True and Unbiased Variance

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True and Unbiased Variance
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Definition

• Population vs. Sample
• Parameter vs. Statistic

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Example

• The median earnings of dentists in the U.S. is
  90,000 a year.

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Population and Parameter

• Population All Units of Interest
• Parameter Population Characteristic

All Dentists in the U.S.
Yearly Median Earnings 90,000
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Example

• A sample of 500 randomly selected American
  dentists had yearly median earnings about
• 120,000.

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Sample and Statistic

• Sample a Subset of the Population
• Statistic Characteristic Computed from Sample
  Data

500 Dentists in the U.S.
Yearly Median Earnings 120,000
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Sampling Variability

• Parameters are fixed, while statistics vary.

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Parameter and Population

• The median earnings of dentists in the U.S. is
  90,000 a year.

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Generalization

• Generalize Characteristic from Sample to
  Population
• Representative Sample
• Random Sample

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Sample Size 500

• A sample of 500 randomly selected American
  dentists had yearly median earnings about
• 120,000.

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Sample Size 5,000

• A sample of 5,000 randomly selected American
  dentists had yearly median earnings about
  91,000.

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Estimate the Parameters

• Statistics based on large samples are likely to
  be closer to the population parameters.

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Random Sampling Model
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Parameters
Population Mean
Population Variance
Population Standard Deviation
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Statistics
Sample Mean
Unbiased Variance
Unbiased Standard Deviation
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• Monte Carlo Simulations

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Population and Parameters

• A Normally-Distributed Population with a Mean of
  0 and a Standard Deviation of 1

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Sample Size

• Six Random Samples of Size n 3
• Six Random Samples of Size n 10
• Six Random Samples of Size n 30

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Statistics and Sample Size

• Statistics based on large samples are likely
  to be closer to the population parameters.

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Sample Means
Closer to the population mean of 0
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True Variance

• Describe Data

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True Standard Deviation
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True Standard Deviations
Closer to the population standard deviation of 1
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Unbiased Variance

• Generalize beyond Data

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Unbiased Standard Deviation
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Unbiased Standard Deviations
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Unbiased SD
For Smaller Sample Size n 3
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Unbiased SD
Closer to the population standard deviation of 1