Population vs Standard: Deviation and Covariance
Population Deviation vs Standard Deviation
How is the population deviation related to the standard deviation?
Population Deviation (of a data set x_i):
σx = √( Σ(x_i - mean(x)) / n)
where mean(x) is the arithmetic mean of the data set over x_i
Standard Deviation:
sx = √( Σ(x_i - mean(x)) / (n - 1))
n is the size of the data set x_i.
Suppose we can calculate the standard deviation by multiplying a factor (let's call it ß for the purpose of this example) to the population deviation.
ß * σx = sx
ß * √( Σ(x_i - mean(x)) / n) = √( Σ(x_i - mean(x)) / (n - 1))
ß * √( Σ(x_i - mean(x))) / √n = √( Σ(x_i - mean(x))) / √(n - 1)
ß * √( Σ(x_i - mean(x))) / √( Σ(x_i - mean(x))) = √n / √(n - 1)
ß = √n / √(n - 1)
ß = √(n/(n - 1))
Hence:
sx = √(n/(n - 1)) * σx
and
σx = sx * √((n-1)/n)
Example:
x = {4, 7, 10, 16, 38}
n = 5
σx = 12.16552506
sx = 12.16552506 * √(5/4) = 13.60147051
Population Covariance vs Standard Covariance
For the data sets x_i and y_i, population covariance:
cov_σ = 1/n * Σ((x_i - mean(x)) * (y_i - mean(y)))
And the sample covariance:
cov_s = 1/(n - 1) * Σ((x_i - mean(x)) * (y_i - mean(y)))
We will use the similar tactic above to find a relationship between population covariance and sample covariance:
ß * cov_σ = cov_s
ß * 1/n * Σ((x_i - mean(x)) * (y_i - mean(y))) =
1/(n - 1) * Σ((x_i - mean(x)) * (y_i - mean(y)))
ß * Σ((x_i - mean(x)) * (y_i - mean(y))) / Σ((x_i - mean(x)) * (y_i - mean(y))) =
n/(n - 1)
ß = n/(n - 1)
Hence:
cov_ s = n/(n - 1) * cov_σ
and
cov_σ = (n - 1)/n * cov_s
Example:
x = {4, 5, 6, 8}
y = {-2, -1, 2, 0}
n = 4
mean(x) = 5.75
mean(y) = -0.25
cov_σ = 1.1875
cov_s = 1.1875 * 4/3 = 1.5833333333
Hope you find this helpful,
Eddie
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