• Sample Variance = Adjusted Sample Variance
• Sample Standard Deviation = Adjusted Sample Standard Deviation

Sample Variance - Intuition

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read: Sample Mean

Let’s say we want the variance of the height of 1 trillion people.

There are 2 methods:

1. population variance - measure the height of all 1 trillion people, acquire population mean, use population variance formula
2. sample variance - get a sample of 100 people, acquire sample mean, use sample variance formula

The first method will get you the actual variance of height as it is the definition of variance over a population. However, it is impractical to measure 1 trillion people.

The second method is much easier, we only need to measure the height of 100 people. However, this sample variance may not accurately reflect the population variance. How well the sample variance reflects to population variance is through calculating sample variance’s:

• expected value - denoted as 𝐄(sample variance) = ?
• variance - denoted as 𝐕𝐚𝐫(sample variance) = ?

we want:

• 𝐄(sample variance) = population variance
• 𝐕𝐚𝐫(sample variance) = 0

Not surprisingly, the more we samples we take from population to calculate to sample variance:

• the closer 𝐄(sample variance) becomes the population variance
• the closer 𝐕𝐚𝐫(sample variance) becomes 0

below goes through the mathematics on why this is the case

Sample Variance - Definition / Formula

• sample variance (𝑠²) = [ 𝛴_{1≤𝑖≤𝑛} (𝑋_𝑖- 𝑋̅)²] / (𝑛 - 1)

where:

• each 𝑋_𝑖is a random sample drawn from a population
• 𝑛 is the sample size
• 𝑋̅ - is the sample mean

Sample Standard Deviation - Definition / Formula

• sample standard deviation (𝑠) = 𝑟𝑜𝑜𝑡(𝑠²)

where:

• each 𝑋_𝑖is a random sample drawn from a population
• 𝑛 is the sample size
• 𝑋̅ - is the sample mean

Sample Variance - Expected Value

𝐄(sample variance 𝑠²) = 𝜎²

proof

• 𝐄(𝑠²) = 𝐄([ 𝛴_{1≤𝑖≤𝑛} (𝑋_𝑖- 𝑋̅)²] / (𝑛 - 1)) # substitution of sample variance formula
• (𝑛 - 1)𝐄(𝑠²) = 𝐄[𝛴_{1≤𝑖≤𝑛} (𝑋_𝑖²- 2𝑋_𝑖𝑋̅ + 𝑋̅²)]
• (𝑛 - 1)𝐄(𝑠²) = 𝐄[𝑋_𝑖²·𝛴_{1≤𝑖≤𝑛}1] - 𝐄[2𝑋̅·𝛴_{1≤𝑖≤𝑛}𝑋_𝑖] + 𝐄[𝑋̅²·𝛴_{1≤𝑖≤𝑛}1]
• (𝑛 - 1)𝐄(𝑠²) = 𝐄[𝑛𝑋_𝑖²] - 𝐄[2𝑋̅·𝑛·𝑋̅] + 𝐄[𝑋̅²·𝑛] # 𝛴_{1≤𝑖≤𝑛}𝑋_𝑖 = 𝑛·𝑋̅
• (𝑛 - 1)𝐄(𝑠²) = 𝑛·𝐄[𝑋_𝑖²] - 2𝑛·𝐄[𝑋̅²] + 𝑛·𝐄[𝑋̅²]
• (𝑛 - 1)𝐄(𝑠²) = 𝑛·𝐄[𝑋_𝑖²] - 𝑛·𝐄[𝑋̅²]
• [(𝑛 - 1)/𝑛]·𝐄(𝑠²) = 𝐄[𝑋_𝑖²] - 𝐄[𝑋̅²]
  • 𝐄[𝑋̅²] = 𝑉𝑎𝑟[𝑋̅] + 𝐄[𝑋̅]²
  • 𝐄[𝑋̅²] = 𝑉𝑎𝑟[(1/𝑛)𝛴_{1≤𝑖≤𝑛}𝑋_𝑖] + 𝜇²
  • 𝐄[𝑋̅²] = (1/𝑛²)𝑉𝑎𝑟[𝛴_{1≤𝑖≤𝑛}𝑋_𝑖] + 𝜇²
  • 𝐄[𝑋̅²] = (1/𝑛²)[𝛴_{1≤𝑖≤𝑛}𝑉𝑎𝑟(𝑋_𝑖)] + 𝜇²
  • 𝐄[𝑋̅²] = (1/𝑛²)[𝛴_{1≤𝑖≤𝑛}𝜎²] + 𝜇²
  • 𝐄[𝑋̅²] = (1/𝑛²)[𝑛𝜎²] + 𝜇²
  • 𝐄[𝑋̅²] = (1/𝑛)[𝜎²] + 𝜇²
• [(𝑛 - 1)/𝑛]·𝐄(𝑠²) = 𝐄[𝑋_𝑖²] - [(1/𝑛)[𝜎²] + 𝜇²]
  • 𝐄[𝑋_𝑖²] = 𝑉𝑎𝑟[𝑋_𝑖] + 𝐄[𝑋_𝑖]²
  • 𝐄[𝑋̅²] = 𝜎² + 𝜇²
• [(𝑛 - 1)/𝑛]·𝐄(𝑠²) = 𝜎² + 𝜇² - [(1/𝑛)[𝜎²] + 𝜇²]
• [(𝑛 - 1)/𝑛]·𝐄(𝑠²) = 𝜎² - (1/𝑛)[𝜎²]
• (𝑛 - 1)·𝐄(𝑠²) = 𝑛𝜎² - 𝜎²
• (𝑛 - 1)·𝐄(𝑠²) = (𝑛 - 1)𝜎²
• 𝐄(𝑠²) = 𝜎²

the expected value / mean of the sample variance 𝑠² is population variance 𝜎²

Sample Standard Deviation - Expected Value

𝐄(sample standard deviation 𝑠) = 𝜎

proof

• 𝐄(𝑠) = 𝐄(√( [ 𝛴_{1≤𝑖≤𝑛} (𝑋_𝑖- 𝑋̅)²] / (𝑛 - 1) )) # substitution of sample standard deviation formula
• 𝐄(𝑠) = √𝐄( [ 𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖- 𝑋̅)²] / (𝑛 - 1) )
• 𝐄(𝑠) = √𝐄(𝑠²)
• 𝐄(𝑠) = √𝜎²# see Expected Value of Sample Variance
• 𝐄(𝑠) = 𝜎

the expected value / mean of the sample standard deviation 𝑠 is population standard deviation 𝜎

Sample Variance - Variance

𝐕𝐚𝐫(sample variance 𝑠²) = ?

proof

• 𝐕𝐚𝐫(𝑠²) = 𝐕𝐚𝐫([ 𝛴_{1≤𝑖≤𝑛} (𝑋_𝑖- 𝑋̅)²] / (𝑛 - 1)) # substitution of sample variance formula
• (𝑛 - 1)²𝐕𝐚𝐫(𝑠²) = 𝐕𝐚𝐫[𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖²- 2𝑋_𝑖𝑋̅ + 𝑋̅²)]
• (𝑛 - 1)²𝐕𝐚𝐫(𝑠²) = 𝐕𝐚𝐫[𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖²) - 𝛴_{1≤𝑖≤𝑛}(2𝑋_𝑖𝑋̅) + 𝛴_{1≤𝑖≤𝑛}(𝑋̅²)]
• (𝑛 - 1)²𝐕𝐚𝐫(𝑠²) = 𝐕𝐚𝐫[𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖²) - 2𝑋̅𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖) + 𝑛𝑋̅²]
• (𝑛 - 1)²𝐕𝐚𝐫(𝑠²) = 𝐕𝐚𝐫[𝛴_{1≤𝑖≤𝑛}(𝑋_𝑖²) - 2𝑋̅𝑛·𝑋̅ + 𝑛𝑋̅²] # 𝛴_{1≤𝑖≤𝑛}𝑋_𝑖 = 𝑛·𝑋̅
• (𝑛 - 1)²𝐕𝐚𝐫(𝑠²) = 𝐕𝐚𝐫[𝛴_{1≤𝑖≤𝑛}𝑋_𝑖²] - 𝑛²𝐕𝐚𝐫[𝑋̅²]
  • TODO
• 𝐕𝐚𝐫(𝑠²) = ?

Sample Standard Deviation - Variance

𝐕𝐚𝐫(sample standard deviation 𝑠) = ?

proof

• 𝐕𝐚𝐫(𝑠) = 𝐕𝐚𝐫(√( [ 𝛴_{1≤𝑖≤𝑛} (𝑋_𝑖- 𝑋̅)²] / (𝑛 - 1) )) # substitution of sample standard deviation formula
• TODO
• 𝐕𝐚𝐫(𝑠) = ?

Sample Variance - Standard Deviation / Standard Error

𝐒𝐄(𝑠²) = 𝐒𝐭𝐝(𝑠²) = ?

Sample Standard Deviation - Standard Deviation / Standard Error

𝐒𝐄(𝑠) = 𝐒𝐭𝐝(𝑠) = ?

Subpages

• Sample Variance - Why (n - 1)？
• Sampling Distribution of Sample Variance／Standard Deviation - Sample Variance／Standard-Deviation Distribution