Standard error of the mean, simply called standard error (SEM or SE), is a fundamental statistical concept that plays a pivotal role in quantifying the uncertainty associated with sample estimates and is essential for making valid inferences in research.

At its core, standard error reflects the variability or dispersion of sample data around the population parameter. It quantifies the average amount by which sample statistics are expected to differ from the true population parameter. A smaller standard error implies that the sample estimate is more likely to be close to the true value.

Standard error is useful for hypothesis testing and constructing confidence intervals. In hypothesis testing, it helps determine whether the observed difference between groups or the sample mean and a hypothesized value is statistically significant. A larger standard error can make it more challenging to detect significant differences.

Standard error is used to calculate the margin of error when constructing confidence intervals. A narrower confidence interval (smaller SE) provides a more precise estimate of the population parameter.

## Standard error formula: #

The standard error formula depends on the data type and the statistic being considered.

When population parameters are known, the formula used for calculating standard error is:

SE = σ√n

where,

- SE standard error
- σ is population standard deviation
- n is the number of elements in the sample.

When the population parameters are unknown, the following formula is used:

SE = s/√n

where,

- SE is standard error
- s is sample standard deviation
- n is the number of elements in the sample.

This formula can only be used to estimate standard error. It takes the sample standard deviation as a point estimate for the population standard deviation.

## List of recommended resources #

### For a broad overview #

Video: Sampling and Standard Error

This video lecture by MIT OpenCourseWare, as part of the course Introduction to Computational Thinking and Data Science, discusses sampling and how to analyze real data.

What Is Standard Error? How to Calculate (Guide with Examples)

This guide by Scribbr provides an overview of the statistical concept of standard error, how it differs from standard deviation, and how it is calculated.

### For in depth understanding #

Sample Size and Power Calculations

This lecture, prepared as part of a series of lectures produced by the Poverty Action Lab and the Massachusetts Institute of Technology, discusses standard errors and its significance in hypothesis testing.

This chapter published by UC Berkeley Statistics provides an in-depth understanding of the concept of standard error in statistics.

### Case study #

Global Data Set on Education Quality (1965-2015)

This paper written by Nadir Altinok, Noam Angrist and Harry Anthony Patrinos presents the largest globally comparable panel database of education quality. The data set includes the standard errors for the estimates, enabling explicit quantification of the degree of reliability of each estimate.

Using Poverty Maps to Improve the Design of Household Surveys: The Evidence from Tunisia

This paper written by Gianni Betti, Vasco Molini and Dan Pavelesku proposes a new method for improving the design effect of household surveys based on a two-stage design. The report wishes to study how improvement of the design effect can result in more precise survey estimates (smaller standard errors and confidence intervals) or reduction of the necessary sample size, that is, a reduction in the budget needed for a survey.

### References #

Standard Error of the Mean (SEM)

What is the Standard Error of a Sample ?