Recover an unbiased variance and distinguish data spread from uncertainty of a mean.
Notes and key methods
For a random sample of size n, \(s^2=[\sum x^2-(\sum x)^2/n]/(n-1)\) is an unbiased estimator of population variance. An estimate of the standard error of the mean is \(s/\sqrt n\).
A sample mean varies between samples. The central limit theorem supports a normal approximation for sufficiently large independent random samples, but it does not imply that the individual observations have become normal.
Check the conditions
Use independent random observations.
The unbiased sample variance has denominator n−1.
Three worked examples
Example 1 · Sampling, unbiased variance and standard errors
A random sample has \(n=12,\ \sum x=192,\ \sum x^2=3160\).
Find the sample mean.
Find the unbiased estimate of population variance.
Find the estimated standard error of the mean.
Find the unbiased variance estimate after every observation is multiplied by 3.
Show the reasoning
(a)
\(\bar x=192/12=16\)
Answer: \(16\).
(b)
\(s^2=(3160-192^2/12)/(12-1)=8\)
Answer: \(8\).
(c)
\(\sqrt{8/12}=0.816496580927726\)
Answer: \(0.816496580927726\).
(d)
\(s_Y^2=3^2s_X^2=72\)
Answer: \(72\).
Example 2 · Sampling, unbiased variance and standard errors
A random sample has \(n=13,\ \sum x=221,\ \sum x^2=3805\).
Find the sample mean.
Find the unbiased estimate of population variance.
Find the estimated standard error of the mean.
Find the unbiased variance estimate after every observation is multiplied by 3.
Show the reasoning
(a)
\(\bar x=221/13=17\)
Answer: \(17\).
(b)
\(s^2=(3805-221^2/13)/(13-1)=4\)
Answer: \(4\).
(c)
\(\sqrt{4/13}=0.554700196225229\)
Answer: \(0.554700196225229\).
(d)
\(s_Y^2=3^2s_X^2=36\)
Answer: \(36\).
Example 3 · Sampling, unbiased variance and standard errors
A random sample has \(n=14,\ \sum x=252,\ \sum x^2=4601\).
Find the sample mean.
Find the unbiased estimate of population variance.
Find the estimated standard error of the mean.
Find the unbiased variance estimate after every observation is multiplied by 3.
Show the reasoning
(a)
\(\bar x=252/14=18\)
Answer: \(18\).
(b)
\(s^2=(4601-252^2/14)/(14-1)=5\)
Answer: \(5\).
(c)
\(\sqrt{5/14}=0.597614304667197\)
Answer: \(0.597614304667197\).
(d)
\(s_Y^2=3^2s_X^2=45\)
Answer: \(45\).
Common mistakes
Do not report the sample standard deviation as the standard error.
Increasing n reduces the uncertainty of the mean, not the population variance.