Standard Deviation vs Standard Error (3 Key Concepts To Know)


In statistics, we describe data sets using both standard deviation and standard error of the mean (also called standard error).  Although these two terms sound similar, there are some key differences to know about.

So, what is the difference between standard deviation and standard error?  The standard deviation (S) of a sample data set tells you the dispersion of values about the mean. The standard error (SE) of a sample data set gives you an idea of how far the sample mean is from the true population mean. The two are related by the equation SE = S/√N, where N is the sample size.

So, if you want to use a sample data set to estimate the population mean, you can decrease your error by using a larger sample size N.  For example, quadrupling the sample size N will halve the standard error.

In this article, we’ll talk about the standard deviation and the standard error of the mean for a data set, along with the relationship between the two terms.  We’ll also look at some examples to make the concepts clear.

Let’s get started.

Standard Deviation vs Standard Error

First, remember that it is often difficult to measure every possible data point in a population.  For example, it would be difficult to find the exact age of every person in a city.

However, we can take a sample of the population to make estimates about the population as a whole.  As long as the sample is unbiased and representative of the population, this method can give us reasonable estimates.

Let’s say we have a sample from a population with the following characteristics:

  • The sample size is N (N data points in the sample)
  • The mean is M (add up all of the data points and divide by N)
  • The standard deviation is S (you can learn how to calculate standard deviation here).
  • The standard error is SE (this is based on S and N)

The standard deviation tells us how the data points are spread around the sample mean.  The standard error of the mean (standard error) tells us how close the sample mean is to the true population mean.

More specifically:

  • The standard deviation is a measure of dispersion.  That is, it tells us how spread out the sample data is about the sample mean.  A higher standard deviation means the data is more spread out.
  • The standard error of the mean is a measure of closeness.  That is, it tells us how close the sample mean is to the true population mean. A smaller standard deviation and a larger sample size indicates a smaller standard error.

The table below compares standard deviation and standard error side-by-side.

StatisticStandard
Deviation
Standard
Error
Alternate
Name
Sample
Standard
Deviation
Standard
Error Of
The Mean
UsesTells us how
spread out
data is
around
the mean.
Tells us how
close the
sample mean
is to the true
population
mean.
FormulaS is the
square root
of: the sum
of squared
differences
from the
mean,
divided by
(N – 1).
SE=S/√N
This table summarizes some of the differences
between standard deviation and standard error.

Before we dive in deeper to connect standard deviation and standard error, let’s define each one.

What Is Sample Standard Deviation (Standard Deviation)?

Sample standard deviation (here, we will use S to represent sample standard deviation) is a measure of dispersion for a sample data set.  This sample data set is taken from a larger population (ideally, it is unbiased and representative of the population).

The sample standard deviation can be used to estimate the population standard deviation.

(You learn about the difference between sample and population standard deviation here).

 Sample standard deviation can also tell us about the spread of data points about the mean in a sample:

  • A large value for standard deviation means that the data is spread far out, with some of it far away from the mean.  That is, on average, a given data point is far from the mean.
  • A small value for standard deviation means that the data is clustered near the mean.  That is, on average, a given data point is close to the mean.
  • A zero value for standard deviation means that all of the data has the same value (which is also the value of the mean).  This situation is rare, but it is possible.

You can learn more about what standard deviation tells us in this article.

What Is Standard Error Of The Mean (Standard Error)?

Standard error (here, we will use SE to represent standard error of the mean) is a measure of how far the sample mean is likely to be from the true population mean.  So, when we use the sample mean to estimate the true population mean, the standard error tells us how good the estimate is likely to be.

  • A large value for standard error means that the sample mean could be quite far from the true population mean.
  • A small value for standard error means that the sample mean is likely to be quite close to the true population mean.
  • A zero value for standard error means that the standard deviation is zero, which means that all of the data has the same value (which is also the value of the mean).

The equation that relates the standard error, the standard deviation, and the size N of a sample is:

  • SE = S / √N

So, the standard error of the mean depends on two things:

  • The size N of the sample data set.
  • The standard deviation S of the sample data set (which also relies on N).

The equation tells us how changes in these values will change the standard error of the mean:

  • A smaller sample standard deviation S will lead to a smaller standard error.
  • A larger sample size N will lead to a smaller standard error.

We cannot control the true standard deviation of a population. However, we can control the sample size that we take from the population.

The following table shows how an increase in N leads to a decrease in the standard error.

Increase
In N
(Sample
Size)
Decrease
In SE
(Standard
Error)
x4x1/2
x9x1/3
x16x1/4
x25x/15
x100x1/10
This table shows how increasing
the sample size decreases
the standard error.

In more general terms, the sample error for any statistic is the standard deviation of the sampling distribution for that statistic.

To get a sampling distribution for a statistic, we:

  • Take repeated samples from the population (they are randomly chosen and not always the same).
  • Calculate the value of the statistic for each sample and record each one.
  • Combine the values of all the sample statistics into a sampling distribution.

This sampling distribution has its own mean and standard deviation.  A smaller standard deviation for this sampling distribution means that the mean is a better estimate for the true population parameter.

Central Limit Theorem

The central limit theorem tells us that when we add up independent random variables, their sum approaches a normal distribution (the more variables we add up, the closer we get to a normal distribution).

When a distribution is normal (or close to normal), much of the data is clustered close to the mean (about 68% is within one standard deviation of the mean).  The entire data set is arranged about the mean in a predictable pattern (see the normal distribution “bell curve” below).

standard-normal-distribution
Here is a standard normal distribution with its signature “bell curve” shape.

The central limit theorem also applies to sampling distributions (which we reviewed earlier).  As we take more random samples from a population (and calculate the mean of each one), the close the sampling distribution gets to a normal distribution.

As we take more samples, the sampling distribution approaches a normal distribution.  We can then find a confidence interval at any level we want (90%, 95%, 99%, etc.) to get a range of values that is likely to contain the true population parameter.

Examples Of Standard Deviation & Standard Error

It will be much easier to see how standard deviation and standard error work (and how they are connected) with some examples.

Example 1: Standard Deviation & Standard Error

Consider the data set:

  • A = {1, 3, 5, 5, 6}

Following the steps in this article to find the sample standard deviation:

For step 1, we calculate the sample mean.  The sample size is n = 5, so:

  • Mean = M = (1 + 3 + 5 + 5 + 6) / 5 = 20 / 5 = 4

Now we will use a table to calculate the necessary values for steps 2 and 3:

Data
Value
Value
Minus
Mean
(X-M)
Squared
Difference
1-39
3-11
511
511
624
This table gives the values, differences
from the mean, and squared
differences for the data set.

For step 4, the sum of the square differences (3rd column in the table above) is 9 + 1 + 1 + 1 + 4 = 16.

For step 5, we divide by n – 1.  Here n = 5, so n – 1 = 4, and so 16 / (n – 1) = 16 / 4 = 4.

For step 6, we take the square root of 4 to get 2.

So, the sample standard deviation is S = 2.

Now, we can find the standard error SE by the equation given earlier:

  • SE = S / √N
  • SE = 2 / √5

So, the standard error of the mean is approximately 0.894.

Example 2: Standard Deviation & Standard Error

Consider the data set:

  • A = {1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9 9, 9}

Following the steps in this article to find the sample standard deviation:

For step 1, we calculate the sample mean.  The sample size is n = 16, so:

  • Mean
  • =M
  • = (1 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 8 + 9 + 9 + 9) / 16 =  80 / 16 = 5.

Now we will use a table to calculate the necessary values for steps 2 and 3:

Data
Value
Value
Minus
Mean
(X-M)
Squared
Difference
1-416
1-416
1-416
2-39
2-39
3-24
4-11
500
500
611
724
839
839
9416
9416
9416
This table gives the values, differences
from the mean, and squared
differences for the data set.

For step 4, the sum of the square differences (3rd column in the table above) is 142.

For step 5, we divide by n – 1.  Here n = 16, so n – 1 = 15, and so  142 / (n – 1) = 142 / 15 = 9.467.

For step 6, we take the square root of 9.467 to get 3.077.

So, the sample standard deviation is S = 3.077.

Now, we can find the standard error SE by the equation given earlier:

  • SE = S / √N
  • SE =  3.077 / √16
  • SE =  3.077 / 4

So, the standard error of the mean is approximately 0.769.

Conclusion

Now you know the difference between the standard deviation and the standard error of the mean (standard error) of a sample data set from a population.

You can learn about uses of standard deviation in this article.

You can learn more about how to interpret standard deviation here.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon

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