Probability & Statistics The Variance and Standard Deviation

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Probability Statistics

• The Variance and
• Standard Deviation

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The Most Important Number

• In the last section we looked at three related
  concepts
• The mean of a sample,
• The mean of a population, µ
• The mean or expected value of a random variable,
  E(X)
• The mean is probably the single most important
  number that can be used to describe a sample,
  population, or probability distribution of a
  random variable.

x
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The Second Most Important Number

• In this section, we learn about the next most
  important number the variance.
• Closely associated with the variance, and
  oftentimes used more, is the standard deviation.

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Spread

• Roughly speaking, the variance measures the
  dispersal or spread of a distribution about its
  mean.
• The probability distribution whose histogram is
  drawn on the left has a smaller variance than
  that on the right.

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Variance of Probability Distribution
Let X be a random variable with values x1, x2, ,
xN and respective probabilities p1, p2,, pN. The
variance of the probability distribution is
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Standard Deviation of a Probability Distribution
The standard deviation of a probability
distribution is
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Unbiased Estimate

• If the average of a statistic, given that the
  statistic were computed for each sample, equals
  the associated parameter for the population, then
  that statistic is said to be unbiased.

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Sample Variance and Standard Deviation

• The unbiased variance for a sample is
• The unbiased standard deviation for a sample is

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Example 1

• Two golfers recorded their scores for 20
  nine-hole rounds of golf.
• a.) Compute the sample mean and the variance of
  each golfers score.
• b.) Who is the better golfer? (Note The lower
  the score, the better.)
• c.) Who is the more consistent golfer?

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Alternative Definitions

• Two alternative definitions for variance are
• and, for a binomial random variable with
  parameters n, p, and q,

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Example 2

• Compute the variance and standard deviation of
  the probability distribution in the table at
  right.

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Example 3

• Find the variance when a fair coin is tossed 5
  times and X is the number of heads.

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More Formulas for SampleVariance and Standard
Deviation (Use When Data Values Listed
Individually)
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Example 4

• The table at right gives the number of books (in
  millions) in the 10 largest public libraries in
  the United States.
• Determine the mean and the standard deviation
  for the number of books.

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Chebychev's Inequality

• Chebychev's Inequality
• Suppose that a probability distribution with
  numerical outcomes has expected value and
  standard deviation Then the probability
  that a randomly chosen outcome lies between -
  c and c is at least

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Example 5

• A drug company sells bottles containing 100
  capsules of penicillin. Due to bottling
  procedure, not every bottle contains exactly 100
  capsules. Assume that the average number of
  capsules in a bottle is 100 and the standard
  deviation is 2. If the company ships 5000
  bottles, estimate the number of bottles having
  between 95 and 105 capsules, inclusive.

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Example 6

• An electronics firm determines that the number
  of defective transistors in each batch averages
  15 with standard deviation 10. Suppose that 100
  batches are produced.
• Estimate the number of batches having between 0
  and 30 defective transistors.

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Summary

• The variance of a random variable is the sum of
  the products of the square of each outcome's
  distance from the expected value and the
  outcome's probability.
• The variance of the random variable X can also be
  computed as E(X 2) - E(X) 2.
• A binomial random variable with parameters n and
  p has expected value np and variance np(1 - p).

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Summary

• The square root of the variance is called the
  standard deviation.
• Chebychev's Inequality states that the
  probability that an outcome of an experiment is
  within c units of the mean is at least
  , where is the standard deviation.