The difference between a standard deviation and a standard error can seem murky. Let’s clear that up in this post!
Standard deviation (SD) and standard error (SE) both measure variability. High values of either statistic indicate more dispersion. However, that’s where the similarities end. The standard deviation is not the same as the standard error.
Here are the key differences between the two:
- Standard deviation: Quantifies the variability of values in a dataset. It assesses how far a data point likely falls from the mean.
- Standard error: Quantifies the variability between samples drawn from the same population. It assesses how far a sample statistic likely falls from a population parameter.
Let’s move on to graphical examples of both statistics so you can understand the differences intuitively. Then you’ll learn how to calculate both the standard deviation and standard error.
Learn more about measures of variability.
Examples of Standard Error vs. Standard Deviation
In the following examples, I use graphs to highlight the differences between standard deviation and standard error. Remember that a SD is the variability within a sample and compares data points to the mean. Conversely, a SE is the variability between samples and compares sample estimates to population parameters.
For these examples, I use statistical software to sample values randomly from a normal distribution with a mean of 100 and standard deviation of 15, which is the distribution of IQ scores.
Imagine you draw a random sample of 10 people and measure their IQs. You can plot their scores on an individual values plot. Visually, we can see the spread of the data points around the mean in the graph below. The red diamond is the sample mean.
The standard deviation mathematically measures the variability. More specifically, it assesses the distances between each data point and the sample mean.
Learn more about the standard deviation.
Now, imagine we draw ten random samples, and each one has ten observations. Even though the samples are all subsets of a common population, their means are bound to differ due to sampling error.
The graph below displays ten random samples drawn from the same population.
The red diamonds indicate the sample means. As you can see, the means fluctuate up and down between the samples.
The standard error of the mean measures the variability between sample means.
Learn more about the standard error of the mean.
Standard Deviation vs. Standard Error in Distributions
A crucial point is that while both statistics quantify variability in a distribution of values, they apply to different distributions. Let’s drill down on that aspect a bit more.
Suppose you draw a single random sample and graph its distribution of values with the curve below.
Each point on the curve represents a data value. The peak represents the mean, while the width is the sample variability. The standard deviation quantifies the width for a distribution of data values. Wider curves indicate that data points fall further from the mean and correspond to higher standard deviations.
Similarly, the standard error also measures the width of a distribution, but which distribution?
Imagine you draw many random samples from the same population, calculate their means, and graph those means in the distribution below.
Statisticians refer to this type of distribution as a sampling distribution. In this type of distribution, each point on the curve is a sample mean rather than an individual data value. The central peak is a population parameter (e.g., the population mean). When n > 1, sampling distributions are narrower than the distribution of individual values. Learn more about sampling distributions.
The standard error quantifies the width of a sampling distribution. Smaller SEs correspond to narrower curves, indicating that sample means tend to fall relatively close to the population mean. That’s fantastic when you’re using a sample to estimate the properties of a population! Learn more about the differences between sample statistics and population parameters.
In this manner, standard errors evaluate the precision of a sample’s estimate. Smaller SEs represent greater precision.
Confidence intervals and margins of error also evaluate the precision of sample estimates, and they do so by incorporating the standard error in their calculations.
Differences Between Calculating the SD and SE
Let’s quickly cover the differences between finding these two statistics. Read my articles about the standard deviation and the standard error for more in-depth information about both.
The sample standard deviation (s) formula below quantifies the difference between each data point and the sample mean.
In a nutshell, the formula finds the average squared difference between the data points and the sample mean, and then takes the square root of that. For more information about how this formula works, read about calculating the standard deviation.
Finding the standard error of the mean involves taking the standard deviation above and dividing it by the square root of the sample size, as shown in the formula below.
These formulas lead to the final difference between the standard deviation and the standard error, the sample size’s effect on the two statistics.
The standard deviation does not tend to increase or decrease as the sample size (N) increases. N is in the denominator, but as it increases the numerator also increases, producing no net tendency to change.
However, the standard error tends to decrease as N increases. This decrease occurs because s is in the numerator and tends to stay constant while N increases in the denominator. Hence, the standard error quantifies how larger sample sizes produce more precise estimates!
Summary of the Differences
Finally, the table provides a quick overview of the differences between the standard deviation and standard error.
|Within a sample
|Defines width of a
|Distribution of individual values
|Assesses distances between
|Data values and sample mean
|Sample statistics and population parameter (i.e., accuracy)
|As sample size increases, there is
|No tendency to change
|A tendency to decrease.