Standard Error (SE) Measure

A Standard Error (SE) Measure is a dispersion measure or an estimate of the standard deviation of a statistic.

• Context:
  • It can qualify how well a Population Sample represents the Statistical Population.
  • It can (typically) be calculated as the square root of the variance divided by the sample size.
  • It can be related to a Relative Standard Error Measure.
  • ...
• Example(s):
  • A Standard Error of the Sample Mean (SEM): [math]\displaystyle{ SE_x= s/\sqrt{n} }[/math], where s is the sample standard deviation and n is the sample size.
  • A Standard Error of difference between samples: [math]\displaystyle{ X_1 }[/math] and [math]\displaystyle{ X_2 }[/math]: [math]\displaystyle{ SE_{x1-x2} = \sqrt {s^2_1 / n_1 + s^2_2 / n_2} \text{, where} s_1,\;s_2,\;n_1,\text{ and }n_2 }[/math] are the respective sample standard deviations and sample sizes
  • ...
• Counter-Example(s):
  • Statistical Variance.
  • Standard Deviation of a Statistical Population.
• See: Sample Standard Deviation, Estimator, Errors And Residuals in Statistics, Standard Deviation, Sampling Distribution, Statistic, Sample Mean, Population (Statistics), Regression Analysis.

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References

2023

• (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/Standard_error Retrieved:2023-10-4.
  • The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM). The standard error is a key ingredient in producing confidence intervals.

    The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

    Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

    In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).

2016

• (Statrek Website, 2016) ⇒ http://stattrek.com/statistics/dictionary.aspx?definition=standard%20error
  • The standard error is a measure of the variability of a statistic. It is an estimate of the standard deviation of a sampling distribution. The standard error depends on three factors:
    • N: The number of observations in the population.
    • n: The number of observations in the sample.
    • The way that the random sample is chosen.

If the population size is much larger than the sample size, then the sampling distribution has roughly the same standard error, whether we sample with or without replacement. On the other hand, if the sample represents a significant fraction (say, 1/20) of the population size, the standard error will be noticeably smaller, when we sample without replacement.

2016

• (CM website, 2016) ⇒ http://changingminds.org/explanations/research/statistics/standard_error.htm
  • If you measure a sample from a wider population, then the average (or mean) of the sample will be an approximation of the population mean. But how accurate is this?

If you measure multiple samples, their means will not all be the same, and will be spread out in a distribution (although not as much as the population). Due to the central limit theorem, the means will be spread in an approximately Normal, bell-shaped distribution.

The standard error, or standard error of the mean, of multiple samples is the standard deviation of the sample means, and thus gives a measure of their spread. Thus 68% of all sample means will be within one standard error of the population mean (and 95% within two standard errors).

What the standard error gives in particular is an indication of the likely accuracy of the sample mean as compared with the population mean. The smaller the standard error, the less the spread and the more likely it is that any sample mean is close to the population mean. A small standard error is thus a Good Thing.

When there are fewer samples, or even one, then the standard error, (typically denoted by SE or SEM) can be estimated as the standard deviation of the sample (a set of measures of x), divided by the square root of the sample size (n): [math]\displaystyle{ SE = stdev(x_i) / sqrt(n) }[/math]

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/standard_error Retrieved:2015-5-3.
  • The standard error (SE) is the standard deviation of the sampling distribution of a statistic, most commonly of the mean. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate. For example, the sample mean is the usual estimator of a population mean. However, different samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and variance). The standard error of the mean (SEM) (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples (of a given size) drawn from the population. Secondly, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analyzed at the time. In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the underlying errors.

2006

• (Dubnicka, 2006k) ⇒ Suzanne R. Dubnicka. (2006). “Introduction to Statistics - Handout 11." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
  • QUOTE: The standard error of an estimator X, denoted SE(X) is simply the standard deviation of X. Thus, the standard error of X is SE(X) = qVar(X).

2003

• (Davison, 2003) ⇒ Anthony C. Davison. (2003). “Statistical Models." Cambridge University Press. ISBN:0521773393

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