In the upcoming slides, I will discuss point estimation in statistics. I will discuss what it is and provide examples that demonstrate how it works.

Any measurement or parameter performed on a data set has to be treated as if all possible observations have not been accounted for. Parameters that are estimated can include mean, standard deviation and, by extension, variance as well. Point estimation is the process of finding a value of any such parameter as the ones just mentioned from random samples of the population. It involves using sample data in order to calculate a single value which will serve as an estimate for a parameter.

\(\text{Population Variance: }\sigma^2 = \frac{\Sigma_{i=1}^{N}(x_i-\mu)^2}{N}\)

\(\text{Sample Variance: }s^2 = \frac{\Sigma_{i=1}^{n}(x_i-\bar{x})^2}{n-1}\)

For the population variance, \(N\) is the actual size of the population and \(\mu\) is the population mean. For the sample variance, \(n\) is the size of the sample and not the actual population size and \(\bar{x}\) is the sample mean. In the sample variance formula, \(n-1\) is the number of degrees of freedom to use. Because this is an estimate, the sample variance must use one less degree of freedom because it is using the estimated value of the mean.

\(\text{Population Mean: }\mu=\frac{\Sigma_{i=1}^{N}x_i}{N}\)

\(\text{Sample Mean: }\bar{x}=\frac{\Sigma_{i=1}^{n}x_i}{n}\)

For the popultaion mean, \(N\) is the actual size of the population whereas for sample mean, \(n\) is the size of the sample and not the actual population size. Using \(n\), we are approximating what the mean could be using the size given to us for the sample.

set.seed(1234)
sampleMean = mean(rnorm(100))
sampleVariance = var(rnorm(100))

• The population mean is 0.
• The sample mean for 100 observations is -0.1567617
• The population variance is 1.
• The sample variance for 100 observations is 1.0654107

As demonstrated by the plots in the previous few slides, we are able to conclude that sample parameters are approximations of the actual population paramaters. When calculating the sample mean or variance, the values are not exact, but are pretty close, to the values of the actual population mean or variance. Point estimation allows us to use the parmeters as a way of getting close to the actual values when we don’t have the entire set of observations for the population to test on.