 Hello and welcome to the session. In this session we shall study distributions with small variability but different centers. Here we will study about two distributions with small variability but there is a difference in their centers. Although the distributions might seem to overlap each other but the heights of bars in dot-plots show the difference between the two distributions. Let us consider the following illustration. A class consists of 17 boys and 17 girls. They were asked how many times they go for an outing with their families in a month. Then girls in boys data was recorded separately and we can see that the variation in girls data is less. Similarly, variation in boys data is also less. So we can say that both the data have similar variability. Also the data seems to be more or less similar to each other thus data might seem to overlap with similar variation. So is the data more or less same for both boys and girls? How can we find out? For this we can see the center for both the series. Now let us draw a dot-plot for both series. First we shall draw dot-plot for girls. As we can see that two occurs three times in the girls series. So we mark two three times. Three occurs three times. So we mark three three times. Similarly four occurs five times. So we mark four five times. Five occurs three times. So we mark five three times and six also occurs three times. So we mark six three times. So here we get the required dot-plot for girls. Now we shall draw the dot-plot for boys. Here in the boys series we see that three occurs once. So we mark three once. Four occurs three times. So we mark four three times. Five occurs two times. So we mark five twice. And we also see that six occurs four times. Seven occurs three times. And eight occurs four times. So we mark six four times. Seven three times. And we mark eight four times. And therefore we get the required dot-plot for boys. And clearly we can see the difference in height of the bars in two graphs and in the girls dot-plot the center is four whereas in the boys dot-plot graph the center is given by six. So we can say that they have similar variability. Their centers are different. Looking at the distribution of the data sets we can observe that there is some overlap in the two data sets. Some boys and girls have both gone and outing with their families three, four, five or six times. Like one boy and three girls went for outing three times. Three boys and five girls went for four times in a month. Two boys and three girls went for five times. And four boys and three girls went for outing six times in a month. Now we use mean as the measure of center, absolute mean deviation as the measure of variation to compare two data sets. Now we shall first calculate mean for girls series. And we know that mean is given by sum of all the observations divided by the total number of observations. So here we have four plus two plus two plus four plus three plus two plus five plus six plus four plus three plus six plus four plus five plus three plus four plus six plus five divided by the total number of observations which is equal to seventeen which is equal to sixty eight by seventeen that is equal to four. For girls series we got the mean which is equal to four. Now we shall find out mean absolute deviation with the help of this mean. And now we shall calculate mean absolute deviation with the help of the following table. First we shall find deviation which is given by value minus mean. So we have mean as four and therefore we get four minus four which is zero, two minus four minus two, two minus four minus two, four minus four zero, three minus four minus one, two minus four minus two, five minus four one, six minus four two, four minus four zero, three minus four minus one, six minus four two, four minus four zero, five minus four one, three minus four minus one, four minus four zero, six minus four two, five minus four is one. and absolute deviations are given by taking modulus of deviations so we get 0. Modulus of minus 2 is 2 and similarly we get all the values mean absolute deviation is given by sum of absolute deviations from mean divided by the total number of observations which is given here as 17 and here sum of absolute deviations from mean is equal to 2 plus 2 plus 1 plus 2 plus 1 plus 2 plus 1 plus 1 plus 2 plus 1 which is equal to 18 divided by 17 and we get 1.06 so for both series we get mean as 4 and mean absolute deviation is equal to 1.06 similarly we shall find mean and mean absolute deviation for boys and we know that mean is given by sum of all the observations divided by total number of observations and if we have all the numbers in the boys series that is 4 plus 3 plus 5 plus 6 plus 8 plus 7 plus 4 plus 8 plus 6 plus 7 plus 6 plus 8 plus 4 plus 6 plus 7 and we divided by 17 that is the number of observations and we get the sum as 102 divided by 17 and therefore we get the mean as 6. Now we shall calculate mean absolute deviation for boys with the help of this mean. We know that the value of mean is given as 6 here we shall calculate deviations which is given by value minus mean so we get 4 minus 6 that is minus 2, 3 minus 6 minus 3, 5 minus 6 minus 1, 6 minus 6 0, 8 minus 6 2, 7 minus 6 is 1, 5 minus 6 is minus 1, 8 minus 6 will be 2, 4 minus 6 is minus 2, 8 minus 6 is 2, 6 minus 6 0, 7 minus 6 1, 6 minus 6 0, 8 minus 6 2, 4 minus 6 minus 2, 6 minus 6 0 and 7 minus 6 is 1 and absolute deviations are given by taking modulus of deviations so we get 2, 3, 1, 0, 2, 1, 1, 2, 2, 2, 0, 1, 1, 0, 2, 2, 0 and 1. We know that mean absolute deviation is given by sum of absolute deviations from mean divided by total number of observations which is given here as 17. Here sum of absolute deviations from mean is given by 2 plus 3 plus 1 plus 2 plus 1 plus 1 plus 2 plus 2 plus 1 plus 2 plus 2 plus 1 and we get 22 divided by 17 which is equal to 1.3 so here for boys mean is equal to 6 and mean absolute deviation is equal to 1.3 so we can say mean absolute deviation in both the series is around 1 as it is 1.06 for girls and 1.3 for boys so we can say that mean absolute deviation in both the series is around 1 and the difference of means is equal to 6 minus 4 that is 2 so here we can say that difference in the centers is approximately twice the variation in each distribution this completes our session hope you enjoyed this session