What is the Standard Error?

The standard error is an estimate of the standard deviation of a statistic. This lesson shows how to compute the standard error, based on sample data.

The standard error is important because it is used to compute other measures, like confidence intervals and margins of error.

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 10 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.

Standard Error of Sample Estimates

Sadly, the values of population parameters are often unknown, making 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, and it provides an unbiased estimate of the standard deviation of the statistic. The table below shows how to compute the standard error for simple random samples, assuming the population size is at least 10 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 of This Lesson

Problem 1

Which of the following statements is true.

I. The standard error is computed solely from sample attributes.
II. The standard deviation is computed solely 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 can be computed from a knowledge of sample attributes - sample size and sample statistics. The standard deviation cannot be computed solely from sample attributes; it requires a knowledge of one or more population parameters. The standard error is a measure of variability, not a measure of central tendency.