Download 3.3 MEASURES OF SPREAD (1) We use the variance to measure

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
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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]
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