In the previous lessons, weâ€™ve covered various ways to draw inferences about a population from a sample. Weâ€™ll now zero in on a specific type of inference known as estimation, which focuses on determining the approximate values of particular population parameters (e.g. the mean, the proportion) based on sample data. Estimation specifically aims to quantify these unknown parameters.

In addition to thinking of estimation as a specific type of statistical inference, you can also think of it as a fundamental component of the larger inferential process. Once you calculate estimates for a certain parameter, you can then use these values in other inferential procedures, like hypothesis testing and building predictive models. These processes, taken together, are what will eventually enable you to draw broad conclusions about your population based on the sample of data youâ€™ve chosen to study.

## The Value of Estimation

Estimation is often singled out for study among the different types of statistical inference because itâ€™s fundamental and widely applicable in both academic research and practical decision-making. Hereâ€™s a deeper look into its value:

*Direct Practical Application*

Businesses, policymakers, and researchers often need specific figures to guide their actions. Estimation provides concrete numerical values or ranges that are immediately useful for decision-making.

*Foundation for Other Inference Methods*

Estimation serves as a basis for more complex statistical methods, such as:

**Hypothesis Testing â€“**Estimation provides the necessary parameters (like means and variances) to test hypotheses about population characteristics.**Predictive Modeling â€“**Estimations of population parameters are often the starting points for creating models that predict future trends and behaviors.

*Simplicity and Accessibility*

Estimation is relatively straightforward compared to other statistical methods. This makes it more accessible to people without extensive statistical backgrounds. Because of its simplicity, estimation enables wider application and understanding of statistical principles. Business executives, for instance, can quickly grasp and utilize estimates for operational and strategic decisions even without highly technical statistics backgrounds.

Even members of the public who arenâ€™t statistics specialists can understand reports and studies that use estimation. This encourages better communication and greater transparency between those who produce statistical data and those who use it or are affected by it. These might be researchers and analysts communicating with their stakeholders, businesses with their customers, or policymakers with their constituents.

*Essential in Data-Driven Decision Making*

Now that we live in the era of big data and analytics, estimation often catalyzes the transformation of raw information into actionable insights. It helps organizations summarize vast amounts of data into meaningful statistics. It also informs action, guiding businesses and organizations toward more data-driven strategies rather than those based on guesswork or intuition.

## Two Types of Estimates

There are two main types of estimates in statistics: point estimates and interval estimates. Each type serves a different purpose and provides different information about the population parameter being estimated. Letâ€™s examine both in detail now:

*Point Estimates*

A point estimate is a single statistic thatâ€™s used to approximate an unknown population quantity. It is derived directly from the sample data and serves as the best guess for the unknown parameter.

Point estimates are straightforward and simple, since they provide a single value as the estimate. Good point estimates are also highly efficient; they should be unbiased and produce the smallest possible variance among all unbiased estimators. This means they come as close to the true population parameter as possible on average.

Some examples of common point estimates include the following:

**Sample Mean****â€“**Used to estimate the population mean .**Sample Proportion****â€“**Used to estimate the population proportion .**Sample Variance****â€“**Used to estimate the population variance .

While point estimates are easy to understand, however, they also have particular shortcomings that limit their usefulness. For one, point estimates provide a single value without any indication of how much this value might vary from the true population parameter. This lack of context can be misleading, and it may cause users to overestimate the valueâ€™s accuracy. The accuracy of a point estimate heavily depends on the sample, since different samples from the same population can yield different point estimates. Thereâ€™s no way to know from the point estimate alone how representative the sample is.

*Interval Estimates*

An interval estimate provides a range of values within which the population parameter is expected to lie, along with a specified level of confidence. This type of estimate accounts for the uncertainty inherent in using a sample to estimate a population parameter.

Interval estimates provide a lower and upper bound, between which youâ€™re most likely to find the true population parameter. They were developed to contend with the abovementioned limitations of point estimates, as they provide a way to express the uncertainty that point estimates donâ€™t typically account for.

## Introducing Confidence Intervals

A confidence interval is a specific type of interval estimate that includes a confidence level. The confidence level is meant to quantify the degree of certainty that the interval contains the true population parameter. Confidence levels are typically chosen based on convention and the needs of the analysis. 95% is the most common confidence level, but 90% and 99% confidence are also used on occasion.

Confidence intervals are the most common type of interval estimates used in statistical inference. In essence, their purpose is to create a range with specific upper and lower limits within which you can capture or trap the unknown population parameter. Thus, you might start with a certain point estimate and know for sure that it cannot exactly equal the population quantity. But if you create a confidence interval, youâ€™ll be able to say with some prespecified level of certainty that the population quantity can be found within the defined range.

To illustrate, if you have a point estimate and an unknown population quantity , then you might calculate an interval estimate where the confidence limits are as follows:

- is the lower limit of the estimate.
- is the upper limit of the estimate.

If you wanted to represent that graphically for clarity, it might look like this:

For any given analysis, you can also generally expect that the following factors will affect confidence intervals:

**Sample Size â€“**Larger samples yield more precise estimates, resulting in narrower confidence intervals.**Variability in Data â€“**More variability in the data leads to wider confidence intervals.**Confidence Level â€“**Higher confidence levels produce wider intervals because they require more certainty that the interval contains the true parameter.

## A Practical Example of Estimation

Letâ€™s look at a concrete example of estimation at work, which also applies what weâ€™ve previously learned about the concepts of statistical sampling and inference. Imagine a cafÃ© wants to know the average amount customers spend per visit, and to this end, it surveys a random sample of 200 customers.

The survey might yield the following data:

- The average spending from the sample is $5.50. This is an example of a point estimate.
- The business might then calculate that the true average spending is between $5.20 and $5.80 with 95% confidence.

Based on the estimates, the business infers that the average spending for all customers likely falls within the estimated range. This information might subsequently affect the strategic decisions the business makes about pricing, marketing, and inventory.

## The Diverse Applications of Estimation

Estimationâ€™s practical value and foundational role in other statistical methods make it a major cornerstone of statistical inference. Because it provides specific numerical insights and ranges, estimation is especially useful for businesses, policymakers, and researchers aiming to develop informed solutions to pressing problems.

Having established the fundamentals of estimation, weâ€™ll soon move on to the particulars of hypothesis testing and calculating confidence intervals.