An estimate is an approximation or a calculated value derived from incomplete or uncertain data. In the context of statistics, an estimate refers to the calculated values, either a single point (point estimate) or an interval (interval estimate), used to infer unknown population parameters based on sampled data.
Definition
- General Approximation: An estimate is a rough calculation or judgement made without the necessity for exact data. It is often based on informed guesses, previous experience, or partial data.
- Statistical Estimate: In statistics, an estimate refers to the use of sample data to calculate an approximation of a population parameter. This can be further divided into:
- Point Estimate: A single value that serves as an approximation of an unknown population parameter.
- Interval Estimate: A range of values, bounded by a lower and an upper limit, that is likely to contain the population parameter.
Types of Statistical Estimates
Point Estimate
A point estimate is a single number calculated from the sample data that best approximates the population parameter. Common point estimates include the sample mean (\(\bar{x}\)), sample proportion (\(p\)), and the sample variance (\(s^2\)).
Interval Estimate
An interval estimate provides a range of values within which the population parameter is expected to fall. This range is calculated with a certain level of confidence, expressed as a confidence interval (e.g., 95% confidence interval). The interval estimate accounts for the variability and uncertainty inherent in sample data.
Formulas
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Point Estimate of Mean:
$$ \hat{\mu} = \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i $$where \( \hat{\mu} \) is the point estimate of the population mean, \(\bar{x}\) is the sample mean, and \(n\) is the sample size. -
Interval Estimate (Confidence Interval for Mean):
$$ \bar{x} \pm z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right) $$where \(\bar{x}\) is the sample mean, \(z_{\alpha/2}\) is the critical value from the standard normal distribution, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
Historical Context
Estimates have been essential in various fields far back into history, including commerce, engineering, and navigation. The formal methods of statistical estimation emerged in the 18th and 19th centuries alongside the development of probability theory by pioneers such as Pierre-Simon Laplace and Carl Friedrich Gauss.
Applicability
Estimates are vital in numerous fields for planning, forecasting, decision-making, and hypothesis testing. They provide a mechanism to make informed decisions without requiring complete certainty:
- Economics: Estimating GDP growth, inflation rates, and other economic indicators.
- Engineering: Estimating costs, materials, and structural integrity in construction.
- Finance: Estimating returns, risks, and valuations for investments.
- Medicine: Estimating treatment effects and disease prevalence from clinical trials.
Comparisons and Related Terms
- Estimator: A rule or formula used to compute an estimate from sample data.
- Bias: The systematic error in an estimate; a biased estimate consistently over- or underestimates the population parameter.
- Precision: The degree to which repeated estimates under unchanged conditions show the same results; lower variability implies higher precision.
FAQs
What is the difference between an estimate and an estimator?
How do we interpret a confidence interval in an interval estimate?
References
- Rosner, Bernard. Fundamentals of Biostatistics. Cengage Learning, 2015.
- Hogg, Robert V., and Craig Allen Tannis. Probability and Statistical Inference. Pearson, 2013.
- Montgomery, Douglas C., and George C. Runger. Applied Statistics and Probability for Engineers. Wiley, 2018.
Estimating is a quintessential tool in mathematics and statistics, aiding in making informed conjectures and decisions based on limited data. Understanding the distinctions and applications of point and interval estimates allows professionals of various domains to handle data-driven predictions confidently.
