Point estimation refers to the use a single sample statistic to estimate an unknown population parameter.
A point estimate is a single value.
It is different from an interval estimate which is a range of values that may contain the population parameter.
| Sample Statistic (Point Estimate) | Population Parameter | |
|---|---|---|
| \(Mean\) | \(\bar{x}\) | \(\mu\) |
| \(Standard\) \(Deviation\) | \(s\) | \(\sigma\) |
| \(Variance\) | \(s^2\) | \(\sigma^2\) |
| \(Proportion\) | \(\hat{p}\) | \(p\) |
We want to find the average height (in cm) of all students in a school using a sample of 15 randomly-selected students.
The sample mean, \(\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\), is the point estimate for the population parameter, \(\mu\).
heights = c(182, 178, 168, 180, 177, 184, 164, 185,
173, 171, 175, 185, 169, 165, 168) ## sample data
mean(heights) ## calculate sample mean
## [1] 174.9333
The mean height for our sample of 15 students is about 175cm, so our best guess for the mean height of all students in the school is 175cm.
The faithful data set contains a sample of 272 waiting times between Old Faithful eruptions.
Using this sample data, how much does wait time vary between eruptions?
We we will find the sample standard deviation of waiting times, \(s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}\), to estimate the standard deviation of all waiting times, \(\sigma\).
Here, \(s\) is our point estimate for \(\sigma\).
The standard deviation for our sample data is about 14 minutes, so we can guess that waiting times between all Old Faithful Eruptions vary by about 14 minutes.
The HairEyeColor data set contains information from a sample of 592 statistic students at the University of Delaware.
Using this sample, we want to estimate the proportion of all students at the University of Delaware who have hazel eyes.
The proportion of students in the sample data with hazel eyes, \(\hat{p} = \frac{x}{n}\), is our point estimate for the proportion of all students with hazel eyes, \(p\).
The sample proportion of statistics students with hazel eyes is \(\frac{93}{592}\) or about 0.16, so we can estimate that the proportion of all students with hazel eyes is 0.16.
The Orange data set contains 35 age (days) and circumference (mm) measurements for a sample of orange trees.
Using this data, can we use an orange tree’s circumference to estimate its age?
To do this, we will need to create a linear regression model from this data. The point estimate in this case would be the age calculated by plugging a specific circumference into the model.
With this model we can now estimate an orange tree’s age using its circumference. For example, if any orange tree has a circumference of 75 mm, we can guess that it is 603 days old.
Libretexts. (2022, April 9). 9.2: Point estimation. Statistics LibreTexts. link
Point estimate: Definition, examples. Statistics How To. (2024, November 18). link
Shen, J. (2024, March 21). Point estimates vs. interval estimates in linear regression analysis — mainly my study notes. Medium. link