In simple terms, the standard error is a measure of how much the sample mean (average) is likely to vary from the true population mean. It tells us how uncertain or "noisy" our estimate of the population mean is based on a sample.

Formula for the standard error of the mean (SE):

SE = (σ / √n)

where:

SE is the standard error of the mean.

σ (sigma) is the population standard deviation, which measures how spread out the values are in the entire population.

√n represents the square root of the sample size (n), which tells us how many observations are in our sample.

Let's break it down with an example:

Imagine you want to find out the average height of all the students in a school, but it's impractical to measure every student. So, you randomly select 30 students and measure their heights.

You find that the average height of these 30 students is 160 centimeters.

You know from previous research (or a larger data set) that the population standard deviation (σ) of student heights in the school is 10 centimeters.

Now, you can use the formula for the standard error:

SE = (10 / √30) ≈ 1.83 centimeters

The standard error in this case is approximately 1.83 centimeters. It tells you that based on your sample of 30 students, the average height of all students in the school is likely to fall within 1.83 centimeters of 160 centimeters. In other words, you can be reasonably confident that the true average height of all students is somewhere between approximately 158.17 and 161.83 centimeters.

So, the standard error helps you understand how much your sample mean might differ from the actual population mean, providing a measure of the uncertainty associated with your estimate. Smaller standard errors indicate a more precise estimate, while larger standard errors suggest more variability and less confidence in your estimate.