as you now see on the screen. On the other hand, if the curve is flat topped, we say that the curve is platicurtic. Students, the normal curve is a curve which is neither very peaked nor very flat and this is the one which is taken as a basis for comparison. The normal curve itself is called a mesocurtic distribution and it is like a bell shaped curve as you now see on the screen. Students, the concept of the normal distribution is of the utmost importance in the theory of statistics and I will discuss this concept with you in a lot of detail, when we discuss probability distributions. It is of a moderate amount of hump and a beautiful bell shaped curve that is the kind that is called the normal distribution. Superimposing the three curves on the same graph, we get a picture as the one that you now see. The leptocurtic curve is the tallest one, the mesocurtic is the intermediate one and the platicurtic is the flat topped distribution. Students, peakedness or ketosis concept, I have discussed with you. Now, the next question is how we measure the degree of peakedness? As you now see on the screen, there is a formula which involves the quartiles and the percentiles and it is known as the percentile coefficient of ketosis. It is defined as quartile deviation divided by p90 minus p10. Students, this measure of ketosis is denoted by k and for the mesocurtic distribution, which is neither very peaked nor flat, its value comes out to be 0.263. Also, this measure, its value can range from 0 to 0.5 and for a leptocurtic distribution, it is less than 0.263, lies somewhere between 0 and 0.263 and for the platicurtic distribution, it is greater than 0.263. In other words, lies somewhere between 0.263 and 0.5. I am sure that you will understand that the closer your answer is to 0.263, the closer your curve will be to the normal curve. The next concept that I am going to discuss with you students is a very important concept and I would like to encourage you to listen to it and to study it very carefully. This is the concept of moments. What do I mean by moments? A moment designates the power to which deviations are raised before averaging them. I know. Let me explain this to you bit by bit. I have just talked to you. There are two or three terms involved. Let us first consider the term deviation. When we say that we are taking the deviation of the x values from a certain value, it is simply a morphoom that we are considering the difference between our x values and that particular value. So, if we say deviations from the mean, that we are talking about the column that we might have of x minus x bar, that is, if we have 10 x values and we have computed their mean, then we can measure 10 other deviations and they will be denoted by x minus x bar. For example, if the x values are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10, then what will be their mean? 5.5. And when we are talking about deviations, then what are we saying? 1 minus 5.5, 2 minus 5.5, 3 minus 5.5 and so on. This is the first thing. Now, I had told you that what is the moment? The power, a certain power to which the deviations are raised before averaging them. If we, I have just talked about these 10 deviations, if we take their first power and then sum them all up or divide them by the number of deviations, then this will be the first moment about the mean. As you now see on the screen, the first moment about the mean is given by sigma x minus x bar raised to 1 divided by n. We can say that the first moment about the mean is equal to sigma x minus x bar over n. In an earlier lecture, I said to you that the sum of the deviations of the observations from the mean is always equal to 0. And in this example which I have just discussed with you, you can see that this will be 1 minus 5.5 is minus 4.5 and the last value which is 10, 10 minus 5.5 is 4.5. Similarly, 2 minus 5.5 is minus 3.5, but 9 minus 5.5 is 3.5. So, your upper deviations are negative and your lower deviations are positive. And when you are adding, then your sum is 0. If sigma x minus x bar is equal to 0, then it is obvious that the first moment about the mean will also be 0. Students, in this first moment we denote it as m 1 and this is a small m. And this is of course, the situation when we are dealing with sample data. If we collect data from the entire population or compute these things, then our notation will be mu 1 for the first moment, mu 2 for the second and so on. Abhi me ne second moment ki baat ki students, second moment se meri kya muradhe? According to the definition that I gave earlier, we are talking about the power to which the deviations have to be raised before averaging them. So, if I talk about m 2, the second moment about the mean, it is given by sigma x minus x bar whole square over n. Students, ye formula jo abhi screen pe dekhaha ye to koi neya formula hi nahin ye. Do you not remember the variance when you squared the deviations of the x values from the mean, found their sum and divided by n? Yes indeed the second moment about the mean is exactly the same thing as the variance. And if you take the positive square root of this quantity, then of course, you obtain the standard deviation. Now that I have discussed the first and second moments, students, we are in a position to formally define the arath moment about the mean. The arath moment about the mean is the arithmetic mean of the arath power of the deviations of the observations from the mean. Symbolically, m r is equal to sigma x minus x bar whole raise to r over n. Ye sare moments jo main abhi tak aap ke sath discuss ke ye, these are moments about the mean and students, they are also called central moments. But then we can define moments in another way as well. Ye saroori to nahi hai ke aap mean he subtract kare. It can be any arbitrary value or it can be the number 0. So, let me define for you the moments about an arbitrary value alpha. As you now see on the screen, the arath moment about an arbitrary origin alpha is given by m dash r is equal to sigma x minus alpha whole raise to r divided by n. Students, aap ne note kia ke ab jo moment main define kia uske liye main notation is temal ki hai m dash r. Ye hi basic farkh hai between the moments about the mean and the moments about any other arbitrary value. We will denote the moments about the mean by m r and the moments about arbitrary value by m dash r. Aap se kushter pehle main aap ke sath discuss kia that the first moment about the mean is equal to 0. Now, let us consider what the first moment about alpha will be equal to. As you now see on the screen, the first moment about alpha is equal to sigma x minus alpha over n and if we open the bracket, we obtain sigma x over n minus alpha which is equal to x bar minus alpha. So, this is a simple relationship between the first moment about alpha and the arithmetic mean x bar. Isitara, we can develop a number of relationships and a short while later, I will be discussing with you formally certain relationships that exist between moments about the mean and moments about a and relationships which are very useful in computing the moments that we require. Lekin iss se pehle main aap ke sath moments about arbitrary origin ka ek special case discuss karna chaatin. Students, if you put alpha equal to 0, you obtain what we call moments about 0 or moments about the origin and as you now see on the screen, the alpha rth moment about 0 will be equal to sigma x raise to r over n. Obviously, if we put alpha equal to 0 in our basic expression for m dash r, the quantity x minus alpha will become simply x and the formula will become sigma x raise to r over n. So, in this, you note a point that when I was talking about moments about alpha, I said moments about an arbitrary origin and when I put alpha equal to 0, I said moments about the origin. Now, you know that if you visualize a graph paper, then the x equal to 0 will represent the origin. So, moments about 0 are also called moments about the origin. Let us apply the concept of moments to a simple example. As you now see on the screen, suppose we have data regarding the examination marks of just 9 students and the marks of the marks are 45, 32, 37 and so on. If we desire to compute the first 4 moments about the mean for this data, the first step of course, is to find the arithmetic mean and once we have done that, we will construct the columns of x minus x bar, x minus x bar whole square, x minus x bar whole cubed and x minus x bar whole raise to 4. Dividing the sums of these columns by the total number of observations, we will obtain m 1, m 2, m 3 and m 4. The important point to note is that the second moment is expressed in square units. The third moment is expressed in the cubes of the units and the fourth moment in the fourth powers of the units. So, that in this example, since we are dealing with the marks of the students, hence we say that m 2 is equal to 25.78 marks squared. Similarly, m 3 is equal to 20.67 marks cubed and so on. Students, you have noted that I computed the first 4 moments or what about the fifth, sixth and so on. Actually, if we go like this, there will be no end to it because obviously, r can be given any value, but a very interesting and important point to note is that the first 4 moments about the mean contain a lot of useful information regarding our data set. I would like to discuss with you the formula of the moments in the case of grouped data. Of course, they were for the case of the raw data and if we have grouped data, then that is, a frequency distribution students, the formula will be very similar to the case of the raw data with the only difference being that the frequency f is inserted in the formula as you saw in earlier cases. So, as you now see on the screen, the r th moment about the mean is given by sigma f into x minus x bar whole raise to r over n and the r th moment about alpha is given by sigma f into x minus alpha whole raise to r over n. Students, in the case of the frequency distribution of a continuous variable, you will recall that x represents the midpoints of the classes. And as I explained in an earlier lecture, this assumption that all the values lying in a particular class are equal to the midpoint of the class, this introduces a certain amount of error and that is called grouping error. May be you remember that I told you that in the case of the arithmetic mean, this error is not significant and it can be ignored. But students, in the case of the variance and the higher moments, this error becomes quite important and it is advisable to apply some sort of a correction for this error. W F Shepard has introduced certain corrections which are known as Shepard's corrections for grouping error. And as you now see on the screen, the Shepard's corrections are given by the formula a. M 2 corrected is equal to M 2 uncorrected minus h square over 12. M 3 corrected is the same thing as M 3 uncorrected. In other words, the third moment about the mean does not need any correction and M 4 corrected is equal to M 4 uncorrected minus h square over 2 into M 2 uncorrected plus 7 over 240 times h raised to 4. All these corrections are valid in the case of frequency distributions of continuous variables and these distributions should be of the form that they should tail off to 0 at each end. It is to be noted that in all these formulae h will represent the uniform class interval. Shepard's corrections ko ham abhi thori der ke baad ek example pe apply karenge. Usse pehle students, I would like to convey to you the mathematical relationships that exist between the moments about the mean and the moments about alpha. As you now see on the screen, M 1 is equal to 0 as stated earlier. M 2 is equal to M 2 dash minus M 1 dash whole square. M 3 is given by M 3 dash minus 3 times h times M 2 dash into M 1 dash plus twice M 1 dash whole cubed and M 4 is equal to M 4 dash minus M 3 dash into M 1 dash plus 6 times M 2 dash into M 1 dash square minus 3 times M 1 dash raised to 4. I will not be discussing the mathematical derivation of these relationships in this course, but of course, if any of you is interested, you are welcome to study the mathematics behind these formulae and they are very much in your own textbook. What I would like to discuss with you is that there is an easy way of remembering these relationships with which will be useful to you when you would like to compute the moments for real life data sets. As you now see on the screen, the two points to remember are number 1 in each of these relations, the sum of the coefficients of the various terms on the right hand side equals 0 and number 2 each term on the right is of the same dimension as the term on the left. Alright, let us now apply all these concepts to an example. Suppose that we have data regarding the marks obtained by students in a test, the total marks were 20 and the students are obtaining marks between 5 and 15, the number of students obtaining these marks are 1, 2, 5, 10 and so on. Suppose that we are interested in computing the first four moments about the mean for this data. If we wish to do so directly, the first step is to compute x bar and for this data set students x bar comes out to be 10.06. Now the point is that x bar equal to 10.06, this is not a very convenient number to work with because in all those columns that you have to construct x minus x bar, x minus x bar square cubed and force power, they will involve a lot of decimals. So, what we do is that first we compute the moments about alpha, where alpha is a convenient number. To work with and then we utilize those relationships that I can wait to you to compute the moments about the mean. So, in this example we consider alpha equal to 10, which is the x value against the maximum frequency 51 and using this number we construct the column of capital D, which is the same thing as x minus 10. By doing so, we obtain minus 5, minus 4, minus 3 and so on and then multiplying this column by f we obtain f d. Similarly, we obtain f d square f d cubed and f d raised to 4. Dividing the sums of these columns by 131, the total number of students we obtain m 1 dash, m 2 dash, m 3 dash and m 4 dash as you see on the screen. Applying the relationships that exist between moments about the mean and moments about alpha, we obtain m 2 equal to 2.64, m 3 equal to 0.08 and m 4 equal to 28.30. Applying the shepherds corrections for grouping error, m 2 corrected comes out to be 2.56, m 3 corrected is the same as m 3 uncorrected and m 4 corrected comes out to be 27.01. Students we have been talking about moments for the last 10-15 minutes and you must be wondering why are we going through all these lengthy calculations. The reason is that as I said earlier moments give you a lot of information about your distribution and in this connection I will now introduce the concept of moment ratios and once we are through that you will be able to understand clearly what the important role of moments is in describing a frequency distribution. Ye jum nai moment ratios ki baat ki iss se murad wo quantities hain jin mein both the numerator and the denominator consist of moments and the most important moment ratios are b 1 and b 2 and as you now see on the screen, b 1 is given by m 3 square over m 2 cubed and b 2 is equal to m 4 over m 2 square both of these quantities are pure numbers and b 1 is used to measure the skewness of the distribution whereas b 2 is used to measure the kurtosis or the peakedness of the distribution. Aayi pehle b 1 pe concentrate karte hain. Students it has been mathematically shown that for a symmetrical distribution b 1 will be equal to 0 but for skewed distributions b 1 will be positive. So in any data set if your b 1 comes out to be 0 or approximately 0 you can conclude that your distribution is approximately symmetric. This means that the distribution is skewed. Now as far as the direction of skewness is concerned students you should look at the sign of the third moment about the mean that is m 3. If m 3 is a positive number it means that your distribution is positively skewed. If m 3 is negative it implies that the distribution is negatively skewed. b 1 cho hain that is m 3 square over m 2 cubed or squaring ki wajase b 1 to bahar hal positive hi aega. Next, let us talk about b 2. It has been mathematically proved that for the normal distribution which is neither very peaked nor very flat b 2 is equal to 3 and students therefore this number 3 acts as the basis for us to decide whether our distribution is leptocortic, mesocortic or platicortic. If for any data set your b 2 comes out to be equal to 3 you can say that your distribution is like the normal curve, the bell shaped curve which is neither very peaked nor flat. But if your b 2 comes out to be greater than 3 this implies that your distribution is leptocortic that is the one which is more peaked than the normal distribution. And if your b 2 comes out to be less than 3 then your distribution is flat that is platicortic. Students b 1 or b 2 ki discussion se aap pe ye waje ho jaan ha chahiye that the third moment and the fourth moment play a very important role in indicating the skewness and the kurtosis of your distribution. After all m 3 occurs in the numerator of b 1 and m 4 is in the numerator of b 2. What about the dispersion and the center of your distribution? Do you not remember that a short while ago I indicated to you that the second moment about the mean is none other than the variance and the positive square root of the variance is the standard deviation the most important measure of dispersion. What about the center? You will be interested to realize that the first moment about zero is none other than the arithmetic mean. After all how will you define the first moment about zero? Sigma x minus 0 over n which is exactly the same as sigma x over n minus the arithmetic mean. So, in this manner students moments indeed play a very very important role in describing frequency distributions. Next time I will discuss with you quite a different and a very interesting concept the concept of correlation and regression. In the meantime I would like to encourage you to practice all these formulae that you have learnt today and to make yourself at home with the concept of moments. Best of luck and Allah Hafiz.