Have you asked yourself how statisticians determine parameters such as the mean age of an entire country's population? It is obvious that they can't get data from every single member of the population to calculate this statistic.
However, they can gather data from small samples from the population, find their mean, and use that as a guide to guessing the parameter for the whole population. This is called point estimation.
This article will address what point estimation is, various methods of estimation, and their formulas. It will also show you some examples of point estimation.
Definition of Point Estimation
By now, you should be familiar with the concepts of population, sample, parameter, and statistics. Serving as a brief reminder:
The population is the group in which you’re interested in studying and for which the results are statistically inferred;
A parameter is a characteristic of the population that you want to study and can be represented numerically;
A sample is a small group of elements from the population in which you have an interest that it is representative;
A statistic is a characteristic of the sample that is represented by a numerical value.
With this said, you can then more clearly understand the concept of point estimation:
Point estimation is the use of statistics taken from one or several samples to estimate the value of an unknown parameter of a population.
This is the reality of a statistical study: it is almost certain that researchers won't know the parameters of the population they are interested in.
Hence, the importance of the sample (or samples) used in a statistical study having as close as possible some or the main characteristics of the population, that is, the sample is representative.
Formulas for Point Estimation
Different population parameters will have different estimators, which in turn will have different formulas for their estimation. Later in the article, you’ll see the some of the more frequently used ones. Let's take a look at some of the terminology and notation used.
The result of a point estimation of a parameter is a single value, usually referred to as the estimator, and it will usually have the same notation as the population parameter it represents plus a hat '^'.
In the table below, you can see examples of estimators and parameters and their respective notations.
Parameter | Notation | Point Estimate | Notation |
Mean | \(\mu\) | Sample mean | \(\hat{\mu}\) or \(\bar{x}\) |
Proportion | \(p\) | Sample proportion | \(\hat{p}\) |
Variance | \(\sigma^2\) | Sample variance | \(\hat{s}^2\) or \(s^2\) |
Table 1. Statistical parameters,
Methods of Point Estimation
There are several point estimation methods including the method of maximum likelihood, the method of least square, the best-unbiased estimator, among others.
All of these methods allow you to calculate estimators that respect certain properties that give credibility to the estimator. These properties are:
Consistent: here you want the sample size to be large so that the value of the estimator is more accurate;
Unbiased: you expect the values of the estimators of samples you might draw from the population to be as close as possible to the true value of the population parameter (a small standard error).
The estimators shown in the previous table are unbiased regarding the parameters they estimate. To learn more about this topic, read our article on Biased and Unbiased Point Estimates.
When the two properties above are met for an estimator, you have the most efficient or best-unbiased estimator. Of all consistent, unbiased estimators, you would want to choose the one that is most consistent and unbiased.
Next, you will learn about two estimators that you will need to be familiar with, which are the sample mean and the estimator for the proportion. These are the best-unbiased estimators for their respective parameters.
Point Estimate of the Mean
Now, to the first estimator. This is the sample mean, \(\bar{x}\), of the population mean, \(\mu\). Its formula is
\[\bar{x}=\frac{\sum\limits_{i=1}^{n}x_i}{n},\]
where
As you have already read, this is the best unbiased estimator of the population mean. This is an estimator based on the arithmetic mean.
Let's look at an example of the application of this formula.
Given the values below, find the best point estimate for the population mean \(\mu\).
\[7.61, 7.17, 9.06, 6.305, 7.805, 7.11, 9.705, 6.11,8.56, 7.11, 6.455, 9.06\]
Solution:
The idea is simply to calculate the sample mean of this data.
\[\begin{align} \bar{x}&=\frac{\sum\limits_{i=1}^{n}x_i}{n} \\ &= \sum\limits_{i=1}^{n}\frac{x_i }{n} \\ &=\frac{7.61}{12} +\frac{7.17}{12}+\frac{9.06}{12}+\frac{6.305}{12}+\frac{7.805}{12} \\ & \quad +\frac{7.11}{12}+\frac{9.705}{12}+\frac{6.11}{12}+\frac{8.56}{12} \\ & \quad +\frac{7.11}{12}+\frac{6.455}{12}+\frac{9.06}{12} \\ &=\frac{92.06}{12} \\ &=7.67 \end{align} \]
The best point estimate for the population mean \(\mu\) is \(\bar{x}=7.67\).
Another estimator related to the mean is of the difference between of two means, \( \bar{x}_1-\bar{x}_2\). You may be interested in this estimator when you want to compare the same numerical characteristic between two populations, for example, comparing the average height between people who live in different countries.
Point Estimate of Proportion
The population proportion can be estimated by dividing the number of successes in the sample \(x\) by the sample size (n). This can be expressed as:
\[ \hat{p}=\frac{x}{n}\]
What does "numbers of successes in the sample" mean?
When you want to calculate the proportion of the characteristic you are interested in, you will count all the elements in the sample that contain that characteristic, and each of these elements is a success.
Let's look at an example of the application of this formula.
A survey was conducted using a sample of \(300\) teacher trainees in a training school to determine what proportion of them view the services provided to them favorably. Out of \(150\) trainees, \(103\) of them responded that they viewed the services provided to them by the school as favorable. Find the point estimation for this data.
Solution:
The point estimation here will be of the population proportion. The characteristic of interest is the teacher trainees having a favorable view about the services provided to them. So, all trainees with a favorable view are successes, \(x=103\). And \(n = 150\). that means
\[ \hat{p} = {x\over n} = {103\over 150} = 0.686.\]
The researchers of this survey can establish the point estimate, which is the sample proportion, to be \(0.686\) or \(68.7\%\).
Another estimator related to the proportion is of the difference of two proportions, \( \hat{p}_1-\hat{p}_2\). You may be interested in this estimator when you want to compare proportions of two populations, for example, you may have two coins and suspect that one of them is unfair because it is landing on a head too frequently.
Example of Point Estimation
There are some important elements associated with a point estimation problem:
Data coming from the sample – after all, no data, no estimation;
An unknown parameter of the population – the value you’ll want to estimate;
A formula for the estimator of the parameter;
The value of the estimator given by the data/sample.
Look at examples where you see all these elements present.
A researcher wants to estimate the proportion of students enrolled at a university who frequent the library of their respective college at least three times a week. The researcher surveyed \(200\) students of the science faculty who frequent their library, \(130\) of whom frequent it at least \(3\) times a week. She also surveyed \(300\) college students from the humanities faculty who frequent their library, of whom \(190\) frequent it at least \(3\) times a week.
a) Find the proportion of students who frequent the science faculty library at least \(3\) times a week.
b) Find the proportion of students who frequent the humanities faculty library at least \(3\) times a week.
c) Which group of students goes to their library the most?
Solution:
a) \(x=\)number of students of the faculty of sciences that frequent their library at least \(3\) times a week, so \(x=130\); and \(n=200.\) For the sciences group,
\[\hat{p}=\frac{130}{200}=0.65.\]
b) \(x=\)number of students of the faculty of humanities that frequent their library at least \(3\) times a week, so \(x=190\); and \(n=300.\) For the humanities group,
\[\hat{p}=\frac{190}{300}=0.63.\]
c) The proportion of science students who frequent their library is greater than the proportion of humanities students who frequent their library. According to this information, you can say that it is more science students who frequent their library.
Point Estimation vs. Interval Estimation
As you may have realized after reading this article, point estimation gives you a numerical value that is an approximation of the population parameter that you would actually like to know.
But the disadvantage of this estimation method is that you don't know how close or how far away from the true value of the parameter the estimator is. And this is where interval estimation comes in, which will consider what is called the margin of error, that information that allows you to appreciate the distance of the estimator to the parameter.
As you can imagine, it is in your interest that the estimated values of the parameters be as close as possible to the true values of the parameters, as this makes the statistical inferences more credible.
You can learn more about interval estimation in the article Confidence Intervals.
Point Estimation - Key takeaways
- Point estimation is the use of statistics taken from one or several samples to estimate the value of an unknown parameter of a population.
- Two important properties of estimators are
Consistent: the larger the sample size, the more accurate the value of the estimator;
Unbiased: you expect the values of the estimators of samples to be as close as possible to the true value of the population parameter.
When those two properties are met for an estimator, you have the best-unbiased estimator.
The best-unbiased estimator for population mean \(\mu\) is the sample mean \(\bar{x}\) with the formula\[\bar{x}=\frac{\sum\limits_{i=1}^{n}x_i}{n}.\]
The best-unbiased estimator for population proportion \(\mu\) is the sample proportion \(\hat{p}\) with the formula\[\hat{p}=\frac{x}{n}.\]
The disadvantage of point estimation is that you don't know how close or how far away from the true value of the parameter the estimator is, that's when the interval estimator is useful.
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