Confidence Intervals

Definition

The confidence interval CI is the range in which a parameter (e.g. the mean value) lies with a certain probability.

If several samples are taken from a population, it is very likely that each sample will have a different mean value. Now you actually want to know the mean value of the population and not that of the sample. The confidence interval indicates the range in which the true mean value of the population lies with a certain probability.

Attention! The above explanation is widely used because it is easy to understand, but it is not considered correct by all experts. Correct, but more complicated is the following definition:

The 95% confidence interval (CI) is an interval calculated from sample data that’s one from an infinite sequence, 95% of which include the population parameter. In the long run, 95% of such intervals include the true mean. ¹

Who

For example, those who are involved in the following professions. ²

Biology – plant height at maturity, animal weight at maturity
Business – revenue, cost, profit (margin?)
Finance – interest rates, stock/bond returns
Housing – Average area of a house, average age of a house
Manufacturing – average diameter of a machined part, average life of an electronic device
Market Research – average age of customer, average household income of customer
Medicine – cholesterol, blood pressure
Polling – percentage of people who support a law,
Population Studies – percent of population with a health condition,
Websites – percent of visitors from U.S., revenue per visitor in an industry or niche

What

In statistics, parameters of the population are often estimated based on a sample, for example the mean or the variance . But these are only estimates and the true value in the population will lie somewhere around these estimates. Now it is very useful to define a range or interval in which the true value will lie with a high probability, this range is called confidence interval. ¹

Why

Confidence intervals are an important concept in statistics.  They help us to make estimates by using the data we have available, and they tell us how much confidence we have in the calculated range.

So, what is a confidence interval used for?  Confidence intervals are used to give a range as an estimate for an unknown population parameter. Confidence intervals are used in various fields, such as biology, business, finance, housing, manufacturing, market research, medicine, polling, population studies, and websites. ²

See Theoretical Knowledge Vs Practical Application.

How

Many of the References and Additional Reading websites and Videos will assist you with understanding and using confidence intervals.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

¹ “Confidence Interval”. 2022. datatab.net. https://datatab.net/tutorial/confidence-interval.

² “What Is A Confidence Interval Used For? (10 Real Life Examples) – JDM Educational”. 2022. jdmeducational.com. https://jdmeducational.com/what-is-a-confidence-interval-used-for-10-real-life-examples/.

Additional Reading

“A Basic Explanation Of Confidence Intervals”. 2019. census.gov. https://www.census.gov/programs-surveys/saipe/guidance/confidence-intervals.html.

A confidence interval is a range of values that describes the uncertainty surrounding an estimate. We indicate a confidence interval by its endpoints; for example, the 90% confidence interval for the number of people, of all ages, in poverty in the United States in 1995 (based on the March 1996 Current Population Survey) is “35,534,124 to 37,315,094.” A confidence interval is also itself an estimate. It is made using a model of how sampling, interviewing, measuring, and modeling contribute to uncertainty about the relation between the true value of the quantity we are estimating and our estimate of that value.

“Confidence Intervals”. 2022. mathsisfun.com. https://www.mathsisfun.com/data/confidence-interval.html.

A Confidence Interval is a range of values we are fairly sure our true value lies in.

“Understanding Confidence Intervals | Easy Examples & Formulas”. 2020. Scribbr. https://www.scribbr.com/statistics/confidence-interval/.

When you make an estimate in statistics, whether it is a summary statistic or a test statistic, there is always uncertainty around that estimate because the number is based on a sample of the population you are studying.

The confidence interval is the range of values that you expect your estimate to fall between a certain percentage of the time if you run your experiment again or re-sample the population in the same way.

The confidence level is the percentage of times you expect to reproduce an estimate between the upper and lower bounds of the confidence interval, and is set by the alpha value.

Videos

In statistics, parameters of the population are often estimated based on a sample, e.g. the mean or the variance. But these are only estimates and the true value in the population will lie somewhere around these estimates. Now it is of course very useful to define a range or interval in which the true value will lie with a high probability, this range is called confidence interval.