Introduction

Point estimation, is a technique used in statistics to estimate parameters of a population.

Let’s suppose I have a sample of data that represents anything numeric regarding a population, but I do not know the exact value of such parameter. Point estimation would allow to me to come to an estimation of what the true value may be.

This is useful, because it allows us to explore statistical venues that may not be clear, or extremley difficult to reach the true value.

Sample Mean

One of the estimators is the sample mean:

\[ \bar{x} = \frac{1}{n} \sum{i=1}^{n}x_i \] The sample mean is used to estimate the mean of a sample of population.

Sample Variance

The sample variance:

\[ s^2 = \frac{1}{n-1} \sum{i=1}^{n}(x_i = \bar{x})^2 \] The sample variance is the calculation of variability in a population, and is used as an estimator of such.

Example: Construction Survey Part 1

Assume there is a construction company that was hired to build a road in a neighborhood. The manager surveyed 100 residents, asking what is a comfortable width for the road in meters:

Example: Construction Survey Part 2

Now, the construction company wants to take the sample mean of the survey, to build the road.

The mean represents the average width, that was preferred by the neighbors.The Red line in the plot represents the population mean.

## [1] "The sample mean is: 19.4m"

## $x
## [1] "Resident"
## 
## $y
## [1] "Road Width in Meters"
## 
## $title
## [1] "Road Width Survey"
## 
## attr(,"class")
## [1] "labels"

Example: Construction Survey Part 3

Now, the construction company wants to take the sample variance of the survey, to build the road.

The variance represents the average of the squared differences between each width, and the mean.

## [1] "The sample variance is: 35.4141414141414m"

## $x
## [1] "Resident"
## 
## $y
## [1] "Road Width in Meters"
## 
## $title
## [1] "Road Width Survey"
## 
## attr(,"class")
## [1] "labels"

Calculations Part 1

In a previous slide, I showed the equations of the population mean, and variance. Despite the complex equations, this presentation was written in R and Latex/ Because we have a sample size of 100, there are a ton of variables to input and calculate manually, and it is much quicker to do it in a programming language.

To calculate the mean, I used the mean() function in R.

mean = mean(df$road_width_df)
paste0("Mean = ", mean, "meters")
## [1] "Mean = 19.4meters"

Calculations Part 2

To calculate the variance, I used the variance() function in R.

variance = var(df$road_width_df)
paste0("Variance = ", variance, "meters squared")
## [1] "Variance = 35.4141414141414meters squared"

Sources