Point estimation is a statistical tool that uses sample data to estimate an unknown population parameter.

For example, a sample mean can estimate a population mean, a sample proportion can estimate a population proportion, and so on.

A population is the entire data set in which we want more information about.

A sample is a smaller group of data within that set that gets selected from the population.

Studying entire populations is often impractical, which in turn is why statisticians instead look at samples to infer population parameters.

Sample means are one of the most common uses of point estimates.

The sample mean is used to estimate what the population mean is.

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \] In this formula:

• \(\bar{x}\) is the sample mean
• \(n\) is the sample size
• \(x_i\) is each observation in the sample

Suppose there was a population of 50 commuting adults and we wanted to get an average weekly commute time of them all.

First we get a sample of say 5 people:

2, 4, 1, 5, and 10 hours

And then using the sample mean formula:

\[ \bar{x} = \frac{2+4+1+5+10}{5} = 4.4 \] The point estimate of the population mean comes out to 4.4 hours per week.

The following R code creates the bar chart for the sample.

commute_data <- data.frame(
  person = c("A", "B", "C", "D", "E"),
  hours = c(2, 4, 1, 5, 10)
)

ggplot(commute_data, aes(x = person, y = hours)) +
  geom_col() + 
  labs(
    title = "Weekly Commute Time of Sampled Persons",
    x = "Person",
    y = "Hours"
  )

The sample mean can be shown on a bar chart as the estimate of the population mean.

Point estimation allows us to use sample data to estimate unknown population parameters.

The sample mean is one common point estimate used to estimate a population mean.

Larger and more representative samples generally provide more useful estimates.