Standard error is a statistical measure of how accurately a set of data likely represents the actual population. For instance, if you took a sample of pupils' exam scores, you could use it to estimate the scores of all pupils in the group. How likely is it that your estimate accurately represents the distribution of scores across the whole group? That's what standard error tells us.
Standard error overview and symbol
Standard error doesn't actually have a proper symbol per se, it is simply denoted as
\[SE.\]
And calculating the standard error of a sample is actually very simple! You just need to cast your mind back to when you learnt about standard deviation, \(\sigma\). The formula for standard error is simply
\[SE = \frac{\sigma}{\sqrt{n}}\]
Where \(n\) is the sample size.
Here's a quick reminder of how to find the standard deviation. All you need is a list of the data points in the set.
The formula for the standard deviation for a sample data set is
\[\sigma = \sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}.\]
Let's take a look at an example to see how you can find the standard error of a sample data set.
Find the standard error for the following data set.
Solution:
First, calculate the mean of the data set.
\[\begin{align} \bar{x} &= \frac{2+3+2+5...}{8} \\ &= 4 \end{align}\]
Next, calculate the standard deviation.
\[\begin{align}\sigma &= \sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}} \\ &= \sqrt{\frac{ (2-4)^2+(3-4)^2+(2-4)^2...}{7}} \\ &= 1.69 \end{align}\]
Finally, calculate the standard error.
\[\begin{align} SE &= \frac{\sigma}{\sqrt{n}} \\ &= \frac{1.69}{\sqrt{8}} \\ &= 0.6 \end{align}\]
Standard error interpretation
So what does standard error actually mean? Well, imagine you take lots of samples of a population. You will find that each sample has a slightly different mean, and this collection of means will itself form a distribution. The standard deviation of this distribution of means is the standard error of the original population.
So, if you work out that the standard error of a group of pupils' heights is \(15\,\text{cm}\), that means that there is a \(68\%\) that your sample mean is within \(15\,\text{cm}\) of the true mean of the group of students. This is due to something called the \(68\)-\(95\)-\(99.7\) rule.
The \(68\)-\(95\)-\(99.7\) rule says that in a normally distributed data set, \(68\%\) of data points fall within one standard deviation of the mean, \(95\%\) of data points fall within two standard deviations of the mean, and \(99.7\%\) of data points fall within three standard deviations of the mean.
Standard error vs. standard deviation
So how is standard deviation different from standard error? Well, standard deviation is a measure of how spread out the data points in a data set are from the mean. The more spread the data points are, the greater the standard deviation.
Standard error on the other hand, as described earlier, is simply a measure of how close to the true mean of the population the mean of your sample is likely to be. The larger the standard error, the further it is likely your mean is from the true mean.
Standard error example
Let's take a look at some examples to make sure you've got everything in hand.
Find the standard error of the following sample data set.
Solution:
First, calculate the mean of the sample.
\[\begin{align} \bar{x} &= \frac{20+25+15...}{12} \\ &= 20.42 \end{align}\]
Next, calculate the standard deviation of the sample.
\[\begin{align} \sigma&= \sqrt{\frac{\sum(x_1-\bar{x})^2}{n-1}} \\ &= \sqrt{\frac{(20-20.42)^2+(25-20.42)^2+(15-20.42)^2...}{11}} \\ &= 2.91 \end{align}\]
Finally, calculate the standard error.
\[\begin{align} SE&= \frac{\sigma}{\sqrt{n}} \\ &= \frac{2.91}{\sqrt{12}} \\ &= 0.84 \end{align}\]
Given the following sample, is it likely that the mean of this sample is within \(0.1\) of the true mean of the population?
| 0.2 | 0.3 | 0.1 | 0 | 0.4 |
| 0.3 | 0.5 | 0.2 | 0.1 | 0.2 |
Solution:
First, calculate the mean of the sample.
\[\begin{align} \bar{x} &= \frac{0.2+0.3+0.1...}{10} \\ &= 0.23 \end{align}\]
Next, calculate the standard deviation of the sample.
\[\begin{align} \sigma&= \sqrt{\frac{\sum(x_1-\bar{x})^2}{n-1}} \\ &= \sqrt{\frac{(0.2-0.23)^2+(0.3-0.23)^2+(0.1-0.23)^2...}{10}} \\ &= 0.15 \end{align}\]
Finally, calculate the standard error.
\[\begin{align} SE&= \frac{\sigma}{\sqrt{n}} \\ &= \frac{0.15}{\sqrt{10}} \\ &= 0.05 \end{align}\]
\(0.1\) is two standard errors. There is a \(95\%\) chance that the mean falls within two standard errors of the mean, therefore:
Yes, it is likely that the mean of this sample is within 0.1 of the true population mean.
Standard Error - Key takeaways
- Standard error is a statistical measure of how accurately a set of data likely represents the actual population.
- The formula for standard error is \(SE=\frac{\sigma}{\sqrt{n}}\).
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