# Standard error

The standard error of a statistic (such as the mean, variance, median etc.) is a measure of variability; a higher standard error indicates that the data is more spread out. More specifically, standard error is the standard deviation of a sampling distribution. A sampling distribution is a probability distribution of a statistic obtained from a large number of samples (of the same size) of a specific population.

As a simple example, imagine a population of 5 students whose weights are 100 lbs, 110 lbs, 120 lbs, 130 lbs, and 140 lbs. The population mean can be computed as:

Given that 5 samples of 2 students each are taken, and their sample means calculated, below is a possible sampling distribution:

Sample | Sample mean |
---|---|

100; 110 | 105 |

100; 140 | 120 |

130; 140 | 135 |

100; 110 | 105 |

120; 130 | 125 |

Notice that some of the sample means are closer to the population mean than others. From this information, we could calculate the standard error of the mean for this sampling distribution, which would provide an estimate of how far the mean of the sample is likely to be from the population mean. Different sample sizes will provide different sampling distributions. In this case, the population is very small, but in most cases the population will likely be too large to determine population statistics, and an appropriate sample size will have to be selected. In general, the larger the sample size, the more accurately a sample statistic will represent the corresponding population statistic.

## Standard error of the mean

Standard error can be calculated for various statistics. The standard error of the mean (SEM) is one commonly computed statistic that is used to compute other statistics such as confidence intervals and p-values. Others include the standard error of the variance, median and more. The formula used to compute the SEM is dependent on whether the population standard deviation or sample standard deviation is known. In most cases, the standard deviation of the population is not known (not feasible, prohibitively expensive, etc.), so the sample standard deviation is used instead. However, if the population standard deviation is known, an exact SEM can be computed using the formula

where σ is the population standard deviation and n is the sample size.

In cases where the population standard deviation is not known, the standard error can be computed using the formula

where s is the sample standard deviation and n is the sample size. Notice that in either case, the larger the sample size, the smaller the standard error will be. As n gets closer to the population size N, the standard error of the mean will decrease. If the sample size is equal to the population, the standard error of the mean will be 0.

Example

15,000 students attend a local university. 5 random samples of the heights of 100 students are taken, and the mean of each sample is computed. The sampling distribution of the mean for a sample size of 100 is shown below.

Sample | Sample mean (in) |
---|---|

1 | x_{1} = 69.4 |

2 | x_{2} = 66.2 |

3 | x_{3} = 67.8 |

4 | x_{4} = 71.4 |

5 | x_{5} = 69.5 |

Find the SEM.

The population standard deviation is not known, so we use the following formula:

The sample standard deviation is computed using the formula

where x_{i} is the ith element of the sample, x is the sample mean, and n is the sample size.

The mean of the sampling distribution is:

Thus, the sample standard deviation is

and the SEM can be computed as:

This value means that the sample mean will typically vary by ~0.87 inches relative to the population mean.

Since the standard error is the standard deviation of a sampling distribution, it can be used to determine confidence intervals. Height follows a normal distribution, and for a normal distribution, 95% of values will fall within 1.96 (~2) standard deviations from the mean. Thus, given that the standard error of the mean is 0.87, we can estimate that 95% of the means from any given sample will be within 2 standard deviations (0.87 × 2 = 1.74) from the population mean. In other words, for any given sample mean, x, the interval

x ± 1.74 in.

will include the population mean 95% of the time.