What is the difference between standard deviation and standard error?

By / August 6, 2021

To understand the difference between standard deviation and standard error, we will first understand the meaning of statistic & parameter in the area of descriptive statistics.

Table of Contents hide
1 Parameter
2 Statistic
3 Estimator
4 Standard Error
5 Difference between Standard Error and Standard Deviation
6 References
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Parameter

It is a numerical description of a population characteristic. For example, the population mean is one of the characteristics of parameter, and the population mean is denoted by \mu.

Statistic

It is a numerical description of a particular sample characteristic. For example, the sample mean is one of the characteristics of statistic, and it is denoted by \bar{x}.

Mathematically: A function of random variables X_1, X_2, X_3, ... ,X_n is called as statistic. For example, mean of random variables, X_1, X_2, X_3, ... ,X_n, denoted as \bar{X} is again a random variable.

Estimator

An estimator is a rule for calculating an estimate of a given quantity based on sample data. For example, the sample mean is a commonly used estimator of the population mean.

Standard Error

Whenever we are trying to find the standard deviation of a statistic, then the outcome is again going to be a function of the random variables, and this standard deviation is called as standard error.

Difference between Standard Error and Standard Deviation

Standard error is a function of statistic whereas standard deviation is a function of unknown parameter. It is called unknown parameter because, the calculation of population parameter is nearly impossible as of huge data size. For example, let us consider \mu be the parameter representing the population mean, which is usually unknown. The standard deviation is defined as

\sigma = \sqrt{ \frac {1}{n} \sum_{i=1}^n { {(x_i - \mu)}^2 }}

Since \mu is unknown, we can not compute standard deviation using population mean. As we can not find \sigma. The only solution is to estimate \mu, by the mean of given sample observations. The sample mean is defined as \bar{x} = \frac {1}{n} \sum_{i=1}^n {x_i}. The standard error is defined as 

\sigma = \sqrt{ \frac {1}{n} \sum_{i=1}^n { {(x_i - \bar{x})}^2 }}

The standard error will be always be a function of observed values. The standard error is always refer to standard deviation which can always be computed on the basis of given sample of data.

References

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