# Standard Error

A mathematical tool used in statistics to measure variability

## What is Standard Error?

Standard error is a mathematical tool used in statistics to measure variability. It enables one to arrive at an estimation of what the standard deviation of a given sample is. It is commonly known by its abbreviated form SE. Standard error is used to estimate the efficiency, accuracy, and consistency of a sample. In other words, it measures how precisely a sampling distribution represents a population.

It can be applied in statistics and economics. It is especially useful in the field of econometrics, where researchers use it in performing regression analyses and hypothesis testing. It is also used in inferential statistics, where it forms the basis for the construction of the confidence intervals.

Some commonly used measures in the field of statistics include:

• Standard error of the mean (SEM)
• Standard error of the variance
• Standard error of the median
• Standard error of a regression coefficient

### Calculating Standard Error of the Mean (SEM)

The SEM is calculated using the following formula:

#### Where:

• σ Population standard deviation
• n Sample size, i.e., the number of observations in the sample

In a situation where statisticians are ignorant of the population standard deviation, they use the sample standard deviation as the closest replacement. SEM can then be calculated using the following formula. One of the primary assumptions here is that observations in the sample are statistically independent. Where:

• s Sample standard deviation
• n Sample size, i.e., the number of observations in the sample

### Importance of Standard Error

When a sample of observations is extracted from a population and the sample mean is calculated, it serves as an estimate of the population mean. Almost certainly, the sample mean will vary from the actual population mean. It will aid the statistician’s research to identify the extent of the variation. It is where the standard error of the mean comes into play.

When several random samples are extracted from a population, the standard error of the mean is essentially the standard deviation of different sample means from the population mean.

However, multiple samples may not always be available to the statistician. Fortunately, the standard error of the mean can be calculated from a single sample itself. It is calculated by dividing the standard deviation of the observations in the sample by the square root of the sample size.

### Relationship between SEM and the Sample Size

Intuitively, as the sample size increases, the sample becomes more representative of the population.

For example, consider the marks of 50 students in a class in a mathematics test. Two samples A and B of 10 and 40 observations, respectively, are extracted from the population. It is logical to assert that the average marks in sample B will be closer to the average marks of the whole class than the average marks in sample A.

Thus, the standard error of the mean in sample B will be smaller than that in sample A. The standard error of the mean will approach zero with the increasing number of observations in the sample, as the sample becomes more and more representative of the population, and the sample mean approaches the actual population mean.

It is evident from the mathematical formula of the standard error of the mean that it is inversely proportional to the sample size. It can be verified using the SEM formula that if the sample size increases from 10 to 40 (becomes four times), the standard error will be half as big (reduces by a factor of 2).

### Standard Deviation vs. Standard Error of the Mean

Standard deviation and standard error of the mean are both statistical measures of variability. While the standard deviation of a sample depicts the spread of observations within the given sample regardless of the population mean, the standard error of the mean measures the degree of dispersion of sample means around the population mean.