What is the Standard Error?
The standard error is an estimate of the standard deviation of the sampling distribution of a statistic.
This lesson shows how to compute the standard deviation of the sampling distribution of a statistic, based on sample size and on population parameters. And it shows how to compute the standard error, based on sample size and on sample statistics.
Both measures are important because either can be used to compute other useful measures, like confidence intervals and margins of error But the standard error may be more useful since it is computed from readily-available sample statistics; whereas the standard deviation is computed from population parameters, which are often unknown.
Notation
The following notation is helpful, when we talk about the standard deviation and the standard error.
| Population parameter | Sample statistic |
|---|---|
| N: Number of observations in the population | n: Number of observations in the sample |
| Ni: Number of observations in population i | ni: Number of observations in sample i |
| P: Proportion of successes in population | p: Proportion of successes in sample |
| Pi: Proportion of successes in population i | pi: Proportion of successes in sample i |
| μ: Population mean | x: Sample estimate of population mean |
| μi: Mean of population i | xi: Sample estimate of μi |
| σ: Population standard deviation | s: Sample estimate of σ |
| σp: Standard deviation of p | SEp: Standard error of p |
| σx: Standard deviation of x | SEx: Standard error of x |
Standard Deviation of Sample Estimates
Statisticians use sample statistics to estimate population parameters. Naturally, the value of a statistic may vary from one sample to the next.
The variability of a statistic is measured by its standard deviation. The table below shows formulas for computing the standard deviation of statistics from simple random samples. These formulas are valid when the population size is much larger (at least 20 times larger) than the sample size.
| Statistic | Standard Deviation |
|---|---|
| Sample mean, x | σx = σ / sqrt( n ) |
| Sample proportion, p | σp = sqrt [ P(1 - P) / n ] |
| Difference between means, x1 - x2 | σx1-x2 = sqrt [ σ21 / n1 + σ22 / n2 ] |
| Difference between proportions, p1 - p2 | σp1-p2 = sqrt [ P1(1-P1) / n1 + P2(1-P2) / n2 ] |
Note: In order to compute the standard deviation of a sample statistic, you must know the value of one or more population parameters. For example, to compute the standard deviation (σx) of the sample mean, you need to know the standard deviation (σ) of the population.
Standard Error of Sample Estimates
Sadly, the values of population parameters are often unknown, which often makes it impossible to compute the standard deviation of a statistic. When this occurs, use the standard error.
The standard error is computed from known sample statistics. The table below shows how to compute the standard error for simple random samples, assuming the population size is at least 20 times larger than the sample size.
| Statistic | Standard Error |
|---|---|
| Sample mean, x | SEx = s / sqrt( n ) |
| Sample proportion, p | SEp = sqrt [ p(1 - p) / n ] |
| Difference between means, x1 - x2 | SEx1-x2 = sqrt [ s21 / n1 + s22 / n2 ] |
| Difference between proportions, p1 - p2 | SEp1-p2 = sqrt [ p1(1-p1) / n1 + p2(1-p2) / n2 ] |
The equations for the standard error are identical to the equations for the standard deviation, except for one thing - the standard error equations use statistics where the standard deviation equations use parameters. Specifically, the standard error equations use p in place of P, and s in place of σ.
Test Your Understanding
Problem 1
Which of the following statements is true.
I. The standard error of sample mean is computed from sample attributes .
II. The standard deviation of a sample mean is computed from sample attributes.
III. The standard error is a measure of central tendency.
(A) I only
(B) II only
(C) III only
(D) All of the above.
(E) None of the above.
Solution
The correct answer is (A). The standard error of a sample mean can be computed from a knowledge of sample attributes - sample size and the sample standard deviation. The standard deviation of a sample mean cannot be computed solely from sample attributes; it requires a knowledge of the population standard deviation, which is a population parameter - not a sample statistic. The standard error is a measure of variability, not a measure of central tendency.