 Hello and welcome to the session. In this session we will discuss measures of dispersion. The dispersion or scatter in the data is measured on the basis of the observations and the types of the measure of central tenancy used there. Measures of central tenancy is not sufficient to give complete information about a given data. For this purpose we use a factor which is called variability and the number that is the single number which describes variability is called measure of dispersion. Here we will discuss measure of dispersions like range and mean deviation. First we have range. Range of a series is equal to the maximum value minus the minimum value. The range of data gives us a lot of idea of variability or scatter but it does not tell about the dispersion of the data from a measure of central tenancy. Now we shall discuss mean deviation which is very important measure of dispersion. Mean deviation is basically the mean of the absolute values of the deviations from the central value. Now we shall discuss mean deviation for ungrouped data. Mean deviation about the mean denoted by this is given by 1 upon n summation i goes from 1 to n modulus xi minus x bar. Here we have xi are the observations. X bar is the mean. N is the total number of observations. Then mean deviation about the median denoted by this is given by 1 upon n summation i goes from 1 to n modulus xi minus m where this m is the median. xi are the observations and n is the total number of observations. Consider this data given to us. Here we have total number of observations that is n is equal to 10. Now the mean x bar is given by summation xi upon n that is sum of all these observations which comes out to be equal to 120 upon n that is 10 is equal to 12. So we get the mean for this data is 12. Now the values xi minus x bar are given by 3 5 minus 2 1 minus 5 6 minus 3 minus 6 2 minus 1. Then the values modulus xi minus x bar are given as 3 5 2 1 5 6 3 6 2 1. Now we have mean deviation about the mean is equal to summation modulus xi minus x bar that is sum of these values which comes out to be equal to 34 upon n that is 10 and this is equal to 3.4. So we get mean deviation about the mean for this data is 3.4. Then we need to find the mean deviation about the median for that purpose we will not calculate the mean x bar but we will calculate the median m. Next we discuss mean deviation for group data. We know that the data can be grouped as discrete frequency distribution and continuous frequency distribution. First we shall discuss mean deviation for discrete frequency distribution. The distribution in which we are given different observations with the corresponding frequencies is the discrete frequency distribution. In this case mean deviation about the mean is given by 1 upon n summation i goes from 1 to n f i into modulus xi minus x bar. Here we have the mean x bar is given by 1 upon n summation i goes from 1 to n x i into f i. x i are the observations f i are the corresponding frequencies then n is the total number of observations and this n is equal to summation f i i goes from 1 to n that is sum of the given frequencies. Next the mean deviation about the median m for discrete frequency distribution is given by 1 upon n summation i goes from 1 to n f i into modulus xi minus m. Here we have the median is m. Consider this data given to us in which we are given the observations xi and the corresponding frequencies f i. Now here we have n is equal to summation f i that is sum of these frequencies which is equal to 66. Now let's calculate the mean x bar which is equal to summation xi f i upon n this is equal to 528 upon 66 and this is equal to 8. So we get x bar that is the mean is equal to 8. Now the values for modulus xi minus x bar is given by 5 3 1 1 3 5 and then f i into modulus xi minus x bar is given by 30, 24, 15, 25, 24, 20. Now summation f i into modulus xi minus x bar that is sum of these values is equal to 138. Now mean deviation about the mean is equal to 1 upon n that is 1 upon 66 multiplied by summation f i into modulus xi minus x bar that is 138 and this is equal to 2.09. Hence we get this is the mean deviation about the mean for this given data. Likewise to calculate the mean deviation about the median we need to find the median for the given data. Now let's discuss how we find mean deviation for continuous frequency distribution. A continuous frequency distribution is a series in which the data are classified into different class intervals without gaps along with the respective frequencies. Mean deviation about the mean for continuous frequency distribution is given by the same formula as for discrete frequency distribution that is equal to 1 upon n summation i goes from 1 to n f i into modulus xi minus x bar but in this case we compute x bar that is the mean by standard deviation method that is x bar is given by a plus summation i goes from 1 to n f i into d i upon n multiplied by h. Here this a is the assumed mean h is the common factor n is equal to summation f i that is sum of all the frequencies then d i is given by xi minus a upon h. So in this way we calculate the mean that is x bar for continuous frequency distribution. Mean deviation about the median is given by the same formula as for discrete frequency distribution that is 1 upon n summation i goes from 1 to n f i into modulus xi minus m. Here the median m is given by the formula l plus n upon 2 minus c upon f multiplied by h. Here l is the lower limit of the median class h is the width of the median class f is the frequency of the median class and c is the cumulative frequency of the class just preceding the median class. And we know that the median class is the class interval whose cumulative frequency is just greater than or equal to n upon 2. Consider this continuous frequency distribution given to us in which we have given the class intervals and the corresponding frequencies xi is the midpoints of the class intervals given by 100, 110, 120, 130, 140, 150. Now see if that is the cumulative frequencies are given by 9, 22, 47, 77, 90, 100. Now here we have n is equal to summation f i that is sum of these frequencies which is equal to 100 then n upon 2 is equal to 50. Now median class is the class interval whose cumulative frequency is just greater than or equal to 50. So as you can see cumulative frequency for the class interval 125 to 135 is 77 which is greater than 50. So we say the interval 125 to 135 is the median class and so from here we have l that is the lower limit of the median class is equal to 125 h width of the median class is 10 f that is the frequency of the median class which is 30 then c which is the cumulative frequency of the class just preceding the median class that is for this class the cumulative frequency would be 47 so c is equal to 47. So now the median m is equal to l that is 125 plus n upon 2 that is 50 minus c that is 47 upon f that is 30 multiplied by h which is 10 this comes out to be equal to 126. Now the values for modulus xi minus m are given by 26, 16, 6, 4, 14, 24 and values for f i into modulus xi minus m by 234, 208, 150, 120, 182 and 240. Now mean deviation about the median m is equal to 1 upon n that is 1 upon 100 multiplied by submission f i into modulus xi minus m that is sum of these values which comes out to be 1134 so this is equal to 11.34 so this is the mean deviation about the median for the given continuous frequency distribution. This completes the session hope you understood the concept of mean deviation and how we find the mean deviation for ungrouped data and group data.