Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Calibration Curves: Linear Least Squares01:20

Calibration Curves: Linear Least Squares

A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
For data that follow a straight line, the standard method for fitting is the linear...
Correlation of Experimental Data01:23

Correlation of Experimental Data

Dimensional analysis simplifies complex physical problems and guides experimental investigations, but it does not provide complete solutions. It identifies the dimensionless groups that influence a phenomenon, but experimental data is needed to establish the specific relationships and validate theoretical predictions.
For example, a spherical particle moving through a viscous fluid experiences drag. Dimensional analysis shows that the drag force depends on the particle's diameter, velocity, and...
Estimating Population Mean with Unknown Standard Deviation01:22

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...
Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
One-Way ANOVA: Equal Sample Sizes01:15

One-Way ANOVA: Equal Sample Sizes

One-Way ANOVA can be performed on three or more samples with equal or unequal sample sizes. When one-way ANOVA is performed on two datasets with samples of equal sizes, it can be easily observed that the computed F statistic is highly sensitive to the sample mean.
Different sample means can result in different values for the variance estimate: variance between samples. This is because the variance between samples is calculated as the product of the sample size and the variance between the...
Variance01:15

Variance

The deviations show how spread out the data are about the mean. A positive deviation occurs when the data value exceeds the mean, whereas a negative deviation occurs when the data value is less than the mean. If the deviations are added, the sum is always zero. So one cannot simply add the deviations to get the data spread. By squaring the deviations, the numbers are made positive; thus, their sum will also be positive.The standard deviation measures the spread in the same units as the data.

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

How Not to Do WLS Fitting in Calibration with Heteroscedastic Data.

Analytical chemistry·2026
Same author

Systematic errors in isothermal titration calorimetry: The role of feedback power and effects of mixing and diffusion on concentrations.

Analytical biochemistry·2025
Same author

Calibrating ITC instruments: Problems with weak base neutralization.

Analytical biochemistry·2024
Same author

Goodness-of-Fit Tests in Calibration: Are They Any Good for Selecting Least-Squares Weighting Formulas?

Analytical chemistry·2022
Same author

A (partial) resolution of binding enthalpy discrepancies in ITC studies of Ba<sup>2+</sup> crown ether complexation: The importance of calibration.

Analytical biochemistry·2021
Same author

Estimating Real-Time qPCR Amplification Efficiency from Single-Reaction Data.

Life (Basel, Switzerland)·2021

Related Experiment Video

Updated: Jul 7, 2026

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Published on: July 3, 2020

Least squares with non-normal data: estimating experimental variance functions.

Joel Tellinghuisen1

  • 1Department of Chemistry, Vanderbilt University, Nashville, TN 37235, USA.

The Analyst
|January 30, 2008
PubMed
Summary

The method of least squares (LS) is valid even without normally distributed data. This approach accurately estimates variance functions (VFE) from replicate statistics, even with non-normal error distributions.

More Related Videos

A Workflow for Lipid Nanoparticle (LNP) Formulation Optimization using Designed Mixture-Process Experiments and Self-Validated Ensemble Models (SVEM)
13:54

A Workflow for Lipid Nanoparticle (LNP) Formulation Optimization using Designed Mixture-Process Experiments and Self-Validated Ensemble Models (SVEM)

Published on: August 18, 2023

Design and Optimization Strategies of a High-Performance Vented Box
14:23

Design and Optimization Strategies of a High-Performance Vented Box

Published on: June 9, 2023

Related Experiment Videos

Last Updated: Jul 7, 2026

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
04:35

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach

Published on: July 3, 2020

A Workflow for Lipid Nanoparticle (LNP) Formulation Optimization using Designed Mixture-Process Experiments and Self-Validated Ensemble Models (SVEM)
13:54

A Workflow for Lipid Nanoparticle (LNP) Formulation Optimization using Designed Mixture-Process Experiments and Self-Validated Ensemble Models (SVEM)

Published on: August 18, 2023

Design and Optimization Strategies of a High-Performance Vented Box
14:23

Design and Optimization Strategies of a High-Performance Vented Box

Published on: June 9, 2023

Area of Science:

  • Statistics
  • Data Analysis

Background:

  • The method of least squares (LS) is commonly assumed to require normally distributed (Gaussian) errors, which is not always true.
  • Estimating data variance functions (VFE) from replicate statistics is a key application where LS is used with non-normal data.
  • Sampling estimates of variance (sigma^2) from normal data follow a chi-squared distribution, which is highly asymmetrical for small degrees of freedom.

Purpose of the Study:

  • To demonstrate the validity and effectiveness of the method of least squares (LS) for estimating variance functions (VFE) when data errors are not normally distributed.
  • To investigate the properties of LS parameter estimates for variance and standard deviation functions derived from non-normal data, specifically chi-squared distributed variance estimates.

Main Methods:

  • Utilized Monte Carlo computations to simulate linear variance functions and assess LS fitting under non-normal error conditions.
  • Employed proper weighting schemes, specifically s^(-4) for VFE and s^(-2) for standard deviation function estimation (SDFE), where weights are evaluated on calculated functions.
  • Implemented an iterative computational approach to handle the necessary weighting, ensuring convergence and minimizing residual bias.

Main Results:

  • LS variance-function parameters remain unbiased and minimum-variance estimates even with non-normal data, provided proper weighting is applied.
  • LS estimates for standard deviation functions can exhibit bias, but this bias is predictable and correctable, stemming from the inherent bias in the standard deviation estimator (s).
  • The iterative weighting process, crucial due to the uncertainty in variance and standard deviation estimates, typically converges quickly with negligible remaining bias.

Conclusions:

  • The method of least squares (LS) is a robust tool for variance function estimation (VFE) and standard deviation function estimation (SDFE) even when data errors are not normally distributed.
  • Proper iterative weighting based on calculated functions is essential for accurate LS fitting in these non-normal scenarios.
  • LS provides unbiased, minimum-variance estimates for variance functions and correctable estimates for standard deviation functions, highlighting its broad applicability in statistical analysis.