Join Eddie Davila for an in-depth discussion in this video Z transformation, part of Statistics Foundations: Probability.
- [Instructor] Z scores, they're helpful tools. They help us measure the distance between each data point and the mean. By the way, I'm in exercise file 03_06_Begin. Here's the dataset for the weights of 10 men. For this dataset we had a mean of 162.6 and a standard deviation of 35.0. The Z score for the last data point 232 pounds is 1.98. This data point was 1.98 standard deviations above the mean but now let's take this one step further. First, we're going to assume that the weights of men in this population are normally distributed. Knowing this we can now answer this question. What percentage of men weigh more than 235 pounds? To answer this, we'll need two things. First, the Z score for 232 pounds which is 1.98. Second, we need a normal distribution table. On this normal distribution table, we find 1.98. How do we do that? Go to your chart and in along the left side, you're going to look for 1.98. And then along the top, you're going to find 0.08. And so where those two intersect 1.9 and 0.08, we find 0.9761. So what does that mean? It means that according to our mean and standard deviation, 97.61% of all men, weigh 232 pounds or less and therefore, only 2.39% weigh more than 232 pounds. Let's use the Z transformation to answer another question. What's the probability a man weighs between 140 pounds and 190 pounds in this population. Now we need two Z scores. First, the Z score for 190 pounds. Plugging our numbers into the formula, we get a Z score of 0.78. The chart tells us that the Z score for 0.78 is 0.7823 or 78.23%. Now the Z score for 140 pounds plugging our numbers into the formula, we get a Z score of negative 0.65. So, how do we find that on this chart? Well, for this negative value, you find positive 0.65. The chart tells that the value for a Z score of 0.65 is 0.7422. Because the bell curve is symmetrical, we can assume that 0.65 standard deviations from the mean is the same in both the positive and negative direction. Therefore, we can subtract 0.7422 from one to find where we are on the left-hand side of the mean. So, one minus 0.7422 gives us 0.2578 or 25.78%. Remember, we're trying to find out the probability a man in this population weighs between 140 pounds and 190 pounds. Well, the probability a man weighs 190 pounds or less is 78.23%. The probability a man weighs 140 pounds or less is 25.78%. So by subtracting those two numbers, we now know the probability that a man weighs between 140 pounds and 190 pounds is 0.5245 or 52.45%. Means, standard deviations, normal curves, Z scores, and z-score tables. Not only do you know what they mean, with these tools you're now capable of providing answers to some interesting probability questions.
Eddie explains that probability is used to make decisions about future outcomes and to understand past outcomes. He covers permutations, combinations, and percentiles, and goes into how to describe and calculate them. Eddie introduces multiple event probabilities and discusses when to add and subtract probabilities. He describes probability trees, Bayes’ Theorem, binomials, and so much more. You can learn to understand your data, prove theories, and save valuable resources—all by understanding the numbers.