We have three popular measurements of errors when doing forecasting and the first one is called the mean absolute deviation, or MAD, as in short abbreviation. We're going to walk through all three of them in detail and then also give you a little bit of context on how they can be used. The first one is called mean absolute deviation, MAD is actually the equation is on the right-hand side and scores in middle, is the difference between my actual, my A stands for actual and an F stands for forecast, so it's the sum of all the datas that we want to measure in an absolute form, the sum of the absolute value, the absolute difference of actual and forecast and divided by T is the total time period, in this case, usually we'll measure it in months, a number of months that we have the data for. On the left-hand side we have some demand, which is the actuals, we have some actual information and also the forecast that was done for those months. We notice that we start with Month 1, so there's no forecast for month 1. Our forecast starts with month 2. To do this, we need to calculate this absolute difference. What's in the A _t minus F _t and t just indicates the time period that we're in. The first one is from month 2, since I don't have anything for month 1 forecast, I'm going to skip that and look at month 2, which is 250 as the actual and then 255 as the forecast and then the equation that we're going to use, so in Excel, there's absolute value equation called ABS, what you would do is just do, sorry, I'll do this again, you type in again to begin equation using the equation of formula within Excel, you need to type equal and then ABS is the equation for absolute value and you need to give it a number, within this, we're going to calculate the difference between my actual and my forecast so that the absolute error in month 2 is 5, then I can either drag or copy and paste. All right, so I can drag this, if I go to the bottom right, there's a little tiny little green square, I can just drag that down and apply the same formula to all the months. Now I have the absolute error for month 2 through 5. Then that takes care of this, calculating the absolute value on the top, on the numerator side but I need to complete the summation. So I'm going to call this as total. I'm going to do equal again. I'm going to use the sum function and then just sum up the months, the absolute error for the four months, which is 40 and then, right now I have total, this is 40, here is my numerator value and my denominator, my time period, t, t as in this case, is four, because I only have four months worth of forecasts and actual. To calculate mean absolute deviation in this case, is taking 40 divided by four, so my mean absolute deviation is 10 and this is the split, right on average. This is how far away they are between the forecast and my actual values in this exercise. The next one that's also commonly used is called mean squared error. You notice that the equation is very similar. I need to calculate the difference between the actual and forecast. In this case, the difference is that I take a square root, I know I squared instead of taking the absolute value. For the inner numerator and I sum it. Let's do that first. In this case, I'm going to calculate the difference between my actual and my forecast. Then I'm going to raise it to the power two because I need to square it in Excel and hit "Enter". The difference is five, and so I square it. Five square is basically 5 times 5 is 25. Then I can do the same thing. I drag down the formula and apply it to the other four months. Then to complete the numerator, I need to sum up my error values. In this case is 50. Now my T is the same I have four, and here's my MSE, my mean square error is 550 divided by 4, so 137.5. These two mean absolute deviation and mean square error, they're very similar. They're telling me what's the spread between the error terms. But the difference is that in this mean-square error actually punishes my outliers. Which means is more sensitive to the, if I have a big difference between my actual and my forecast. For example, these two, these are a little bigger. The difference is bigger than just five on the first two data points. The difference is 10. Then it amplifies that difference. You'll see this MSE is pretty widely use if you use the regression that we mentioned before to measure the spread of my error term, my forecast and demand. How the difference is my error in my forecast. If you have a need to amplify that to determine the outliers. You wish to minimize these outliers or you wish to place more "Punishment", the weights on the outliers they may seem maybe a good one to use. Otherwise you can stick with the mean absolute deviation to understanding of the spread of the error terms. These two, the error terms is the numeric value. Then the last one we're going to cover, which is very widely used in business. Now I'll explain why, but let me go to how to calculate this first. In the numerator, we'll see that we're going to calculate the absolute, again, the error or the actually the value for the error between the difference between my actual and my forecast. In this case, I'm going to divide it by the actual for the same period. Then I'm going to sum up all these terms. In column D, I call it absolute error. That's where I'm calculating the absolute value between my actual and forecast. Then when I divide it by the actual, this is the absolute percentage error APE. Then that now completes the calculation within the big parentheses. Let's do that, this is the same as before, I use the absolute value ABS right between the two, between demand and forecast. Then to calculate absolute percentage error, I divide it by the actual. I take my absolute error, in this case, phi, and then divide it by 250, so D3 divided by B3. I can do the same thing. If I highlighted two and I can either double-click or I can drag anything applies the same formula to all the data points that you have. Now, we need to complete the numerator calculation with a summation. In this case, my total here, actually, I'm summing up the my absolute percentage error. I have a t, t in this case is still four, and I have four periods. My MAPE is this divided by 4. It's 0.03531136 and so forth. The last one times 100, that is just to convert it to a percentage term. Now, we have the MAPE in decimal is 0.035 and so forth. If we want to convert it to percentage which is easier to understand, I can just hit the percentage sign. Then that is synonymous to multiplying by 100. My MAPE, mean absolute percentage error is 3.5%. You can say my accuracy is the inverse of that, so it's 1 minus my error term. My accuracy is 96.5%. The reason MAPE is very common or very popular in business and any presentation by helping people understand how did I do how did [inaudible] do in the past quarter, past month, is because the percentage gives people a sense of the magnitude of the difference or the error or the accuracy. When I tell people that I am in the past four months, my data, my forecast accuracy was about 96.5 or 97%, about 97% accurate. Then people can grasp that right in there mind, "Oh that's very good." But if I tell people my error mean absolute deviation is 10, there's no comparison by 10. That sounds small. But if my quantity is very small, then the difference of 10 is actually big, by a quantity of 100, these are in hundreds, quarter 10 maybe significant, or if these are in the millions, my commands are in the millions, then difference of 10 is not very significant. The same with mean square error, it doesn't really convey a magnitude, there's no baseline, there is no foundation that people can say, "Well, what is 137 out of, how significant is this number?" But from a business term for executives, for business communication, you may eliminate those issues. You tell people, "I'm 97% accurate." That means I'm 97% confident that my forecast will be right 97% of the time, which is very good. But the other two, mean absolute deviation and mean square errors to have as used, there is more on the improving my my forecast model. Then the MAPE is more on the highest, is very effective in using it to communicate that people know how I did.