The standard error of the mean is a way to measure how spread out values are in a dataset. It is calculated as:
Standard error of the mean = s / √n
where:
- s: sample standard deviation
- n: sample size
This tutorial explains two methods you can use to calculate the standard error of the mean for a dataset in Python. Note that both methods produce the exact same results.
Method 1: Use SciPy
The first way to calculate the standard error of the mean is to use the sem() function from the SciPy Stats library.
The following code shows how to use this function:
from scipy.stats import sem #define dataset data = [3, 4, 4, 5, 7, 8, 12, 14, 14, 15, 17, 19, 22, 24, 24, 24, 25, 28, 28, 29] #calculate standard error of the mean sem(data) 2.001447
The standard error of the mean turns out to be 2.001447.
Method 2: Use NumPy
Another way to calculate the standard error of the mean for a dataset is to use the std() function from NumPy.
Note that we must specify ddof=1 in the argument for this function to calculate the sample standard deviation as opposed to the population standard deviation.
The following code shows how to do so:
import numpy as np #define dataset data = np.array([3, 4, 4, 5, 7, 8, 12, 14, 14, 15, 17, 19, 22, 24, 24, 24, 25, 28, 28, 29]) #calculate standard error of the mean np.std(data, ddof=1) / np.sqrt(np.size(data)) 2.001447
Once again, the standard error of the mean turns out to be 2.001447.
How to Interpret the Standard Error of the Mean
The standard error of the mean is simply a measure of how spread out values are around the mean. There are two things to keep in mind when interpreting the standard error of the mean:
1. The larger the standard error of the mean, the more spread out values are around the mean in a dataset.
To illustrate this, consider if we change the last value in the previous dataset to a much larger number:
from scipy.stats import sem #define dataset data = [3, 4, 4, 5, 7, 8, 12, 14, 14, 15, 17, 19, 22, 24, 24, 24, 25, 28, 28, 150] #calculate standard error of the mean sem(data) 6.978265
Notice how the standard error jumps from 2.001447 to 6.978265. This is an indication that the values in this dataset are more spread out around the mean compared to the previous dataset.
2. As the sample size increases, the standard error of the mean tends to decrease.
To illustrate this, consider the standard error of the mean for the following two datasets:
from scipy.stats import sem #define first dataset and find SEM data1 = [1, 2, 3, 4, 5] sem(data1) 0.7071068 #define second dataset and find SEM data2 = [1, 2, 3, 4, 5, 1, 2, 3, 4, 5] sem(data2) 0.4714045
The second dataset is simply the first dataset repeated twice. Thus, the two datasets have the same mean but the second dataset has a larger sample size so it has a smaller standard error.
Additional Resources
How to Calculate the Standard Error of the Mean in R
How to Calculate the Standard Error of the Mean in Excel
How to Calculate Standard Error of the Mean in Google Sheets