# Standard error

**SEstandard errorsstandard error of the meanstandard error of measurementfinite population correction standard error of the meanerrorerror termRelative Standard ErrorSEM**

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.wikipedia

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### Standard deviation

**standard deviationssample standard deviationsigma**

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.

This derivation of a standard deviation is often called the "standard error" of the estimate or "standard error of the mean" when referring to a mean.

### Variance

**sample variancepopulation variancevariability**

This forms a distribution of different means, and this distribution has its own mean and variance.

This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem.

### Sampling distribution

**distributionsampling**

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.

standard error of that quantity.

### Regression analysis

**regressionmultiple regressionregression model**

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, e.g., confidence intervals).

The standard errors of the parameter estimates are given by

### Confidence interval

**confidence intervalsconfidence levelconfidence**

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, e.g., confidence intervals). when the probability distribution of the value is known, it can be used to calculate an exact confidence interval;

In our case we may determine the endpoints by considering that the sample mean from a normally distributed sample is also normally distributed, with the same expectation μ, but with a standard error of:

### Normal distribution

**normally distributednormalGaussian**

If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.

Of practical importance is the fact that the standard error of is proportional to, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100.

### Unbiased estimation of standard deviation

**anti-biasingsample standard deviationbias**

Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20. See unbiased estimation of standard deviation for further discussion.

When this condition is satisfied, another result about s involving c 4 (n) is that the standard error of s is, while the standard error of the unbiased estimator is

### Quantile

**quantilesquintiletertile**

If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean.

The standard error of a quantile estimate can in general be estimated via the bootstrap.

### Sample size determination

**meadsample sizeSampling sizes**

as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.

When estimating the population mean using an independent and identically distributed (iid) sample of size n, where each data value has variance σ 2, the standard error of the sample mean is:

### 1.96

The following expressions can be used to calculate the upper and lower 95% confidence limits, where \bar{x} is equal to the sample mean, SE is equal to the standard error for the sample mean, and 1.96 is the 0.975 quantile of the normal distribution:

Standard error (statistics)

### Weighted arithmetic mean

**averageaverage ratingweighted average**

Standard error of the weighted mean

Its minimum value is found when all weights are equal (i.e., unweighted mean), in which case we have, i.e., it degenerates into the standard error of the mean, squared.

### Autocorrelation

**autocorrelation functionserial correlationautocorrelated**

where the sample bias coefficient ρ is the widely used Prais–Winsten estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample point pairs.

While it does not bias the OLS coefficient estimates, the standard errors tend to be underestimated (and the t-scores overestimated) when the autocorrelations of the errors at low lags are positive.

### Sampling fraction

When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction":

To correct for this dependence when calculating the sample variance, a finite population correction (or finite population multiplier) of (N-n)/(N-1) may be used.

### Coefficient of variation

**CVrelative standard deviationcoefficients of variation**

Coefficient of variation

Relative Standard Error

### Statistical parameter

**parametersparameterparametrization**

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.

### Mean

**mean valuepopulation meanaverage**

This forms a distribution of different means, and this distribution has its own mean and variance.

### Reduced chi-squared statistic

**reduced chi-squared**

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, e.g., confidence intervals).

### Bias of an estimator

**unbiasedunbiased estimatorbias**

Note: the standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and standard deviation: the standard error of the mean is a biased estimator of the population standard error.

### Student's t-distribution

**Student's ''t''-distributiont''-distributiont-distribution**

Note: The Student's probability distribution is approximated well by the Gaussian distribution when the sample size is over 100.

### Statistic

**sample statisticempiricalmeasure**

### Function (mathematics)

**functionfunctionsmathematical function**

in many cases, if the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated;

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

when the probability distribution of the value is known, it can be used to calculate an exact confidence interval;

### Chebyshev's inequality

**ChebyshevAn an inequality on location and scale parametersBienaymé–Chebyshev inequality**

when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and

### Vysochanskij–Petunin inequality

**Vysochanskiï–Petunin inequalities**

when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and

### Central limit theorem

**limit theoremsA proof of the central limit theoremcentral limit**

as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.