Transcript mean
The Arithmetic Mean Asami Yamamoto The Arithmetic Mean the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the observations (xi) of the list divided by the number of the observations (n) in the list and is denoted by x. Example: If you have 3 numbers then add them and divide them by 3(=n): If you have 4 numbers add them and divide by 4(=n): Therefore… n Xi’s are the values of the random variable, and Pis are their probabilities. For continuous variable in which each Xi occurs only once (P=1/n) n E(X)= Σ Xi Pi i=1 n E(X)= Σ Xi×1/n i=1 therefore… If X is a random variable, then the expected value of X can be seen as the long-term arithmetic mean that occurs on repeated measurements of X. This is the content of the law of large numbers If three conditions are satisfied, the arithmetic mean of the observations in our sample is an unbiased estimator μ. These three conditions are… 1. Observations are made on randomly selected individuals. 2. Observations in the sample are independent of each other. 3. Observations are drawn from a large population that can be described by a normal random variable. In this case, there is the second fundamental theorem of probability; the law of large numbers. The Law of Large Numbers is a fundamental concept in statistics and probability that describes how the average of a randomly selected sample from a large population is likely to be close to the average of the whole population. For example… The average weight of 10 apples taken from a barrel of 100 apples is probably closer to the "real" average weight than the average weight of 3 apples taken from that same barrel. This is because the sample of 10 is a larger number than the sample of only 3 and better represents the whole group. If you took a sample of 99 apples out of 100 apples, the average would be almost exactly the same as the average for all 100 apples. It can be used for infinite(∞) number, the sample size increases, the arithmetic mean of Xi approaches the expected value of X, E(X). lim n→∞ n = E(X) As a result, we get more data, we can estimate the unknown expected value with the average of the observation, and its value is more reliable.