knitr::kable(data.frame(stringsAsFactors=FALSE, Symbol = c("$s$", "$\\sigma$", "$\\hat{\\sigma}$", "$s^2$", "$\\sigma^2$", "$\\hat{\\sigma}^2$"), What.is.it = c("Sample standard deviation", "Populatio...knitr::kable(data.frame(stringsAsFactors=FALSE, Symbol = c("$s$", "$\\sigma$", "$\\hat{\\sigma}$", "$s^2$", "$\\sigma^2$", "$\\hat{\\sigma}^2$"), What.is.it = c("Sample standard deviation", "Population standard deviation", "Estimate of the population standard deviation", "Sample variance", "Population variance", "Estimate of the population variance"), Do.we.know.what.it.is = c("Yes - calculated from the raw data", "Almost never known for sure", "Yes - but not the same as the sample standard dev…
However, in simple random samples, the estimate of the population mean is identical to the sample mean: if I observe a sample mean of \(\bar{X}\) = 98.5, then my estimate of the population mean is als...However, in simple random samples, the estimate of the population mean is identical to the sample mean: if I observe a sample mean of \(\bar{X}\) = 98.5, then my estimate of the population mean is also \(\hat{\mu}\)=98.5. Figure 7.12: An illustration of the fact that the sample mean is an unbiased estimator of the population mean (panel a), but the sample standard deviation is a biased estimator of the population standard deviation (panel b).