Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
3.3 MEASURES OF SPREAD (1) We use the variance to measure how “spread out” the data is from the mean or expected value. Given n numbers, x1 , x2 , ..., xn , the variance is (x1 − µ)2 + (x2 − µ)2 + · · · + (xn − µ)2 Var = n where µ is the mean. Also we define the standard deviation, denoted by σ, as √ σ = Var. (2) Calculate the variance and standard deviation. Follow the same instructions as for finding the mean and median. The standard deviation is represented by σx on the page “1 - Var Stats L1 (,L2)”. To find the variance, press VARS, select Statistics (option 5), then select σx (option 4), press X 2 and hit ENTER. To clear a list: Scroll up to the list name, press CLEAR, and then press ENTER. Make sure you do this every time before you start a new problem. (3) For a binomial distribution with n trials and probability of success p in a single trial and q = 1 − p, √ Var(X) = npq and σ(X) = npq (4) Chebyshev’s Inequality Let X be a random variable with expected value µ and standard deviation σ. Then the probability that the random variable associated with a random outcome lies between µ − hσ and µ + hσ is as least 1 − h12 , that is 1 P (µ − hσ 6 X 6 µ + hσ) > 1 − 2 . h Guchao Zeng; Department of Mathematics, Texas A&M University, College Station, TX 77843, USA; [email protected] 1