 Hi, folks. This is Dr. Don, and I have a problem out of Chapter 2, Underscriptive Statistics. In this problem, we are interested in the variation of values in a data set, but we do not know what type of distribution the data has. Most often in introductory statistics, we are told the distribution is bell-shaped or normal, but we can still find some useful information about the variation in data values by using Shebyshev's theorem. This theorem can be applied to any type of distribution, but if we know the type, it is better to use the tools for that specific type, such as the empirical rule for bell-shaped distributions. Shebyshev's theorem says that at least 75% of the data will fall between plus and minus 2 standard deviations, and that at least 89.89% will fall between plus and minus 3 standard deviations. Shebyshev's formula says that the portion of any data lying between k standard deviations is at least 1 minus 1 divided by k square. But it is important to remember this formula only works for integer values of k and for k greater than 1. Because the percentages beyond 4 standard deviations are so small, we usually stop at a k of 4. While you can just use that formula to find the upper and lower values or proportion between using basic Excel, I have made an Excel calculator cheat sheet to make solving these types of problems a bit easier and faster. You can download a copy of the worksheet of my website, drdonright.com. I'll show you how to use it with a few example problems. Reading this problem, we are given the mean and standard deviation, and we want to estimate the number of farms with values per acre between $1,600 and $2,000. But we are told the distribution is bell shaped, so we should not use Shebyshev's on this problem. Let's look at another. Okay, here we do not know the type of distribution, but we are told to use Shebyshev's anyway. The sample size n is 12, the mean number of TVs is 4, with a standard deviation of 1 TV. They want to know how many households have between 2 and 6 TVs. We enter the data into the blue cells, n equal 12, mean equal 4, standard deviation equal 1. The calculator updates and we can look for the lower and upper values. They are in the k equal 2 row and looking across, we find that at least 75% of the households will be in that range, and that is 9 households. On another problem, here we are trying to find out how many times a geyser erupted that lasted between 2.26 and 5.26 minutes. We enter the data, n equal 40, the mean equal 3.76, and the standard deviation equal 0.75 minutes. The calculator updates and we can see those values in k equal 2 and the number is 30. One last example, we are given the mean score is 80 and the standard deviation is 4 points, but we are not given the sample size. Not to worry though, because we are given k equal 2 standard deviations we can use the calculator to help us find the proportion between the lower and upper limits required by this problem. We enter the data, mean of 80, standard deviation of 4, and we just leave in as it is because we don't know what it is. Reading across the k equal 2 row, we see at least 75% of the scores fall between 72 and 88. Okay, that's how easy it is with my calculator. Remember you can get it at my drdonright.com website. Hope this helps. And if it does help, please consider subscribing to my YouTube channel, The Stats Files. Just click on the big red button.