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WOLFRAM|DEMONSTRATIONS PROJECT

Illustrating the Law of Large Numbers

population
0's and 1's
N(μ, σ)
random digits of π
first 100 digits of π
proportion of 1's in the {0,1} population
mean of normal population, μ
standard deviation of normal population, σ
sample size
new sample
The first 20 numbers in the sample are {1,0,0,1,0,0,1,0,0,1,1,1,0,1,1,1,1,1,0,0}.
The sample mean of these 20 numbers is 0.550000.
show population mean
The law of large numbers states (informally) that as the number of independent observations drawn from a population with finite mean
μ
increases, the mean of those observed values approaches
μ
. This Demonstration illustrates that behavior by plotting the sample mean as a function of the current sample size
n
, for
n=1
to
n=100
. Random samples can be drawn from a population of 0's and 1's (with any proportion of 1's), a normal population (with a range of
μ
and
σ
available), or from the (first 100,000) digits of
π
. The first 100 digits of
π
are available as a population, with samples of size
n
consisting of the first
n
digits, to allow for classroom illustration with a familiar and frequently referred to as random set of digits.
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