 Hello and welcome to the session. In this session we will use mean and standard deviation of a data set to fit into a normal distribution and to estimate population percentage. So let us discuss about normal distribution and area under normal curve. First of all let us discuss normal curve. Now see the following histogram. Here data is symmetrically distributed as only one highest bar. Also the corresponding bars on the right and left side of the highest bar are equal in size and if you draw a free hand curve starting from the midpoint of the first bar, the highest bar and coming down to the midpoint of last bar we get a bell shaped curve. The tip of distribution is symmetric curve having 1p at the centre so that one part is near image of the other and the highest point is at the centre where mean is equal to median is equal to mode normal curve. So normal curve distribution is a frequency distribution that often occurs when large number of values distribution which is called normal curve the frequencies of the distribution of the portion of the distribution. Now see the middle three bars so most of the portion is around centre a small portion of the population occurs at the extreme small portion is at the end of normal distribution is a theoretical model of population it depends population describes the shape of the curve that is how far will be the two ends of the curve. The normal distribution occurs quite frequently in real life length of newborn babies of manufactured items but by normal distance the number of data values must be variation to be approximately normal. Now let us discuss properties of normal curve graph is symmetric about centre and it is bell shaped. Second property is point reached by the curve at the mean, mean is equal to median is equal to mode. Now let us discuss area under normal curve or a normal curve we take horizontal axis. Now the points taken on the horizontal axis represent values that are certain number from mean. Normal curve here each interval represents one centre deviation and here the centre value is taken which is denoted by mu here the values that is on right side of mean are taken as x bar plus sigma x bar plus 2 sigma then x bar plus 3 sigma and so on where sigma denotes standard deviation of mean are taken as x bar minus 2 sigma and so on under the normal curve above horizontal represents mean divides the graph in two equal parts different values are on the right side of the, on the left side we are on this curve and we can see that about 68% of the data or area lies between sigma that is 34% plus 34% which is equal to of the data lies between x bar minus sigma and x bar about 95% of the data or area lies between minus 2 sigma that is 13.5% plus 34% plus 34% plus 13.5% which is equal to 95% of the data lies between x bar minus 2 sigma plus 2 sigma and 99% of data or area lies between 1 standard deviation of the mean about 95% of the values are within 2 standard deviations of the mean about 99% of the values are within 3 standard deviations of the mean now we know that on both sides of the center of the area lies on right side of the area lies on left side of the mean the area lies of the area next one is sigma of the mean that is between x the area lies on the right of the mean that is between plus 2 sigma at lies on the left side on the right side of the left side of the when we know the mean and those involved the x it can work standard deviation is 20 out of the graph we will find plus 3 sigma sigma is given as 20 sigma will be equal to 230 minus 20 that is equal to 220 then 40 plus 20 which is equal to 2 is equal to 180 and x bar plus 3 sigma is equal to 300 now these values on the horizontal line now we will learn vertical lines at these points these lines by free so we have joined the top of these lines by free hand now suppose we want to find number of valves that can work between now let us see between that is what percent of valves are between 220 to have to check the percentage between sigma and sigma now we know the percentage distribution under this normal curve required percentage between x bar minus sigma and x bar plus 2 sigma equal to 34.5 percent which is equal to number of valves that can work between 220 to 280 hours is equal to 81.5 this is equal to 81.5 upon 100 into 2000 and unsolvable this is equal to the area represents probability represents probability we have discussed about normal and the normal curve and this completes our session hope you all have enjoyed the session