Calculating probabilities in the normal distribution. The normal distribution is probably the most commonly used distribution for Six Sigma. You will learn later that due to something called the central limit theorem, we can often do statistical analysis of samples under the assumption that the sample mean follows a normal distribution. Mathematically, the formula for the distribution si the one on the slide. But we always use tables or technology, rather than applying the formula directly. The area under the normal curve between two x values, represents the probability that the random variability would fall between those two values. To calculate that value, we first determine the z scores of the points. The z score is the number of standard deviations away from the mean the measurement is. Therefore the formula is z equals the quantity x minus mu divided by sigma. Where mu represents the mean of the population and sigma represents the standard deviation. Once z is calculated you can then use the table or technology to determine the area to the side of that z. The table in the certified Six Sigma Yellow Belt book provides the area to the right of the z. However, most tables and technology provide the area to the left, so that's the way I'll explain it for the rest of the module. There is also a PDF table provided in the module for you. Since the total area of the curve is equal to exactly 1, you can count the probability of being to the right of z by subtracting the area to the left from 1. Suppose after you plug in the x mu and sigma, you obtain a z of -1.21. In the table, go down the first column until you get to -1.2. And then go across the columns at the top until you get to 0.01 for the hundredths place. The value there is 0.1131 which means that the fraction of the area under the curve that is to the left of the x is 0.1131. For example, suppose you have a process that is normally distributed with a mean of 12 and a standard deviation of 3, and you want to know the probability of being less than 15. First, we calculate that the z score is 15-12 divided by 3 which is equal to 1. Then we look up 1.0 in the table provided in the module. In first column, go down to 1.0 and then across the top look for 0.00 since the hundreds place is 0. We obtain 0.8413, which is the area to the left of z = 1.00. This means at the probability of being less than 15 on a normal curve with mean 12 and standard deviation 3 is 0.8413. If you want to calculate the probability of being greater than 16.5, you would calculate z equals 16.5 minus 12 divided by 3 equals 1.5. Look up 1.5 in the table by going down to 1.5 and across to 0.00. This will be the area to the left of 16.5. And since you want the probability of being greater than 16.5, you would subtract that value from 1. So from the table the area to the left of 16.5 is 0.9332 and your answer is 1-0.9332 which is equal to 0.0668. If you want to calculate the probability of being between 16.5 and 12. You would need to subtract the area to the left of 12 from the area to the left of 16.5. So this value would be 0.9332-0.5000 = 0.4332. This is even easier to do in Excel. If you want the area to the left of x using Excel you don't even need to calculate your z. Simply enter, =norm.dist(x,mu,sigma, the word "true") and the area to the left of x will automatically be calculated. If you want the area to the right of x instead, then enter =1- =norm.dist(x,mu,sigma,"true") instead. In either case, you put in the correct numbers for your x, mu and sigma and type that full word, true. For example, to find the area to the left of 15 in the previous example, it would be =norm.dist 15, 12, 3, true. And for the area to the right of 16.5, 1-norm.dist(16.5, 12, 3, true).