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Midrange

Midrange

A somewhat easy to compute quantitative estimate of a data set’s central tendency is its midrange, which is defined as the mean of the minimum and maximum values of an ordered data set.
Simply put, the midrange is half of the data set’s range. Similar to the mean, the midrange is sensitive to the extreme values and hence the prospective outliers. However, unlike the mean, the midrange is not sensitive to all the values of the data set that lie in the middle. Thus, it is prone to outliers and...

Estimating Population Mean with Unknown Standard Deviation

Estimating Population Mean with Unknown Standard Deviation

In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the Guinness...

Central Limit Theorem

Central Limit Theorem

The central limit theorem, abbreviated as clt, is one of the most powerful and useful ideas in all of statistics. The central limit theorem for sample means says that if you repeatedly draw samples of a given size and calculate their means, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. In other words, as sample sizes increase, the distribution of means follows the normal distribution more closely.
The sample size, n, that...

Central Tendency: Analysis

Central Tendency: Analysis

Measures of central tendency are tools used in biostatistics to identify the average or center of a dataset. They offer a single representative value for understanding and summarizing data distribution.
The mean is one such measure, calculated by totaling all values in a dataset and dividing by the number of values. For instance, the mean blood pressure reading (120, 130, 140, 150) would be 135. However, the mean can be affected by extreme values or outliers.
The median, another measure,...

Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

Estimating Population Mean with Known Standard Deviation

Estimating Population Mean with Known Standard Deviation

To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form as follows:
(point estimate - error bound, point estimate + error bound)
The...

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Related Experiment Video

Updated: Jun 28, 2026

Topographical Estimation of Visual Population Receptive Fields by fMRI

Topographical Estimation of Visual Population Receptive Fields by fMRI

Published on: February 3, 2015

Fuzzy robust estimation of central location.

C Sârbu¹, H F Pop

¹Department of Analytical Chemistry, Faculty of Chemistry and Chemical Engineering, 'Babeş-Bolyai' University, Arany Janos Str. 11, RO-3400 Cluj-Napoca, Romania.

|October 31, 2008

Summary

A new Fuzzy 1-means algorithm offers a robust method for estimating central location. This fuzzy set theory approach outperforms traditional mean estimators and matches or surpasses other robust methods.

Related Experiment Videos

Last Updated: Jun 28, 2026

Topographical Estimation of Visual Population Receptive Fields by fMRI

Topographical Estimation of Visual Population Receptive Fields by fMRI

Published on: February 3, 2015

Area of Science:

Statistics
Fuzzy Set Theory
Data Analysis

Background:

Estimating central location is crucial in data analysis.
Conventional methods like the ordinary mean can be sensitive to outliers.
Robust estimation techniques are needed for data with anomalies.

Purpose of the Study:

Introduce a novel robust algorithm for central location estimation.
Simplify the mathematical approach using fuzzy set theory.
Compare the new algorithm's performance against existing estimators.

Main Methods:

Developed a Fuzzy 1-means (FM) algorithm based on fuzzy set theory.
Compared FM against ordinary mean, median, trimmed mean, and M-estimators (Huber, Tukey, Hampel, Andrews).
Evaluated algorithm performance on diverse datasets from scientific literature.

Main Results:

The Fuzzy 1-means algorithm provides a mathematically simpler robust estimation method.
FM demonstrated superior performance compared to the ordinary mean estimator.
FM's performance was comparable or superior to established robust estimators.

Conclusions:

The Fuzzy 1-means algorithm is an effective and robust method for central location estimation.
This novel approach offers advantages in simplicity and performance.
FM presents a valuable alternative for statistical analysis, especially with outlier-prone data.