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Variance

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.

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...

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...

Estimating Population Standard Deviation

Estimating Population Standard Deviation

When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and...

What are Estimates?

What are Estimates?

It isn't easy to measure a parameter such as the mean height or the mean weight of a population. So, we draw samples from the population and calculate the mean height or mean weight of the individuals in the sample. This sample data acts as a representative measure of the population parameter. These sample statistics are known as estimates.
The estimate for the mean of a sample is denoted by ͞x, whereas the mean of the population is designated as μ. Further, parameters such as the mean,...

One-Way ANOVA: Equal Sample Sizes

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...

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Related Experiment Video

Updated: Jul 10, 2026

Decomposing the Variance in Reading Comprehension to Reveal the Unique and Common Effects of Language and Decoding

Decomposing the Variance in Reading Comprehension to Reveal the Unique and Common Effects of Language and Decoding

Published on: October 11, 2018

[Variance estimation methods in samples from household surveys].

Maria Cecilia Goi Porto Alves¹, Nilza Nunes da Silva

¹Instituto de Saúde, Secretaria de Estado da Saúde de São Paulo, São Paulo, SP, Brasil. cecilia@isaude.sp.gov.br

Revista De Saude Publica

|November 10, 2007

Summary

This study evaluated variance estimators for urban household surveys in Brazil, finding that Taylor linearization, Jackknife, and BRR replication methods offer similar accuracy. While confidence intervals were slightly lower than nominal, they still allowed for reasonable estimations in complex survey designs.

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Area of Science:

Statistics
Survey Methodology
Demography

Context:

Household surveys are crucial for understanding urban populations.
Complex estimation methods are often necessary due to sample composition.
Accurate interpretation of survey results hinges on understanding sampling errors.

Purpose:

To evaluate the performance of variance estimators in Brazilian urban household surveys.
To compare the accuracy and confidence interval coverage of Taylor linearization, Jackknife, and BRR replication techniques.
To assess the impact of sampling design on variance estimation.

Summary:

Three variance estimation techniques (Taylor linearization, Jackknife, BRR replication) were compared using data from an employment and unemployment survey in São Paulo.
The study employed stratified cluster sampling with repeated samples drawn from three different designs.
Accuracy was assessed via mean square error, and confidence interval coverage was evaluated.

Impact:

All three variance estimators demonstrated comparable accuracy and confidence interval coverage.
Bias was found to be negligible relative to the standard error magnitude.
Despite slightly lower than nominal confidence levels, the estimators facilitate reasonably reliable interval estimations for complex surveys.