Two terms that students often confuse in statistics are standard deviation and standard error.
The standard deviation measures how spread out values are in a dataset.
The standard error is the standard deviation of the mean in repeated samples from a population.
Let’s check out an example to clearly illustrate this idea.
Example: Standard Deviation vs. Standard Error
Suppose we measure the weight of 10 different turtles.
For this sample of 10 turtles, we can calculate the sample mean and the sample standard deviation:
Suppose the standard deviation turns out to be 8.68. This gives us an idea of how spread out the weights are of these turtles.
But suppose we collect another simple random sample of 10 turtles and take their measurements as well.
More than likely, this sample of 10 turtles will have a slightly different mean and standard deviation, even if they’re taken from the same population:
Now if we imagine that we take repeated samples from the same population and record the sample mean and sample standard deviation for each sample:
Now imagine that we plot each of the sample means on the same line:
The standard deviation of these means is known as the standard error.
The formula to actually calculate the standard error is:
Standard Error = s/ √n
where:
- s: sample standard deviation
- n: sample size
What’s the Point of Using the Standard Error?
When we calculate the mean of a given sample, we’re not actually interested in knowing the mean of that particular sample, but rather the mean of the larger population that the sample comes from.
However, we use samples because they’re much easier to collect data for compared to an entire population.
And of course the sample mean will vary from sample to sample, so we use the standard error of the mean as a way to measure how precise our estimate is of the mean.
You’ll notice from the formula to calculate the standard error that as the sample size (n) increases, the standard error decreases:
Standard Error = s/ √n
This should make sense as larger sample sizes reduce variability and increase the chance that our sample mean is closer to the actual population mean.
When to Use Standard Deviation vs. Standard Error
If we are simply interested in measuring how spread out values are in a dataset, we can use the standard deviation.
However, if we’re interested in quantifying the uncertainty around an estimate of the mean, we can use the standard error of the mean.
Depending on your specific scenario and what you’re trying to accomplish, you may choose to use either the standard deviation or the standard error.
As usual Zach, you are brilliant at explaining things!
Thank you! This is so simplified and clear
Hi Zach,
thank you so much for explaining this in such an understandable way.
The only thing I’m still not getting is how you calculated the figure 8.68 for the standard deviation (SD) in the beginning of the example? Also what unit is the SD meant to be? 8.68 what? I know now it’s how spread out the data is in the sample, and I get how to calculate the mean and thanks to the formula also the Standard Error, but am still getting stuck on the SD. Would really appreciate your answer!
Many thanks and kind regards,
M.
If you calculate the difference between each data and the mean value of the sample and squared this value, then summed up all values, and divide them by number of samples. Finally, calculate the square root of the result, you will have how far your data is dispersed from the average value of this sample.
I think what’s confusing for me is…the Standard Deviation, by definition, is an average. To calculate it we divide the sum of squares by n, and then get the square root of that. So, by dividing sum of squares by n (and then getting its square root), you’ve ALREADY figured out your average deviation from the mean. And if your distribution is a distribution of sample means, then your standard deviation represents how much variation there exists between the means of your samples.
So, to divide the standard deviation by n (sq rt of) all over again is to divide an average (standard deviation) by a number that was already used to calculate said average in the first place. To take what’s already an average and divide that by n seems to neglect the fact that we already divided by n to get the average (standard deviation) in the first place. But I’m obviously not seeing something. Would appreciate any help, if possible. Thanks!
Hi Damien…You’re raising a very insightful question about the relationship between **standard deviation** and the **standard error of the mean** (which is what you’re referring to when dividing the standard deviation by the square root of the sample size, \( n \)).
Here’s the key idea to clarify the confusion:
### 1. **Standard Deviation** (SD)
The standard deviation measures the **average deviation of individual data points** from the mean within a single sample. The formula for standard deviation of a sample is:
\[
s = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n – 1}}
\]
This gives you the spread or variability of **individual data points** in a sample.
### 2. **Standard Error of the Mean** (SE)
The standard error of the mean, on the other hand, measures the variability **between sample means** if you were to repeatedly take many samples from the same population. It quantifies how much the sample mean \( \bar{x} \) is expected to vary from the true population mean \( \mu \).
The relationship between the **standard deviation** \( s \) and the **standard error of the mean** is given by:
\[
SE = \frac{s}{\sqrt{n}}
\]
Where:
– \( s \) is the standard deviation of the sample.
– \( n \) is the sample size.
### Why Divide by \( \sqrt{n} \)?
When you take multiple samples from a population, the mean of each sample will differ slightly due to random sampling variation. The larger your sample size, the **less** variation you will see between these sample means. That’s why the standard error (SE) gets smaller as \( n \) increases. The division by \( \sqrt{n} \) reflects the fact that the average of sample means converges more closely to the population mean as the number of samples increases.
### Intuition:
– The **standard deviation** gives us an idea of how spread out the data is within a sample.
– The **standard error** gives us an idea of how much variability there is **between different sample means** if you were to take many samples.
In other words, while the standard deviation tells you about variability within a single sample, the standard error tells you about the expected variability **across different sample means**. Dividing by \( \sqrt{n} \) adjusts for the fact that as sample size increases, we get a more precise estimate of the population mean.
Does this help clear up the confusion? Let me know if you’d like further clarification!