The standard error of the mean in r is an important value in descriptive statistics. It helps to test the confidence level of an observation group. Unfortunately, r does not have a native function to handle it. As a result that you either have to write your own formula or get one from a package.

### Description Of How To Find Standard Error

Unfortunately, r programming does not have a built-in function for finding the standard error. Now, you can find such a formula in a package such as plotrix but it is just as easy to just create a formula to do the job as needed, this formula is as follows.

standard_error = function(x) sd(x) / sqrt(length(x))

- sd(x) is the sample standard deviation and it is calculated from the sample mean.
- length(x) is the sample size, and it is also the length of the vector being used as an argument in this function.

This is a simple little function to write, making it a convenient alternative to downloading an entire package for a single function.

### Explanation

The standard error of the mean in r is a handy tool when dealing with normally distributed data. It can also be handy when working with a regression model. It is a population parameter being based on both the standard deviation and sample size. The standard deviation is calculated based on the mean and we calculate mean values based on the sample size. The standard error is a measurement of sample variance, in that it demonstrates how spread out the sample is. It is a measurement of the average distance of data points from the mean value. Because of its extra relationship to the size of the sample, it can also supply measurement of the accuracy of data collection.

### Examples

Here we have four examples of code that produces the standard error using the formula described above.

> x = c(2, 4, 6, 8, 10, 12, 14, 16,18)

> standard_error(x)

[1] 1.825742

This is a simple example of code calculating the standard error.

> df = data.frame(x = c(1, 2, 3, 4, 5, 6, 7, 8),

+ y = c(1, 4, 9, 16, 25, 36, 64, 81))

> standard_error(df[,2])

[1] 10.32507

This is an example of code calculating the standard error of the mean, showing one approach to accessing a column in a data frame.

> x = rnorm(8)

> x

[1] 0.4570330 0.9488965 -1.5436080 0.1698043 -0.3568552 -0.5294474 0.6506566 -0.8832273

> standard_error(x)

[1] 0.2983383

This is an example of code calculating the standard error of the mean for a random sample using a normal distribution. The random variable is used as the argument in the formula that we created earlier.

> head(cars)

speed dist

1 4 2

2 4 10

3 7 4

4 7 22

5 8 16

6 9 10

> standard_error(cars$dist)

[1] 3.64434

> standard_error(cars$speed)

[1] 0.7477858

This is an example of code calculating the standard error for a more realistic data set. The data frame used here accesses each column using the df$column format.

### Application

The main application of the standard error of the mean is in calculating the confidence interval. The confidence interval is a statistic related to how accurate the sampling of observations has been. It helps to show the necessity of using multiple samples. Furthermore, it mathematically shows why the confidence interval gets better as the sample size increases. Because the standard error of the mean is related to both the population mean and the population size, our formula shows that the standard error gets smaller as the sample size increases. This is the reason it is best to collect lots and lots of data rather than relying on a few data points. The larger your sample size the more accurate your results will be.

The standard error of the mean is an unusual calculation in that while no built-in formula exists, it is easy to create a formula for it. Furthermore, it can be obtained by installing a package. In either case, it is an easy calculation to make and its application in calculating the accuracy of data is important.