Standard Deviation vs Standard Error: Clearing up the Confusion with Visual Examples

Standard deviation and standard error are two statistical concepts that are often confused with each other. Though these two measures are related to variability in the data, they are different.

Standard deviation measures the variability in the dataset. The formula for standard deviation is given below.

Example 1

For example, a dataset with (10,10,10 & 10) has a standard deviation of 0.  That means no variability in the dataset.

Example 2

Let us use another example:

Dataset B: 10, 11, 12 & 14

Dataset C: 10, 100, 1000 & 2000

The standard deviation of dataset B (1.71) is lower than that of Dataset C (929.5). The lower the variability of the data, the lower will be the standard deviation. 

Example 3

Imagine there are 30 students in a class. You draw a first random sample of 5 students and measure their height. You can find out the mean (green line in the picture below) for this sample. And also you can compute the standard deviation for this sample (SD1).

You draw a total of four random samples. Similarly, you compute sample means for each of these samples (highlighted as vertical lines) and standard deviation (SD1, SD2, SD3 & SD4).

Now you have four sample means, as shown at the bottom of the following illustration. For four sample means, you can compute the mean of means. Also, you can find out the standard deviation for these four sample means. This is called the standard error of the mean (SEM).

In other words, Standard Error of Mean is the Standard Deviation of the Sample Means.

The standard error of the mean denotes the precision of the sample mean as an estimator of the true population mean. Due to random sampling error, sample mean varies from sample to sample and this variation is captured by the standard error of the mean. The smaller the standard error of the mean, the more precise the estimate of the true population mean.

The formula for standard error of the mean (SEM) is given below:
SEM = (sample standard deviation) / sqrt(sample size)

In summary, though standard deviation and standard error measure variability, they have different uses and interpretations.